matesesosalesianos.files.wordpress.com · 5 NUMBERS – REVISION EXERCISES 1. Find the LCM and GCD...

1

TABLE OF CONTENTS

Contenido NÚMEROS................................................................................................................................................3

POTENCIAS Y RAÍCES ................................................................................................................................ 11

SUCESIONES Y PROGRESIONES .................................................................................................................... 17

PROPORCIONALIDAD ................................................................................................................................ 25

POLINOMIOS .......................................................................................................................................... 35

ECUACIONES........................................................................................................................................... 41

SISTEMAS DE ECUACIONES ......................................................................................................................... 49

RECTA, PARÁBOLA E HIPÉRBOLA ................................................................................................................. 59

FUNCIONES ............................................................................................................................................ 69

ESTADÍSTICA ........................................................................................................................................... 77

PROBABILIDAD ........................................................................................................................................ 91

ÁREAS Y VOLÚMENES .............................................................................................................................. 99

TRANSFORMACIONES EN EL PLANO. MOSAICOS ........................................................................................... 105

APÉNDICE: LUGARES GEOMÉTRICOS ........................................................................................................... 111

3

NÚMEROS

Naturales : son 0, 1, 2, 3, 4,…

Enteros se obtienen añadiendo los opuestos -1,-2,-3,… y los Racionales dividiendo.

Reales : son el conjunto de los números con decimales.

Orden de las operaciones (jerarquía de las operaciones):

Operaciones con fracciones:

Suma y resta: reducimos a común denominador, y se suma o restan los numeradores.

Multiplicación: se multiplica en línea. División: se multiplica en cruz.

Potencia y raíz: se hace la potencia o raíz del numerador y del denominador.

(*) Castillo de fracciones: si el numerador o el denominador contienen una fracción Ej:

.

(*) Valor absoluto: es el número sin el signo. |-2| = 2, y |+2| = 2.

Tipos de decimales:

Exacto: vienen de una división exacta. Sólo tienen algunos decimales.

Periódico: vienen de una división no exacta. Un grupo de decimales (periodo) se repite

infinitamente. Son puros si toda la parte decimal es periódica. Si no, son mixtos.

Irracionales ( ): no son fracciones, y tienen infinitos decimales no periódicos.

Aproximación: - Truncar: quitamos las cifras decimales que no necesitemos.

- Redondear: truncamos, pero hay que aumentar en uno la última cifra si la

siguiente cifra era mayor o igual que 5.

Error Absoluto: diferencia (sin signo) entre el valor exacto y el aproximado.

Error Relativo: error absoluto dividido entre el valor exacto (se expresa en porcentaje).

Notación científica: escribir el número como un decimal entre 1 y 9, por una potencia de 10.

Regla de los signos

+ · + = +

+ · - = -

- · + = -

- · - = +

1) Paréntesis.

2) Potencias y raíces.

3) Multiplicaciones y divisiones

4) Sumas y restas.

4

NUMBERS

Natural : they are 0, 1, 2, 3, 4,…

Integer they are formed by adding their opposites -1,-2,-3,… and Rationals dividing.

Real : they are the set of all decimal numbers.

Order of operations (precedence rules):

Operations with fractions:

Add and subtract: reduce to common denominator, and add or subtract numerators.

Multiplication: multiply across the top and bottom. . Division: cross multiply.

Power and root: just calculate the power or root of numerator and denominator.

(*) Complex fraction: when the numerator or denominator contains a fraction. E.g.:

.

(*) Absolute value: it is the number without its sign. |-2| = 2, and |+2| = 2.

Types of decimal numbers:

Terminating: they come from a terminating decimal fraction. Decimals stop after a few digits.

Periodic (recurring decimal): they come from a recurring decimal fraction. A group of

decimals (period) repeat forever. We call those which start their recurring cycle

immediately after the decimal point purely recurring. Those that have some extra

digits before their cycles are also called mixed recurring (or eventually recurring).

Irrational ( ): any number which does not stop and does not end with a recurring pattern

(thus they are non-fractional numbers).

Approximation: - Truncate: remove the decimal digits you do not need.

- Round: truncate, but increase in one the last digit if the next one was

greater or equal to 5.

Absolute Error: difference (without sign) between the exact value and the approximation.

Relative Error: absolute error divided by the exact value (expressed in percentage).

Scientific notation: write the number as a decimal between 1 and 9 times a power of 10.

(to some decimal places)

Rules of signs

+ · + = +

+ · - = -

- · + = -

- · - = +

1) Parentheses (brackets).

2) Exponents and roots.

3) Multiplication and divisions

4) Addition and subtraction.

5

NUMBERS – REVISION EXERCISES

1. Find the LCM and GCD of the following numbers:

a) 120 and 150 b) 378 and 528 c) 140 and 350

d) 720 and 1470 e) 79 and 84 f) 240 and 300

g) 168 and 252 h) 80 and 120.

Solutions:

a) LCM=600, GCD=30 b) LCM=24·33·7·11, GCD=6 c) LCM=22·52·7, GCD=70

d) LCM=24·32·5·72, GCD=30 e) LCM=22·3·7·79, GCD=1 f) LCM=24·3·52, GCD=60

g) LCM=23·32·72, GCD=22·3·7 h) LCM=24·3·5, GCD=23·5=40.

2. Find the LCM and GCD of the following numbers:

a) 40, 105 and 160 b) 72, 120 and 210 c) 54, 126 and 180.

Solutions:

a) LCM=25·3·5·7, GCD=5 b) LCM=23·32·5·7, GCD=6 c) LCM=22·33·5·7, GCD=18.

3. Write as a fraction or as a decimal number:

a) b) 2.8 c) d)

e)

f)

g)

h)

Solutions:

a)

b)

c)

d)

e) 3.1875 f) g) h) .

6

OPERATIONS INVOLVING FRACTIONS – PRACTICE

1. Simplify before multiply:

a)

= b)

= c )

=

d)

= e)

= f)

=

g)

h)

i)

2. Calculate. Remember to simplify, whenever it is possible.

a)

= b)

1+2=

c)

= d)

=

e)

= f)

=

3. Work out

a)

= b)

=

c)

= d)

=

e)

= f)

=

4. Work out:

a)

= b)

c)

d)

e)

= f)

Solutions:

1. [a] 1 [b]

[c] 2. [d] -4 [e] -

[f]

[g] -25 [h]

[i]

.

2. [a]

[b]

[c]

[d]

[e]

[f]

.

3. [a]

[b]

[c]

[d]

[e]

[f] 18.

4. [a] 2 [b]

[c]

[d]

[e]

[f] .

7

Numbers - Word Problems

1. A water pitcher (jarra) weighs 0.64 kg when empty and 1.728 kg when

filled with water. How much does the water weigh? [Sol: 1.088kg]

2. Eva is on a diet which states that she cannot consume more than 600

calories in one meal. Yesterday she had lunch: 125 g of bread, 140 g of

asparagus, 45 g of cheese and an apple of 130 g.

If 1 g of bread has 3.3 calories, 1 g of asparagus, 0.32, 1 g of cheese, 1.2, and 1 g of an

apple 0.52. Did Eva follow her diet?

3. Juan has got €200 in the bank. He pays a bill for 5 books, which cost €30 each. Then, he

earns his pay (paga) of the last seven days (€40 per day). Last, he withdraws (sacar) €200

and buys a €320 game console. Represent the situation of his account using an integer.

Does he have money or does he owe money? How much?

4. A gardener (jardinero) fills 1/5 of his vegetable garden with potatoes, 2/3 with

cabbage (col) and the rest, which amounts to 120 square metres, with onions.

What portion of the garden occupy the onions? What is the area of the garden?

5. A car dealer is selling a new model for €12,000, with one sixth of the price to be paid

upfront (por adelantado) and the rest in forty equal monthly instalments (plazos).

How much is paid upfront? How much is each monthly instalment? [Sol: 2000 and 250]

6. A thrifty (ahorrador) individual has €245,000 in his current account. He invests two-fifths

of the money in shares (acciones) in an insurance company.

How much is that? How much is left in the account? [Sol: Invests €98,000. 147,000 left]

7. A shelf (estantería) in a supermarket holds 80 one-quarter litre bottles and 44 one and a

half litre bottles. How many litres are there on the shelf? [Sol: 86 litres]

8. Juan moves forward of a metre with each step. How many steps must he take to

complete a 9 kilometre walk? [Sol: 10,800 steps]

9. Victoria is planning for her holiday. She calculates that if she spends a third of her savings

(ahorros) on a plane ticket and a quarter on a hotel, she will still have €450 left. How

much money does she have?

What fraction of her money will she spend? What fraction will she keep?

How much money does she have?

10. Find the absolute error and relative error when:

a) We say 30 minutes instead of 27 minutes. c) We round 3.66 to tenths.

b) We say €15 instead of €16. d) We truncate 6.7 to units.

120m2

8

Use scientific notation to solve the following problems

11. The distance between the Sun and Earth is approximately 150,000,000 km. Write it using

scientific notation. [Sol: 1.5×108km]

12. The Sun has a Mass of 1,988,000,000,000,000,000,000,000,000,000 kg. Write it using

scientific notation. [Sol: 1.988×1030

kg.]

13. The radius of the sun is 695 500 km. What is its approximate volume written in Scientific

notation? [Hint: V=

] [Sol: 1.409×10

18km

3.]

14. The mass of the Moon is 73,000,000,000,000,000,000,000 kg. What is this written in

Scientific notation? [Sol: 7.3 × 1022 kg]

15. The speed of light in a vacuum is 299 792 458 m/s. What is this written in Scientific

notation? [Sol: 2.997 924 58×108 m/s]

16. Photocopy paper is packaged in reams (“muchas páginas”; 500 sheets). The thickness of

the pack is 41 mm. What is the thickness of one sheet of paper written in Scientific

Notation using meters? [Sol: 8.2×10-2mm=8.2×10-5

m]

17. The rest mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 910 938 kg.

What is this written in Scientific notation? [Sol: 9.10938 × 10-31kg]

18. The human eye blinks (parpadear) about 6.25·106 times each year. About how many

times has the eye of a 14 year old blinked? (Use scientific notation). [Sol: 8.75 × 107]

19. If the average person eats 2.3 slices of pizza per week, how many slices of pizza are

consumed in Alabama (population: 5.8·105) in one week? [Sol: 1.334 × 106 slices]

20. A tiny space inside a computer chip has been measured to be 0.00000256m wide,

0.00000014m long and 0.000275m high. What is its volume? [Sol: 9.856×10-17

m3]

21. The speed of light is 3·108 meters/second. If the sun is 1.5·1011 meters from earth, how

many seconds does it take light to reach the earth? And minutes? [Sol: 500s=8m20s]

(Hint: Use scientific notation and write an equation and to solve it)

9

3º ESO Mathematics Exam (trial exam) IES Extremadura

Bilingual section

NUMBERS.

Prime factors (2 pts.):

1) (1 pt.) Write as a product of prime numbers:

a) 2520= b) 1296= c) 3388= d) 2340=

2) (1 pt.) Compute LCM and GCD:

a) 2520 and 1296 b) 3388, 2520 and 1100.

Calculations (4 pts.)

3) (1 pt.) =

4) (1 pt.)

=

5) (1 pt.)

=

6) (1 pt.)

=

Decimal numbers (4 pts.)

7) (1.5 pts.) Write as a decimal number or as a fraction in lowest terms

a)

= b) 3.36363636...= c)

d) 6.25= e)

=

8) (1 pt.) Use scientific notation:

a) 3897000000000000 = b) 0.0000000009137 =

c) 4.6·104 · 5.1·107= d) 5.2·104 + 3.51·106=

9) (0.5 pts.) Find the absolute and relative errors when we say 30 people instead of 28 people.

10) (1 pt.) Oil (petróleo) reserves in the United States are estimated to be 3.5·1010 barrels.

Consumption amounts to (asciende a) 3.2·109 barrels per year.

At this rate (a este ritmo), how long would US oil reserves last?

What should be the consumption if we want the oil to last 30 years?)

11

POTENCIAS Y RAÍCES

Raíz (radical):

porque 53=125. Es una raíz de índice 3 (raíz cúbica).

porque 24=16. Es una raíz de índice 4 (raíz cuarta).

- Si el índice es par, no hay raíz de números negativos. Si es impar, la raíz sí existe.

Ej.:

NO EXISTE, pero

.

Ejercicio: memoriza y escribe usando raíces 23=8, 33=27, 43=64, 53=125, 63=216, 73=343.

Propiedades de las potencias

Propiedades de los radicales

Índice y exponente se simplifican como fracciones. Ej.:

;

.

Extraer factores: se calcula la raíz de cada factor del radicando. Ej.

Suma/resta: si son iguales, se suman/restan los coeficientes. Ej. .

(*) Escribir con un solo radical: Reducir a índice común. Ej.

.

Misma base: se suman los exponentes. 52·58=510 , 56·5=57.

Mismos exponentes: se multiplican las bases 43·53=203.

Misma base: se restan los exponentes 3

4

7

55

5 , 11)4(7

4

7

555

5

Mismos exponentes: se dividen las bases 8135

15

5

15 4

4

4

4

.

81=8 , 80=1 , (-3)1=-3 , (-3)0=1.

Se da la vuelta a la fracción.

,

.

(y se quita el signo)

Exponente par: se quita el signo del paréntesis. (-5)2=52=25, pero -52= -25.

Exponente impar: el signo sale fuera del paréntesis (-5)3=-53=-125.

Se convierte en radical. El denominador es el índice de la raíz

77 47

4

1622 , 5

4

25

16

25

16

25

16 2

1

.

Producto

División

Exponentes 0 y 1

Exponente

negativo

Base

negativa

Exponente

fraccionario

Potencia de potencia Se multiplican los exponentes (43)2=46.

12

POWERS AND ROOTS

Root (radical):

because 53=125. Is a root of index 3 (cubic root).

because 24=16. Is a root of index 4 (fourth root).

- If the index is even, there is no root for negative numbers. If it is odd, the root does exist.

E.g.:

DO NOT EXIST, but

.

Exercise: memorize and write using roots 23=8, 33=27, 43=64, 53=125, 63=216, 73=343.

Properties of Exponents

Properties of radicals

Exponent and index can be converted or simplified. E.g.

;

.

Take out factors: find the root of each factor in the radicand. E.g.

Add/subtract: if they are equal, add/subtract the coefficients. E.g. .

(*) Write using one radical: Reduce to a common index. E.g.

.

Same base: add the exponents. 52·58=510 , 56·5=57.

Same exponents: multiply the bases 43·53=203.

Same base: subtract exponents 3

4

7

55

5 , 11)4(7

4

7

555

5

Same exponents: divide the bases 8135

15

5

15 4

4

4

4

.

81=8 , 80=1 , (-3)1=-3 , (-3)0=1.

Flip the base upside down.

,

.

(and remove the sign)

Exponent even: remove the sign in parenthesis. (-5)2=52=25, but -52= -25.

Exponent odd: put the sign outside the parenthesis (-5)3=-53=-125.

Turns into a radical. The denominator is the index of the root

77 47

4

1622 , 5

4

25

16

25

16

25

16 2

1

.

Product

Division

Exponents 0 and 1

Negative

exponent

Negative

base

Fractional

exponent

Power to a power multiply the exponents (43)2=46.

13

POWERS AND ROOTS – REVISION EXERCISES

1. Write using only one radical (or find the result)

a)

b)

c)

d)

e)

f)

Solutions: a)

b)

c)

d)

e) 6 f)

.

2. Take out all possible factors.

a) b)

c)

d)

e)

f)

Solutions: a) b)

c) 10

d)

e)

f)

.

3. Move factors to the inside of the radical

a) b)

c)

d) 22·7

e)

f)

Solutions: a) b)

c)

d)

e)

f)

.

4. Simplify (add and subtract radicals)

a) b)

c)

d) = e)

f)

Solutions: a) b)

c) d) e)

f) .

14

OPERATIONS INVOLVING POWERS AND ROOTS – PRACTICE

1. Calculate. Remember to simplify, whenever it is possible.

a)

= b)

=

c)

= d)

110 =

e)

= f)

=

2. Work out

a)

= b)

=

c)

= d)

=

e)

+

= f)

=

3. Work out:

a)

= b)

=

c)

= d) 5-1+

=

e)

+ 2= f)

=

Solutions:

1. [a] -2 [b]

[c] -5. [d]

[e] -

[f]

2. [a] [b]

[c]

[d]

[e] 7 [f] .

3. [a] [b] 3 [c] 7 [d] 0 [e]

[f] 5.

15


Bilingual section

POWERS AND ROOTS.

Powers (5.5 pts):

1) (1.5 pts.) Write using only one power:

a) 158 ·15-2 ·156 = b)

= c) 104·(-3)4·(-2)4= d)

=

e)

f)

g) 52·82·40-6=

2) (1 pt.)

=

3) (1 pt.)

=

4) (1 pt.)

=

5) (1 pt.)

=

Radicals (5 pts.)

6) (1 pt.) Write using only one radical

a)

= b)

= c)

=

7) (1.5 pts.) Take out all possible factors:

a) 500 b) 5 10126 4·10·7·4 c) 5 420

8) (1 pt.) Move factors to the inside of the radical:

a) 53 b) 3 104

9) (1 pt.) Simplify:

a) 134213161314

b) 4058803455

17

SUCESIONES Y PROGRESIONES

Sucesión: es un conjunto de números (términos) ordenado (hay primero, segundo, tercero,…)

Suelen representarse como a1, a2, a3,… Ej. Para (2, 4, 6,…) a1=2, a2=4, a3=6,…

Término general an: fórmula que representa el valor del término en el lugar “n”.

Progresión aritmética: cuando cada término se obtiene sumando siempre el mismo número

al anterior. El número se llama diferencia (d). Ej. (1, 3, 5, 7, …), se obtiene sumando 2.

Término general:

Suma de los n primeros términos:

Progresión geométrica: cuando cada término se obtiene multiplicando por el mismo

número siempre. El número se llama razón (r). Ej. (2, 4, 8, 16,…), multiplicando por 2.

Término general:

Suma de los n primeros términos:

Interés simple: cuando no lo acumulamos al capital

(no sigue generando intereses).

Interés compuesto: cuando sí lo acumulamos al capital

(sí sigue generando intereses).

C=Capital (dinero) final, c=capital inicial, r=porcentaje (rédito), I=Interés, t = tiempo.

18

SEQUENCES AND PROGRESSIONS

Sequence: is a set of ordered numbers (terms) (there is a first one, second one, third one,…)

They are normally represented by a1, a2, a3,… E.g in (2, 4, 6,…) a1=2, a2=4, a3=6,…

General Term an: formula that represents the value of the “nth ” term.

Arithmetic progression: when each successive term is obtained by adding always the same

number to the previous term. This number is called difference (d).

E.g. (1, 3, 5, 7…), is obtained by adding 2.

General term:

Sum of the n first terms:

Geometric progression: when each successive term is obtained by multiplying the previous

term always by the same factor. The number is called ratio (r).

E.g.. (2, 4, 8, 16…), is obtained by multiplying by 2.

General term:

Sum of the n first terms

Simple interest: when we do not cumulate it to the capital

(it does not generate interest any more).

Compound interest: when we do cumulate it to the capital

(it continues generating interest).

B=Balance (final amount), p=principal (starting amount), r=percent (rate),

I=Interest earned, t = time.

19

SEQUENCES – REVISION EXERCISES

1. Fill the chart using progressions:

Terms Description General Term Some terms…

{30,25,20,15,…}. a10=

Multiples of 2, plus 15. a100=

an= -3n+1 a6=

{-10,-7,-4,…} a12=

2. Compute the general term for the following sequences (0.75 pts. each section):

a)

,...16

9,

8

11,

4

13,

2

15. an=

c) {10 000, 5000, 2500, 1250, …}. an= . Compute a6=

3. Compute, and write the general term:

a) For an arithmetic progression, we know that a3=19 and a7=39. Compute S15.

b) Given the geometric progression {3, 6, 12, 24,…}, compute S12=

c) We want to compute (using the correct formula), 500+ 500·1.8+500·1.82+...+500·1.8

20 =

- What type of succession are we adding?

- How many terms are we adding?

- What is the 10th term of the succession?

4. Laura is going to deposit €15,000 in a bank for 10 years. She wants to know how much will she

have after those 10 years, and how much she will have earned.

a) If we use simple interest of 3 %:

b) If we use compound interest of 3 %

Solutions:

[2] a)

, b)

. a6=312.5

[3] a) d=

. an=5n+4. a1=9, a15=79 S15=

. b) r =2. an=3·2n-1. a12=6144. S12=

c) It is a geometric progression. We are adding 1+20=21 terms. r=1.8; an=500·1.8n-1. a10=500·1.8999179.65

S21=

[4] a) Arithmetic progression: d=3% of 15000=450. {15 000,15 450, …} an=450n+14 550. a10=19 050.

She will have €19 000, so she will have earned 19 050-15 000=4 050 (It is €450 nine times)

b) Geometric progression. r=1.03. {15 000,15 000·1.03,…} an=15 000·1.03n-1. a10=15 000·1.03919 571.60

She will have €19571.60, so she will have earned 1 9751.60-15 000=4 571.60€

20

5. Find the general term for the following sequences, and find the missing terms:

a)

, b)

, c)

;

d)

, e)

, f) ;

g) h)

6. Find the general term and sums for the following arithmetic progressions:

a) a4=19, a6=31; S30? b) a1=26, a5=10; S10? c) a3=-71, a53=79; S100?

d) a1=16, a5=56; a5+…+a40? e) 3+ … +239 = ? f) a4=-8, a10=10; a10+…+a30?

g) a3=44, a13=24; a30+…+a50? h) a1=1, a21=7; a40+…+a87?

7. Find the general term and sums for the following geometric progressions:

a) {2, 6, 18,…}; S10? b) {5, 10, 20, …}; S20? c) 0.1+0.4+1.6+…+1638.4

d) 30+30·1.2+…+30·1.220 e) 10+30·0.4+…+10·0.415 f) 60+60·1.05+…+60·1.0511

Solutions:

[5] a)

;

b)

;

c)

;

d)

;

e) ;

f) ;

g) ; h) ;

[6] a) an=6n-5. S30=2640 b) an=30-4n. S10=80 c) an=3n-80. S100=7150

d) an=10n+6. 56+ +406=8316 e) an=4n-1. 3+…+239=7260 f) an=3n-20. 10+ +70=840

g) an=50-2n. (-10)+ +(-50) = - 630 h) an=0.3n+0.7 12.7+ +26.8 =948.

[7] a) an= . S10=59048 b) an= . S20= c) an= . 0.1+…+1638.4=

d) an= . S21= e) an= . S16= f) an= . S12=

60

21

SEQUENCES AND PROGRESSIONS - WORD PROBLEMS

(*) Use the progressions formulas to compute the totals in the problems.

1. Pedro did 40 sit-ups (abdominales) on Tuesday, 50 sit-ups on Wednesday, 60 sit-ups on

Thursday, 70 sit-ups on Friday, and 80 sit-ups on Saturday. If this pattern (pauta)

continues, how many sit-ups will Pedro do on Sunday?

2. The teacher gave 18 gold stickers (pegatinas doradas) to the first student, 24 gold stickers

to the second student, 30 gold stickers to the third student, and 36 gold stickers to the

fourth student. If this pattern (pauta) continues, how many gold stickers will the teacher

give to the fifth student?

How many stickers will he give in total?

3. A new cookbook is becoming popular. The local bookstore ordered 1 copy in May, 5

copies in June, 25 copies in July, and 125 copies in August. If this pattern (pauta)

continues, how many copies will the bookstore order in September?

How many copies will it order in total?

4. Ana put 2 beads (cuentas-abalorios) in the first jar (tarro), 4 beads in

the second jar, 8 beads in the third jar, 16 beads in the fourth jar, and

32 beads in the fifth jar. If this pattern continues,

How many beads will Ana put in the sixth jar? How many in total?

5. Luisa picked 2 flowers from the first bush (arbusto), 4 flowers from the second bush, 8

flowers from the third bush, and 16 flowers from the fourth bush. If this pattern (pauta)

continues, how many flowers will Luisa pick from the fifth bush? How many will be in

total? [ (*) Use a sequences formulas] [Sol: 32; 62 in total]

6. In a theatre, there are 28 chairs in the first row, 32 chairs in the second row, 36 chairs in

the third row and 40 chairs in the fourth row. If this pattern continues: how many chairs

will there be in the twelfth row? If there are 15 rows, what is the capacity of the theatre?

[Sol: 840 chairs]

7. We have a 40m deep well (tenemos un pozo de 40m de profundidad). We have paid €7.5

for the first metre and for each successive metre €2.3 more than for the previous. How

much does the well cost? [Sol: €2094]

8. Juan has €90 in a savings account. The interest rate is 5% per year and is not

compounded. How much interest will he earn in 5 years? How much will he have?

22

9. Certain businessman earns per year a 6% more than the previous year. If the first year he

earned €25,000, how much will he earn the 10th year? How much in those 10 years?

10. We are paying a debt. The first week we pay €5, the next week €9; then €13, €17, and

so on. If we pay in 30 weeks, how much do we owe (debemos)? [Sol: €2010]

11. Leonardo deposited €100 in a savings account earning 5% interest, compounded

annually. How much will he have in 6 years? [Sol: €134]

12. Monica has €80 in a savings account that earns 10% interest, compounded annually. How

much will she have in 3 years? [Sol: €106.48]

13. Angeles has €90 in a savings account that earns 5% annually. The interest is not

compounded. How much will she have in 2 years? [Sol: €99]

14. Compute the principal we have to pay in an account that pays a 5% of compounded

interest if we want to get €1,526.50 in twelve years. And how much if the interest is

simple? [Sol: €460.63; and €954.06 if it is simple]

15. The ending balances in Carissa’s savings account for each of the past four years form the

sequence {$1,000, $1,100, $1,210, $1,331,...}. Is the sequence arithmetic, geometric, or

neither? Find the next two terms of the sequence. [Sol: Geom. a5=1464.1, a6=1610.51]

16. A large pizza at Joe’s Pizza Shack costs $7 plus $0.80 per topping. Write a sequence of

pizza prices consisting of pizzas with no toppings, pizzas with one topping, pizzas with two

toppings, and pizzas with three toppings. Is the sequence arithmetic, geometric, or

neither? How do you know? [Sol: Arithmetic. an=6.20+0.8n]

17. A family purchased furniture on an interest-free payment plan with a fixed monthly

payment. Their balances after each of the first four payments were $1,925, $1,750,

$1,575, and $1,400.

a) Is the sequence of the balances arithmetic, geometric, or neither? Explain how you

know. If it is arithmetic or geometric, state the common difference or common ratio.

b) Continue to find the terms of the sequence of balances until you get a term of 0. After

how many payments will the balance be $0?

23


Bilingual section

SEQUENCES AND PROGRESSIONS.

1. (3 pts.) Fill the chart using progressions (0.25 pts. each cell):

Terms Description General Term Some terms…

{2,4,6,8,…}. a6=

Multiples of 3, plus 5. a15=

an= 4n-7 a20=

{20,15,10,…} a8=

2. (1.5 pts.) Compute the general term of the following sequences (0.5 pts. each section):

a)

,...15

9,

11

11,

7

13,

3

15. an=

b) ,...24,12,6,3 . an= . Compute also, a8=

c) {3000,300,30}. an= . Compute also, a6=

3. (2.5 pts.) Calculations:

a) For certain arithmetic progression, a1=2 and a3=12. Compute the sum of its 20 first terms.

b) Compute 2+ 2·1.05+ 2·1.052+ 2·1.053+…+2·1.0515=

c) Given the geometric progression {3, 6, 12, 24…}, compute S12=

d) Use the formula for Sn to compute: (2+7+12+17+...+102)·(1+4+7+...+46)=

4. (1 pt.) During December, a shop manager has been taking note of his electricity

consumption. The first day, it was 2Kwh; the second, 5Kwh and so on, increasing in

arithmetic progression.

a) What was the consumption the day 31st?

b) What is the consumption of December in total?

5. (2 pts.) Luis is going to deposit €18,000 in a bank for 20 years.

a) How much will he earn if we use simple interest of 5 %?

b) How much will he earn if we use compound interest of 5 %?

25

PROPORCIONALIDAD

- Proporción: igualdad entre dos razones (cociente de magnitudes). El número obtenido se

llama constante de proporcionalidad. Ejemplo:

es una proporción y la constante es 2.5.

Además, 5 y 10 se llaman extremos, y 2 y 25 medios.

Relación entre magnitudes:

Directa: si una aumenta, la otra también. Proporcional cuando al doble de una le

corresponde el doble de la otra; al triple le corresponde el triple,…

Ejemplos. Geometría: Teorema de Thales, figuras semejantes y mapas a escala.

Porcentajes. (*) Para aumento/disminución encadenados es mejor hacerlos

multiplicando.

Inversa: si una aumenta, la otra disminuye. Proporcional cuando al doble de una le

corresponde la mitad de la otra; al triple le corresponde un tercio,…

(*) Compuesta: si está relacionada con más de una magnitud al mismo tiempo.

Ejemplo: el espacio recorrido con la velocidad y la duración de un viaje.

Cálculo del cuarto proporcional y el medio proporcional

Propiedad fundamental: “El producto de los medios es igual al producto de los extremos”.

Ej. para

, 3x=1·25. Y en

, x·x=1·25; x2=25. “x” es el medio proporcional.

Resolución de problemas. Se pueden hacer de dos formas:

1ª: calcular la constante de proporcionalidad (usando una función) “reducción a la unidad”.

2ª: hacer el planteamiento usando la regla de tres, que puede ser compuesta.

- Reparto Proporcional: la cantidad a repartir se corresponde con el total.

(*) Reparto con proporcionalidad compuesta: primero hay que juntar las magnitudes

multiplicándolas (o dividiendo si el reparto es inversamente proporcional).

26

PROPORTIONALITY

- Proportion: statement that two ratios (quotient of magnitudes) are equal. The number

obtained is called proportionality constant. Example:

is a proportion and the constant

is 2.5. Moreover, 5 and 10 are called extremes, and 2 and 25 means.

Relationship between magnitudes (types of variation):

Direct: the greater the 1st, the greater the 2nd. Proportional i If one doubles, the other will

also double, etc,…

Examples. Geometry: Thales Theorem, similar figures and scaled maps.

Percentages. (*) For consecutive percentage increase/decrease it is better to

compute them multiplying.

Inverse: the greater the 1st, the smaller the 2nd. Proportional if one doubles, the other will

become half as large, etc.,…

(*) Compound: when two or more magnitudes are involved in the relationship

Example: the distance travelled with speed and duration of a travel.

Finding the fourth proportional and the mean proportional)

Propiedad fundamental: “El producto de los medios es igual al producto de los extremos”.

E.g. for

, 3x=1·25 And in

, x·x=1·25; x2=25. “x” is the mean proportional.

Problem solving. There are two ways:

1st: find the constant of variation (using a function) “unitary method”.

2nd: set the problem up using the rule of three, which can be compound (double).

- For Proportional Distribution, the amount to share out corresponds to the total.

(*) Distribution with compound proportion: first you have to put the magnitudes together by

multiplying them (or dividing if it is inverse proportion).

27

PROPORTIONALITY - WORD PROBLEMS

1. In a shipment of 400 parts, 14 are found to be defective. How many defective parts

should be expected in a shipment of 1000? [35 parts]

2. A piece of cable 8.5 cm long weighs 52 grams. What will a 10-cm length of the same cable

weigh? [61.18 grams]

3. A snowstorm dumped (depositar) 18 inches of snow in a 12-hour period. How many

inches were falling per hour? [1.5 inches]

4. 2 gardeners fence in a garden in 9 hours. How long would it have taken for 6 equally

productive gardeners? [3 hours]

5. Mary can read 25 pages in 30 minutes. How long would it take her to read a 100 page

book? [120 minutes]

6. An employee working at an electronics store earned $3582 for working 3 months during

the summer. What did the employee earn for the first two months? [$2388]

7. A farmer has enough grain to feed 60 cattle (ganado) for 25 days. He sells 10 cattle. For

how many days will the grain last now? [100 days]

8. A company’s quality control department found and average of 5 defective models for

every 1000 models that were checked. If the company produced 60,000 models in a year,

how many of them would be expected to be defective? [300 defective models]

9. To determine the number of deer in a forest, a forest ranger tags 280 and releases them

back into the forest. Later, 405 deer are caught, out of which 45 of them are tagged.

Estimate how many deer are in the forest. [2520 deer]

10. The ratio of men to women at a class is 6 to 5. How many women students are there if

there are 3600 men? [3000 women]

11. A town has 800 inhabitants. Twelve percent of them have never seen the sea. How many

of the inhabitants have never seen the sea? [96 inhabitants]

12. It takes 20 hours for a tap (grifo) with a flow of 15 litres per minute to fill a tank (depósito)

with water. How long will it take if its flow is reduced to 12 litres per minute? [25 hours]

13. A farmer harvests 25,000 kg. of corn and sells 85% to an animal feed factory. How many

kilos did the feed factory buy? [21,250kg.]

14. A 1 kilo cake contains 150 grams of sugar. What percentage of the cake is sugar? [15%]

28

15. In a football team, 12 players have missed practice. This is 40% of all the players. How

many players are on the team? [Sol: 30 players]

16. Ana lost 12kilos, which is 15% of what she weighed one year ago. How much did she

weigh one year ago? [Sol: 80kg.]

17. A farmer has got food enough for 1200 rabbits for 180 days. If he sells 300 rabbits, how

long will the food last? [240 days.]

18. The price of a bus ticket used to be €2, but today it goes up 5%. How much will a ticket

cost from now on? [€2.10]

19. In a given population, 2,480 people last year had the flu. This year the number is 30%

lower. How many people had the flu this year? [1736 people]

20. A company with 1,675 employees cuts its staff by 8%. How many employees does it have

after the cut? [1206 employees]

21. Three partners make €12,900 in a business. Juan put €18,000, Antonio €15,000 and Marta

€10,000. How much should each of them receive? [€5,400,: €4,500 and €3,000.]

22. Two workers earn 660€ for a job. The first one worked for 4 days and the second one

worked for 7 days. How much should they receive? [Sol: €240 and €420]

Geometric proportionality

23. Use similarity of figures to work out the missing side lengths. Round to tenths.

[1] [2] [3] [4]

[5] [6] [7]

[8] [9] [10]

Sol: 3.9 and 4.8

Sol: 2 and 4.1

Sol: 2.8 and 2.8 Sol: 4 and 5.6

Sol: 4.3 and 6.5 Sol: 7.3 and 11.1

Sol: 2.3 and 3.2 Sol: 4.25 and 7.2

Sol:7.8 and 6.3 Sol:6.4 and 7.6

29

[11] [12] [13]

24. Find the ratio of similarity (r). Then use it to find the surface of the larger base (A) and the

volume of the big pyramid/cone (V). Finally, work out the frustum’s volume (F).

[Calcula la razón de semejanza (r). Luego úsala para encontrar el área de la base mayor (A) y el volumen de la

pirámide/cono grande (V). Por último, deduce el volumen del tronco (F)].

a) b) c)

d) e) f)

g) h) i)

Solutions:

a) r=2, A=32cm2, V=85.6cm

3, F=74.9cm

3. b) r=3, A=63cm

2, V=251.1cm

3, F=241.8cm

3.

c) r=4, A=28.8cm2, V=76.8cm

3, F=75.6cm

3. d) r=2.5, A=34.38dm

2, V=143.28dm

3, F=134.11dm

3.

e) r=2, A=12cm2, V=16cm

3, F=14cm

3. f) r=2.5, A=18cm

2, V=24cm

3, F=21cm

3.

g) r=2.5, A=1.75m2, V=0.63m3, F=0.59m3. h) r=3, A=63m2, V=189m3, F=182m3.

i) r=3.5, A=21.44cm2, V=18.87cm3, F=18.43cm3.

Sol: 2.6 and 6.5 Sol: 4.5 and 2.6 Sol: 7.8, 6 and 7.7 and 3.7

4cm

8cm 8cm

2

10.7cm3 2cm

8cm

1.8cm2

1.2cm3 4cm

12cm

7cm2 9.3cm3

5.5dm2

9.17dm3 2.5dm

6.25dm

2cm

5cm 4.5cm2

3cm3

1.75cm2

0.44cm3

1cm

3.5cm

3m 9m

7m2 7m3

1m

0.4m 0.28m2

0.04m3

2.5cm

5cm

3cm2

2cm3

8

3

10

30

Compound proportion:

25. Two workers channel (canalizar) 100m pipes (tuberías) in 10 days. How long will take 5

workers to channel 350m pipes.

26. 10 men can lay a road 75 Km. long in 5 days. In how many days can 15 men lay a road 45

Km. Long?

27. Wheat (trigo) costing €480 is needed to feed 8 people over (durante) 20 days. What is the

cost of wheat required to feed 12 people over 15 days?

28. In a ship, they have food enough for 400 people and 64 days if they have portions of 1960g.

How much could they eat if there were 140 people but the trip lasted 80 days?

29. In a workshop (taller), spending 8 hours per day, it has taken them 5 days to make 1,000

pieces. How long will take them to make 3,000 pieces working 10 hours per day? [12 days]

30. The dorm (residencia de estudiantes) charges $6300, for 35 students for 24 days, in how

many days will the dorm charges be $3375 for 25 students? [Sol: 18 days]

31. 24 men working at 8 hours per day can do a piece of work in 15 days. In how many days

can 20 men working at 9 hours per days do the same work? [Sol: 16 days]

32. Working 8 hours per day, a glass factory makes 6,000 bottles in 3 days. How long would it

take it to 10,000 bottles working 10 hours per day? [Sol: 4 days]

33. In order to finish a building work in 360 days, 30 workers are needed working 8 hours per

day. How long will it take to finish to 45 workers working 6 hours per day? [Sol: 320 days]

34. A transport company charges €80 for carrying 1500kg of goods a distance of 100km. How

much will charge for carrying 4500kg a distance of 250km? [Sol: €600]

35. Working 8 hours per day a textile factory makes 15,000 pairs of socks in 12 days.

- How many pairs of socks will it produce over the next ten days if it doubles its working

hours (to 16 hours)? [Sol: 25,000 pairs]

- And how many days does it need to make 20,000 socks, working 16 hours? [Sol: 8 days]

36. 20 cows consume 600kg of feed in three weeks. How many kg. of feed do 30 cows

consume in one month (4 weeks)? [Sol: 1200kg]

31

Proportional distribution

37. In a competition, Pedro achieved a 10m shot with a 4kg weight and Oscar a 6m shot with a

3kg weight. Distribute a €174 prize directly to the weight and the distance.

38. A company is going to distribute €2,250 among three employees. Juan is 35 years old and

earns €1,400 monthly. Ana is 24 and earns 1,200, and Luis is 48 and earns €1,600. The

distribution will be directly proportional to their ages and inversely to their salaries.

39. A company is giving €750 gratification to three of their typist. Ana typed 250 pages and but

had 5 typing errors per page, Juan 240 pages and had 4 errors per page, and María 280 but

had 7 errors per page. Distribute the gratification proportionally among them (directly to

the pages and inversely to the typing errors).

40. Distribute a €450 bonus (gratificación) between two employees, proportionally to the

number of hours worked and new clients made and inversely to the extra-money spent.

Luis: 40hours, 20 clients and €100 extra. Julián: 35 hours, 18 clients and €90 extra.

41. Distribute 528€ between two brothers directly to their ages and final marks:

Juan is 15 and got 8 points. Virginia is 16 and got 9 points. [Sol: Juan €240; Virginia €288.]

42. We have paid two teams 6888€ for clearing a forest. There are 12 people in the first team

and they worked for 8 days. In the second, there are 15 people and they worked for 10

days. How much should receive each team? [Sol: 1stteam: €2688; 2nd team €4200.]

43. Luis and Sonia got 430 points in a trivia competition. Luis answered correctly 23 questions

in 10 minutes, and Sonia 30 questions in 15 minutes. Distribute the points proportionally to

the questions and inversely to the time spent. [Sol: Luis: 230 points; Sonia: 200 points]

44. A hospital is sending 28 patients to other hospitals. Distribute the patients proportionally

to the available (disponible) beds and inversely to the distance. Hospital A: 20 beds and is

at 10km. Hospital B: 30 beds and 15km. Hospital C: 36 beds and 12 km.

[Solution: Hospital A: 8 patients; Hospital B: 8 patients, Hospital C: 12 patients.]

Consecutive percent increases/decreases

45. Three years ago, Andrés rented his house for €400 per month. The R.P.I. “Retail Price

Index” (parecido al I.P.C.) has been: 3%, 4% and 3.5% these years. What should be the

rental (precio de alquiler) now? [Sol: €443.48]

46. A factory has to pay a €15,000 invoice. They get a 15% discount for prompt payment

(pronto pago). V.A.T of the operation is 12 %. How much does the factory have to pay?

47. A fridge costs € 336 after a price increase of 40%. What was the original price? [€240]

32

48. We have paid €24 for a skirt that had a 20% off. We have also paid a 21% VAT. What is the

price of the skirt without discount and without VAT? [Sol: 24.79€]

49. Over the period of a year the price of an article first increases by 40%, then decreases by

10% before finally decreasing a further 20%. Calculate the percentage change over the

whole year. [It increases a 0.8% (it is a 100.8% of the total)]

50. Over three decades an expanse of forest changes: 1st decade; it increases by 28%. 2nd: it

decreases by 40%. 3rd: 1970 to 1980 it increases by 15%.By what percentage did the forest

change during all three decades? [Decreases by 11.68% (it is a 88.32% of 100%)]

51. Three years ago, the cost of living went up by 10% and by 8% two years ago. Then, last year

it went down by 5% (this data is not true). How much had the cost of living increased in

these three years? [The amount has increased by 12.86% (and not 10+8-5=13%)]

52. In the sales we bought a painting for € 105, a bicycle for €50.40 and a book for €16.35. If all

the prices were reduced by 30% how much would we have spent on the same items before

the sales? [Painting: €150; Bicycle: €72; Book: €23.36]

53. I have an investment that has returned a 20% increase in performance from the start of the

year until October, and then increased a further 3% in November, what has been the year

to date? [23.6% (it is a 123.6% of the total)]

54. A coat is marked down by 40%, and you have a coupon good for an additional 25% off. If

VAT is 20% and you are charged €400, what is the original price (without VAT)? [€740.74]

55. Your business is growing faster than you had ever imagined. Last year you had 100

employees statewide. This year, you opened several additional locations and increased the

number of workers by 30%. With demand so high, next year you will be opening new stores

nationwide and plan to increase your employee roll by an additional 50%. Determine the

projected number of employees next year. [195 workers]

56. We are making a cone using a base having 8cm radius. Its slant height is 10cm,

and we need 4/5 of a circle for the top.

[The net of a cone is a circle with part missing (and a smaller circle underneath).

You use the slant height itself as the radius of the part-circle].

a) Write an equation relating “fractions of circle” with “radius” and “slant height”.

Find the proportionally constant. [Sol: Fraction =

. C=1]

b) What fraction of a circle do we need for a cone with 20cm slant height and

base radius 10 cm? [Sol: One half of a circle]

c) What is the base radius of the cone we can make using 3/4 of a circle having 9

cm radius (so the slant height is 9cm)? [Sol: 12 cm]

33


Bilingual section

PROPORTIONALITY.

(*) All word problems must be answered using a sentence.

1. (1 pt.) Compute the proportional mean “x” of 75 and 12 (i.e. so that

)

2. (1.5 pts.) A farmer has got 300 animals and fodder (pienso) enough to feed them over 90

days. He decides to sell some of them so that he has fodder enough for 135 days.

a) How many animals does he have now?

b) How many has he sold?

3. (1.5 pts.) In a factory, 3 machines produce 480 pieces in 4 days, working 8 hours per day.

a) How long will take them to produce 200 pieces, working 6 hours per day?

b) How many hours per day do they have to work 5 machines to produce 500 pieces in 2

days?

4. (1.5 pts.) Three friends have a business that makes a profit of €3,500. Distribute it

proportionally to the time and money invested by each of them.

Juan: 3 hours and €20.

Luisa: 5 hours and €15.

Antonio: 4 hours and €10.

5. (1.5 pts.) Distribute a €360 incentive between two workers, inversely proportional to the

number of days they have arrived late to work this year. Ana: 10 days and María: 8 days.

6. (1.5 pts.) A piece of cloth cost 30€ in December. In January they offer it with a 10%

discount, and later they took another 30% off. If they add a 16% VAT to the final Price,

a) Compute the price including VAT.

b) Is the same as if (in total) they had made a 24% discount? What would have been the

price in that case?

7. (1.5 pt.) The price of a house was marked up (subir el precio) a 10% two years ago and a 4%

last year, but this year it was marked down (bajar el precio), a 4%.

If now the price €200.000, how much was it two years ago?

35

POLINOMIOS

- Monomio: producto de un número “coeficiente” por varias letras “parte literal” (se admiten

exponentes en las letras pero no raíces de letras, o letras en el denominador, etc.). El grado

del monomio es la suma de los exponentes.

Producto: juntamos las letras y sumamos los exponentes de las que son iguales. Ej: x2y4·xy3=x3y7.

Suma y resta de monomios semejantes (con la misma parte literal): se suman o restan los

coeficientes. Ej. 2x+5x=7x y 2x-5x=-3x.

- Polinomio: suma y/o resta de varios monomios (términos). Si hay dos términos, es un

binomio; si hay tres, trinomio, etc. Se escriben ordenados, del grado mayor al menor.

El monomio de mayor grado da el grado del polinomio. Su coeficiente se llama principal.

Término independiente: un número que aparece sin letras. Su grado es 0.

Ej: 3x2-5x+1: trinomio de segundo grado, coeficiente principal 3 y término independiente 1.

Valor numérico: resultado de sustituir las letras por un número.

Raíz de un polinomio: cuando el valor numérico es 0.

Operaciones con polinomios: (el polinomio debe ir entre paréntesis)

Signo “+”: se quita el paréntesis y se deja todo como está. Ej.: +(3x2-5x+1)= 3x2-5x+1.

Signo “-“: se cambia el signo de cada término. Ej. -(3x2-5x+1)= -3x2+5x-1.

Producto: se usa la propiedad distributiva. Ej. (x+3)·(x-2)=x2-2x+3x-6.

Puede usarse para sacar factor común. Ej. x2+5x=x(x+5), 4x+6=2·(x+3)

Pueden usarse para factorizar. Ej. x2–4=(x+2)(x-2), x2-5= .

División: el grado del resto debe ser menor que el del divisor.

Para dividir entre x+a puede aplicarse la regla de Ruffini.

Teorema del resto: al dividir entre x-a, el resto es el valor numérico para x=a.

Teorema del factor: un polinomio es divisible entre x-a cuando a es raíz suya.

Fracción algebraica: cociente de dos polinomios. Ej.

. Podemos operar con ellas o

simplificarlas como con las fracciones de números.

Igualdades notables:

(a+b)2=a2+b2+2ab , (a-b)2=a2+b2-2ab

(a+b)·(a-b)=a2-b2.

36

POLYNOMIALS

- Monomial: algebraic expression where a number “coefficient” is multiplied by letters “literal

part” (exponents are allowed on letters but not square roots, nor a letter in a denominator,

etc.). The degree of a monomial is the sum of the exponents.

Product: join the letters and add the exponents of letters alike. Ex: x2y4·xy3=x3y7.

Addition and subtraction of like monomials (with the same literal part): add or subtract

coefficients. For instance: 2x+5x=7x and 2x-5x=-3x.

- Polynomial: addition and/or subtraction of several monomials (terms). If there are two

terms, it is a binomial; if there are three, it is a trinomial, and so on. We write them ordered

from the greatest degree to the least.

The monomial with the greatest degree gives us the degree of the whole polynomial. It is

called the leading term and its coefficient is called leading.

Independent term: a number without letters. Its degree is 0.

Ex: 3x2-5x+1: second degree trinomial, principal coefficient 3 and independent term 1.

Numerical value: the result of replacing letters with numbers in an algebraic expression.

Root of a polynomial: when the numerical value is 0.

Operations involving polynomials: (when the polynomial is in parentheses)

“+” sign: remove the parentheses and leave everything as it is. Ex.: +(3x2-5x+1)= 3x2-5x+1.

“-“ sign: change the sign of the terms. Ex.: -(3x2-5x+1)= -3x2+5x-1.

Product: we use the distributive property. Ex.: (x+3)·(x-2)=x2-2x+3x-6.

It can be used to find the common factor. Ex.: x2+5x=x(x+5), 4x+6=2·(x+3)

They can be used for factoring. Ex.: x2–4=(x+2)(x-2), x2-5= .

Division: the degree of the remainder is less than the degree of the divisor.

To divide by x+a we can use Ruffini’s rule.

Remainder Theorem: the remainder of the division by x-a, is the numerical value for x=a.

Factor Theorem: a polynomial is divisible by x-a when a is a root of the polynomial.

Algebraic fraction: quotient of two polynomials. Ex.:

. We can operate with them or

simplify them as well as we do with numerical fractions.

Special binomial products:

(a+b)2=a2+b2+2ab , (a-b)2=a2+b2-2ab

(a+b)·(a-b)=a2-b2.

37

POLYNOMIALS – REVISION EXERCISES

1. Use the special binomial products (either to multiply or to factorize)

a) = b) = c) =

d) = e) = f) =

g) = h) = i) =

j) = k) = l) =

2. Take out common factors. (*) Use them to simplify the algebraic fractions, if possible.

a) 2x3-5x2+x= b) (x-1)x2+5x(x-1)-3(x-1)= c) 6x8-4x5+2x3=

d) 3x2(2x+5)+6x(2x+5)-9(2x+5)= e)

= f)

=

g)

= h)

= i)

=

3. Compute the following divisions (compute the quotient “q” and the remainder “r”):

a)

b)

c)

d)

e)

f)

g)

h)

i)

Solutions:

1. [a] . [b] . [c]

[d] . [e] . [f] .

[g] . [h] . [i] .

[j] . [k] [l] .

2. [a] x·(2x2-5x+1). [b] (x-1)(x

2+5x-3). [c] 2x

3·(3x

5-2x

2+1)

[d] 3(2x+5)(x2+2x-3). [e] 5x2-4x+1. [f] x2-5x+2

[g]

. [h]

. [i] 3x2+2x-5.

3. [a] q= ; r= . [b] q= ; r= . [c] q= ; r=-3x2+2

[d] q= r= . [e] q= ; r= . [f] q= ; r=0.

[g] q= ; r= . [h] q=x3-2x2+3x-1; r= . [i] q= ; r= .

38

POLYNOMIALS - WORD PROBLEMS

1. Determine an expression representing the total income from selling roses at €6 each and

daffodils (narcisos) at €3 each.

2. Think of a number. Subtract 7. Multiply by 3. Add 30. Divide by 3. Subtract the original

number. The result is always 3. Use polynomials to illustrate this number trick.

3. Let an integer be represented by x. Find, in terms of x, the product of three consecutive

integers starting with x.

4. What expression represents “eight less than the product of five and a number”?

5. Find the area and the perimeter of these polygons. Simplify the answers.

a) b) c)

6. Write a variable expression for the area of a square whose side is x + 8.

7. The side of a cube is represented by x + 1. Find its volume in terms of x.

8. A circular courtyard has an area of 10 – 2x2. There are two rectangular

flower beds in the courtyard. Write an expression that represents the

green lawn area.

9. Two weeks ago James bought 3 cans of tennis balls. Last week he bought

4 cans of tennis balls. This week he bought 2 cans of tennis balls. The tennis balls cost d

dollars per can. Write an expression in simplest form that represents the total amount that

James spent

10. For his birthday, Carlos’s parents give him €5 for each year of his age plus €50. His

grandmother gives him €10 for each year of his age. Let a represent Carlos’s age in years.

Write a polynomial expression for the amount that Carlos receives from his parents.

a) Write a polynomial expression for the amount that he receives from his grandmother.

b) Write a polynomial expression for the total amount that Carlos receives from his parents

and grandmother.

c) How much will Carlos receive when he is 15 years old?

39

11. Lydia took a taxi from her home to school that charged $2 plus $0.50 per mile. Her brother

Luke took a taxi the same distance that charged $3 plus $0.30 per mile. Let d represent the

distance in miles.

a) Write a polynomial expression for the cost of Lydia’s taxi. Then write a polynomial

expression for the cost of Luke’s taxi

b) Find an expression representing the total cost of Lydia and Luke’s taxi rides

c) What is the total cost if the distance is 20 miles?

12. Maria bought 7 CDs at x dollars each and used a coupon for $20 off her purchase of more

than 5 CDs. Ricky bought 4 CDs at x dollars each and redeemed (canjear) a coupon for $10

off his purchase of more than 3 CDs.

a) Write polynomial expressions representing how much each spent after the discount.

b) Write a polynomial representing how much more Maria spent than Ricky.

13. The polynomial expression (300 + 0.4s) – (500 + 0.3s) represents the difference between

two salary options that Chuck has in his new position as a salesperson (vendedor). Write

this difference in simplest form.

14. On a test worth 100 points, Jerome missed 3 questions worth p points each but answered a

bonus question correctly for an extra 5 points. Suni answered 4 questions incorrectly and

did not get the bonus.

a) Write polynomial expressions representing each student’s score on the test.

b) write a polynomial representing how many more points Jerome scored than Suni.

15. Sal’s Pizza Place charges €8 for a large pizza plus €0.75 for each topping (ingredient), while

Greco’s Cafe charges €10 for the same size pizza plus €0.90 for each topping.

Write a polynomial in simplest form that represents how much more a pizza with t

toppings would cost at Greco’s than at Sal’s.

16. The Marshalls’ pool is 5 feet longer than twice its width w. Write two expressions for the

area of the pool. What is the area of the pool if it is 12 feet wide?

17. When the Science Club members charged p dollars to wash each car at their car wash, they

had 8p customers. When they doubled their price, they had 12 fewer customers.

a) Write expressions representing the new price and the new number of customers.

b) Write an expression representing the amount of money they made at the new price.

c) How much money did they raise at the new price if the original price was $5 for each car?

40


Bilingual section

POLYNOMIALS.

1. (1.5 pts.) Given: P(x) = 10x2 +6x-2 , Q(x) = 1– 3x2 –4x3, R(x) = 2x2 – 3 , calculate

P(x) – [Q(x)-2x·R(x)] =

2. (1 pt.) Marta bought 5 CDs at x dollars each and used a coupon for $20 off. Juan bought 4

CDs at x dollars each and redeemed (canjear) a coupon for $10 off.

a) Write polynomial expressions representing how much each spent after the discount.

Marta: Juan:

b) Write a polynomial representing how much more Maria spent than Juan:

3. (1.5 pts.) Take out the common factors and simplify:

a)

=

b)

=

4. (1.5 pt.) Use the special binomials products (either to compute or to factorize):

a) 9x2 - 144= b) (5x2 –6)2 = c) (x3-5)(x3+5) =

5. (3 pts.) Divide: P(x)= - - by Q(x)= -

a) (1 pts.) Quotient: Remainder:

b) (0.5 pts.) Write division and the result using algebraic fractions:

c) (1.5 pt.) Check the result.

6. (1.5 pts.) Use Ruffini’s rule.

a) 2x4-3x3+6x2-x+5 divided by x+1. Quotient: Remainder:

b)

Quotient: Remainder:

41

ECUACIONES

- Ecuación: Igualdad entre expresiones algebraicas que sólo se cumple para algunos valores de

las letras, que se llaman solución. Resolver la ecuación es encontrar las soluciones.

Las letras se llaman incógnitas. El grado es el mayor grado de los términos.

La parte de la izquierda primer miembro, y la de la derecha, segundo miembro.

Resolución de ecuaciones de primer grado (ecuaciones lineales):

Resolución de ecuaciones de segundo grado (ecuación cuadrática):

CASOS POSIBLES:

1. La incógnita siempre aparece con el mismo grado: se hace como con las de primer grado, y luego se hace una raíz. En la raíz cuadrada hay que poner ±.

Ejemplo: x2=9; . Soluciones x=+3 y x=-3.

2. Producto igual a 0: se resuelve cada factor por separado.

Ejemplo: para (x-1)·(x-2)=0, se resuelve x-1=0 y x-2=0. Las soluciones son x=1 y x=2.

3. Se puede factorizar: si no hay término independiente, se saca la incógnita factor común.

Ejemplo: x2-3x=0. Factorizamos x·(x-3)=0. Las soluciones son x=0 y x=3.

4. Ecuación “completa”:

Se calcula más rápido si hacemos primero el discriminante .

Fórmulas de Cardano: cuando a=1, la suma de las soluciones es –b, y su producto es c.

Ecuaciones de Tercer grado o mayor: se puede usar el método de Ruffini. Para las raíces

enteras, sólo hay que probar con divisores del término independiente.

ax2+bx+c=0. Soluciones:

1. Quitar paréntesis.

2. Quitar denominadores.

3. Transponer y agrupar términos

semejantes.

4. Despejar la incógnita.

(reduciendo primero a común denominador, o si es una igualdad de fracciones, se multiplica en cruz).

(se les cambia el signo al cambiarlos de miembro).

(Si un número multiplica a un miembro, pasa al otro

dividiendo, y viceversa)

42

EQUATIONS

- Equation: Equality of two algebraic expressions that is correct for only certain values of the

letters, called solution. To solve an equation is to find its solutions.

Letters are called unknowns. The degree is the greatest of the terms’ degrees.

The left side is the first member, and the right side, the second member.

Solving first degree equations (linear equations):

Solving second degree equations (quadratic equations):

POSSIBLE CASES:

1. The unknown always appears with the same degree: it is solved as the first degree ones and then, one takes a root. With square roots one has to write ±.

Example: x2=9; . Solutions: x=+3 and x=-3.

2. A product equals 0: solve each factor separately.

Example: for (x-1)·(x-2)=0, one solves x-1=0 and x-2=0. Solutions are x=1 and x=2.

3. It is possible to factorise: if there is no independent term, pick out common factor the

unknown.

Example: x2-3x=0. We factorise like x·(x-3) =0. Solutions are x=0 and x=3.

4. “Complete” equation:

It is computed faster if we first work out the discriminant .

Cardano formulae: when a=1, the sum of the solutions is –b, and their product is c.

Third or higher degree equations: we can use Ruffini’s method. For integer roots, choose

only divisors of the independent term.

ax2+bx+c=0. Solutions:

1. Eliminate parentheses.

2. Eliminate fractions.

3. Transpose and combine like terms.

4. Get the unknown by itself.

(reduce first to common denominator, or if it is an equality of fractions, cross multiply.).

(changing the sign when changing to the other side).

(If a number multiplies on one member, goes to the

other dividing it, and vice versa)

43

EQUATIONS – REVISION EXERCISES

1. Solve the following equations (first degree techniques).

a)

b)

c)

d)

e)

= f)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)

2. Solve the following equations (second degree techniques).

a) (x+1)(x-1)-5(x+2)=x-20 b) 2x(x+3)+x(5-3x)=x2+5 c) (x-4)·(3x+1)=x·(x+1)-4

d)

e)

f)

g)

h)

= i)

j)

k)

l)

m) n) (2x-6)2+5=(x+2)(x-2) o)

p)

q)

r)

Solutions:

1. [a] x=2. [b] x=1. [c] x=5. [d] x=-1. [e] x=-1. [f] x=3. [g] x=3. [h] x=5. [i] x=-2. [j] x=4

[k] x=-3 [l] x=1 [m] x=6 [n] x=3 [o] x=5 [p] x=1 [q] x= 2 [r] x=5.

2. [a] x=3. [b] x=5, x=1/2. [c] x=0, x=6. [d] x=2, x=-12/5 [e] x=-2, x=4.

[f] x=3, x=-7/2. [g] x=5, x=-3/8. [h] x=4, x=-1/3 [i] x=5, x=-4. [j] x=2

[k]x=5, x=10 [l] x=-5, x=2. [m] x=-4, x=3. [n] x=3, x=5 [o] x=-2, x=5

[p] x=2, x=-14/5 [q] x=-3, x=6. [r] x=±2.

44

3. Solve the following equations (using first degree equations techniques).

a) x2+3·(5-2x2)=6-4x2 b) 3(15-x3)+4x3-45=3(x3+5)-18 c) 4·(x2-3)-6·(x2-2)+3x2-102=21

d) 4x3+100=5(x3-50)+7 e) 3·(40-2x4)-8·(x4-10)=-8-x4 f) x5-3(2x5+4)+6(2-2x5)=3+2(3x5+10)

4. Solve the following equations (remember NOT to multiply before solving).

a) (x-4)(3x+6) = 0 b) (5-x)·(2x+3)·x=0 c) x·(x2-5x+6)=0

d) (x2+x-2)(x-2)=0 e) (3x2+14x-5)·x2=0 f) (2x2-x-1)·(x-5)=0

g) (x2-4)·(x2-3)=0 h) (x3-343)·(2x2+3x-5)=0 i) (2x3-250)·(x3+64)=0

5. Take out common factors before solving (remember NOT to multiply the polynomials).

a) x3+2x2 - 8x= 0 b) x·(5x-1)+3·(5x-1)=0 c) x·(3x-6)=5·(3x-6)

d) x2(x-1)-5x(x-1)-6(x-1)=0 e) (x2-4)x2+2x(x2-4)-15(x2-4)=0 f) 3x2(2x2+5x-3)+9x(2x2+5x-3)=0

g) x2·(x3+27) – 5·(x3+27)=0 h) (2x3-54)·x2 =-4·(2x3-54) i) (x4+16)·x2+2x(x4+16)=-5(x4+16)

6. Solve using Ruffini’s method:

a) 2x3+16x=10x2+8 b) x3+11x=6·(x2+1) c) x4+x3-7x2-x+6=0

d) x·(x2+5x+6)-5x2=5(6+5x) e) x3+x=10·(4x-7) f) 27·(x-2)=x3

g) 3x-6 = 3x·(x-2) h) x·(x2-6x+5)=6·(x2-6x+7) i) x2·(x3-2x)=3x·(x2-1)-x

Solutions:

3. [a] x=±3. [b] x=2. [c] x=±11. [d] x=7. [e] x=±2. [f] x=-1

4. [a] x=4, x=-2 [b] x=5, x=-3/2,x=0. [c] x=0, x=2, x=3. [d] x=-2, x=1, x=2. [e] x=0, x=1/3, x=-5.

[f] x=-1/2, x=1,x=5. [g] x=±2, x=± . [h] x=7, x=-5/2,x=1 [i] x=1, x=9/4.

5. [a] x=0, x=2, x=-4 [b] x=-3, x=1/5. [c] x=2, x=5. [d] x=-1, x=1, x=6. [e] x=±2, x=3, x=-5.

[f] x=1/2, x=-3,x=0. [g] x=-3, x=± . [h] x=3. [i] No solution.

6. [a] x=1, x=2. [b] x=1, x=2,x=3 [c]x=-1, x=1, x=-3,x=2 [d] x=5, x=-3, x=-2 [e] x=2, x=5, x=-7.

[f] x=-6, x=3. [g] x=1, x=2 [h] x=2, x=3, x=7 [i] x=-2, x=-1, x=0, x=1, x=2.

45

7. Solve using the most suitable method:

a) x3+2x

3-4x2

2 x2-6x+4 b)

c) (2x-4)·(3x-5) 2(x+1)

d) 2x·(x2-9)-10·(x2-9) 0 e)

f)

x3-3x2-3x+9

5+x2+3

2-2x+2

7x-1

10

g) 2x·(x-5) (x-1)·(x+1)+x+1 h)

i)

j) x3=19x+30 k) 3 ·(x2-5)-12·(x2-5) 0 l)

m)

n)

o) (x2-2x)(2x-5)-6x (x+4)(x-4)

p) x3(5x-1)-8(5x-1)=0 q) 4x(x3-2)+8(x3-2)=0 r)

s)

t) u)

v)

w) x·(x+1)+1 0 x)

y) x(x2+4)-5(x2+4)=0 z) x5-17x3+12x2+52x-48 0.

Solutions:

7. [a] x=1, x=2, x=6. [b] x=1, x=3, x=

. [c] x=1, x=3. [d] x=±3, x=5. [e] x=.

[e] x=1, x=

. [f] x=2, x=3, x=

[g] x=0, x=11. [h] x=±3, x=-1, x=2. [i] x=-3, x=2.

[j] x=-3, x=-2, x=5 [k] x=±2, x= [l] x=2 [m] x=1, x=

[n] x=0, x=2, x=

[o] x=-1, x=2, x=4 [p] x=2, x=

[q] x=-2, x=

[r] x=±1, x=7 [s] x=±1

[t] x=0, x=±2 [u] x=±1, x=±3 [v] x=-3, x=1, x=

[w] No solution. [x]x=0, x=

[y] x=5 [z]x=1, x=±2, x=3, x=-4 .

46

EQUATIONS - WORD PROBLEMS

1. The length of a rectangular window is 5 dm more than its width, x. The area of the window

is 36 square decimetres. Write an equation to find the dimensions of the window.

2. Mike’s Fitness Centre charges €30 per month for a membership. All-Day Fitness Club

charges €22 per month plus an €80 initiation fee for a membership. After how many

months will the total amount paid to the two fitness clubs be the same?

How much will it be?

3. A Plumbing (fontanería) Service charges €35 per hour plus a €25 travel charge for a service

call. “P4U Plumbing Repair” charges €40 per hour for a service call with no travel charge.

How long must a service call be for the two companies to charge the same amount?

How much will they charge?

4. The Lone Star Shipping Company charges €14 plus €2 a kg. to ship an overnight (nocturno)

package. Discount Shipping Company charges €20 plus €1.50 a kg. to ship an overnight

package. For what weight is the charge the same for the two companies?

How much will the shipment cost?

5. Julia and Isa are playing games at the arcade. Julia started with $15, and the machine she is

playing costs $0.75 per game. Isa started with $13, and her machine costs $0.50 per game.

After how many games will the two girls have the same amount of money remaining?

How much will they have then?

6. The Wayside Hotel charges its guests €1 plus €0.80 per minute for long distance calls.

Across the street, the Blue Sky Hotel charges its guests €2 plus €0.75 per minute for long

distance calls. Find the length of a call for which the two hotels charge the same amount.

What would be the price?

7. Juan is a part-time student at Horizon Community College. He currently has 22 credits, and

he plans to take 6 credits per semester until he is finished. Juan’s friend Lidia is also a

student at the college. She has 4 credits and plans to take 12 credits per semester. After

how many semesters will Juan and Lidia have the same number of credits?

How many credits will they have?

8. A car leaves Badajoz at 5:30pm, driving at 120km per hour. At the same time, a bus leaves

Mérida (60km further) driving at 100kmph.

a) How long will take the car to catch up with the bus? What time will it be?

b) How long will have travelled each of them?

47

9. Juan’s got 215€ and Pedro’s got 160€. How much should Juan borrow from Pedro to have

twice the money as Pedro? How much will each have? [Solution: Juan should borrow €35].

10. Pedro is 34 years old and his son is 6. In how many years will Pedro be three times older

than his son? How old will each of them be? [Solution: In 8 years. Pedro will be 42 and his son 14].

11. [Clocks] Juan is watching a clock, and realizes that the minute-hand moves at a speed of 12

numbers per hour, and the hour-hand at 1 number per hour.

a. At 12:00a.m., both hands coincide, and not at 3:15p.m., but a little later.

What time after 3:00p.m. do both hands coincide? [Sol: 3+3/11=3h16m21’6s]

b. At 6:00pm. they form a straight angle (their difference is 6 numbers). At

7:05p.m. the angle is not straight; that happens a little later. What time after 7:05p.m. do

they form a straight angle? [Sol: At 7+1/11=7h05m27s]

c. At 9:00p.m. they are perpendicular (so their difference is 3 numbers), but not at 2:25p.m.

What time after 2:00p.m. do they form a right angle? [Sol: At 2+5/112h27m16’36s]

12. Juan is 3 years old and his father is 32. In how many years will the product of their ages be

252? How old will they be then? [Sol. In 4 years time. Juan will be 7 and his father 36]

13. Ana is 10 years old and her mother is 34. How long ago was the product of their ages equal

to 145? How old were them? [Sol: 5 years ago. Ana was 5 and her mother 29].

14. A rectangular swimming pool is twice as long as it is wide. A small concrete (cemento,

hormigón) walkway (pasillo) surrounds the pool. The

walkway is a constant 2 metres wide and has an area of

196 square metres. Find the dimensions of the pool.

(Hint: solve (2x + 4)(x + 4) - (2x)(x) = 196)

Solution: The pool is 15 meters wide and 30 meters long.

48


Bilingual section

EQUATIONS.

Solutions:

1. (1.75 pts.) x2(3x-1)+2x(3x-1)=-5(3x-1) ..

2. (1.75 pts.) (2x2-50)(2x2-4x-6)=0.

3. (2 pts.)

4. (1.75 pts.) x3+16=12x.

5. (1 pt.) Complete, so the solutions are…

a) x=1, x=3: x2 _____·x _____=0 b) x=-1, x=2: 3x2 _____·x _____=0.

6. (1.75 pts.) A car leaves Mérida at 10:20am in direction to Madrid, driving at 120km per

hour. At the same time, a van leaves Madrid (350km away) driving at 90km per hour in

direction to Mérida.

a) How long will it take them to meet on the road? What time will it be?

b) How many km will have travelled each of them?

[*] Set the problem up using a one unknown equation. Write answer (a) using hours and minutes.

49

SISTEMAS DE ECUACIONES

SISTEMA DE ECUACIONES: es cuando tenemos varias ecuaciones de las que buscamos una

solución común. Si las ecuaciones son de primer grado, el sistema se llama lineal.

- Para dar la solución del sistema, hay que dar un valor para cada incógnita.

RESOLUCIÓN DE SISTEMAS DE DOS ECUACIONES Y DOS INCÓGNITAS.

MÉTODOS ALGEBRAICOS:

Se elimina una incógnita y queda una ecuación de una incógnita. La resolvemos.

Después, se sustituye el valor obtenido en una de las ecuaciones y se resuelve lo que quede.

- Método de SUSTITUCIÓN: despejamos una incógnita en una ecuación y sustituimos en la otra.

Lo usamos en sistemas no lineales cuando alguna incógnita aparece con varios exponentes.

- Método de IGUALACIÓN: despejamos la misma incógnita en las dos ecuaciones. Hacemos una

nueva ecuación igualando los resultados.

- Método de REDUCCIÓN: sumamos las ecuaciones de manera que desaparezca una incógnita.

Puede que primero haya que multiplicarlas por algún número.

MÉTODO GEOMÉTRICO:

Cada ecuación lineal se dibuja como una recta del plano (usando una tabla de valores). La

solución es el punto donde se cortan las rectas.

50

EQUATIONS SYSTEMS

EQUATIONS SYSTEM: it is when we have several equations and we are looking for a common

solution. If the equations are of first degree, the system is called linear.

- In order to find a solution for the system, we have to give a value for each unknown.

SOLVING SYSTEMS OF TWO EQUATIONS AND TWO UNKNOWNS.

ALGEBRAIC METHODS:

Eliminate one unknown in order to get a “one variable equation” that we solve.

Then, we substitute the value obtained in one of the equations and solve the remaining

equation.

- SUBSTITUTION method: isolate one variable in one equation and substitute in the other.

We use it in nonlinear systems when any of the unknowns appears with different exponents.

- ALGEBRAIC EQUATION method: isolate the same variable on each equation. Make another

equation equalling the results.

- ELIMINATION (ADDITION) method: add the equations so that one variable disappears. You may

first need to multiply them by some number.

GRAPHING METHOD:

Draw each linear equation as a line on a coordinate plane (using a table of values). The

solution is the point where the lines meet.

51

EQUATION SYSTEMS – PRACTICE

1. Solve the following equation systems.

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)


a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

Solutions:

1. [a] x=1, y=3. [b] x=-3, y=1. [c] x=2, y=5 [d] x=-3, y=10 [e] x=3, y=1 [f] x=5, y=3

[g] x=-2, y=1 [h] x=1, y=4. [i] x=7, y=7 [j] x=1, y=-1 [k] x=5, y=10 [l] x=2, y=-3.

2. [a] x=3, y=2. [b] x=4, y=-1. [c] x=5, y=1 [d] x=4, y=2 [e] x=2, y=-1 [f] x=3, y=6

[g] x=3, y=6 [h] x=2, y=5. [i] x=1, y=-2 [j] x=-1, y=-3.

52


a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

4. Solve the following equation systems. Use substitution

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

Solutions:

3. [a] x=±1, y=2. [b] x=±3, y=2. [c] x=1, y=-3 [d] x=±1, y=5 [e] x=7, y=±5 [f] x=± , y=±3 (4 sol.)

[g] x=

, y=3 [h] x=±4, y=3. [i] x=3, y=2 [j] x=5, y=±4 [k] x=6, y=±4 [l] x=±2, y=±7 (4 sol.)

4. [a] {x=2, y=3}, {x=14/3, y=1/3}. [b] {x=-1, y=-1}, {x=-2/7, y=-8/7} [c] x=±3, y=4.

[d] {x=10, y=-2}, {x=14, y=0}. [e] {x=-4, y=12}, {x=3, y=5}. [f] {x=-1, y=0}, {x=2, y=3}

[g] {x=3, y=-1}, {x=41/9, y=4/3}. [h] {x=-1, y=1}, {x=3, y=5}. [i] {x=2, y=1}, {x=-8/5, y=11/5}.

[j]

53

5. Solve the following equation systems. (Use algebraic equation method –igualación-)

a)

b)

c)

d)

e)

f)

g)

h)

i)

6. Solve the following equation systems. Choose the most suitable method.

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

Solutions:

5. [a] - - - [b] - - - - [c] There is no solution.

[d] - - [e] - - [f] -

[g] [h] -

- [i] -

-

6. [a] - - -

[b] - [c] -

-

[d] -

[e] - - [f] - - -

[g] -

[h] -

[i] (four solutions)

[j] - -

54

EQUATIONS SYSTEMS - WORD PROBLEMS

1. Two years ago, Carlos’ age was triple the age of his son Luis, but in twelve years his age will

only be double that of Luis. Calculate their ages. [Sol: 44 and 16].

2. On a test with 30 answers, students receive 2 points for each correct answer and lose 1

point for each incorrect answer.

A boy who answered all of the test’s questions got a score of 24 points. How many correct

answers and how many incorrect answers did he give? [Sol: 18 correct and 12 incorrect]

3. How old is my youngest son if the sum of triple his age and five times his older brother's

age is 78, and his age is half the age of his older brother’s age? [Sol: youngest: 6, older 12]

4. A trader splits (reparte) 315 litres of oil between 1.5 litre and 0.75 litre bottles. If he fills

300 bottles, how many bottles of each capacity does he use?

5. I paid €45 for a shirt and a pair of trousers which would have cost me €52 had they not

been on offer. The shirt had been discounted 20% and the trousers 10%. What were the

original prices?

6. Find the value of two numbers if twice the lesser number is one more than the greater

number and three times the lesser number minus the greater number is five. [Sol: 4 and 7]

7. In my piggy bank (hucha) I have three times 2€ coins than 1€ coins. In all, I have 84€. How

many coins do I have of each type? [Sol: 36 coins of 2€ and 12 coins of 1€.]

8. In a language school, 430 students studied either English or French last year. This year,

students studying English have increased 18% and students studying French have increased

15%. There are now 502 students studying the two languages. Calculate how many English

students and how many French students there were last year.

9. I changed a lot (lote) of 20 cent coins for €1 coins, and now I have 12 coins less than I did

before I changed them. How many 20 cent coins did I have?

10. Three years ago, Laura’s age was half of Ana’s age, and in seven years their ages will add up

to 50. How old are they? [Sol: Laura is 13 years old and Ana is 23 years old.]

11. Ana and Raúl, the boy she was babysitting, were playing basketball together. Her score was

30 points, and his score was 10 points. Ana wanted to make the game fairer (más justo), so

she called a time-out and modified the rules a bit. Ana explained that, for the rest of the

game, she would get 3 points per basket, and Raúl would get 4 points per basket. Then

they played a bit longer. After the time-out, they both made the same number of baskets

and ended up with a tied score. How many points did each person have at the end?

55

12. A merchant has two types of coffee: one of 6€ per kg, and another of 4€ per kg. He mixes

them, and the price turns out to be 4.5€ per kg. In all, he has 8 kg of coffee.

How much is there of each type of coffee? (*) Set the problem up using an equations system. The first one adding the kilograms and the second one

computing the price of 8kg of coffee.

13. We mix two types of flour, one that costs €0.75 per kg and another that costs €1.15 per kg,

and we obtain 50 kg of a mix which costs €1 per kg. How much of each type of flour is used

in the mix? [Sol.: 18.75kg of the cheaper flour and 21.35kg. of the more expensive flour]

14. A producer’s union mixes two types of coffee, one that costs €5 per kg and another that

costs €7.5 per kg. They obtain 30kg of a mix which costs €6 per kg. How much coffee of

each type does the mix contain?

15. I want to mix two types of wine that costs €5 per litre and €3 per litre to obtain 25 litres of

a mix that will cost €3.80 per litre. How many litres of each type of wine do I need to mix?

16. The freshman (estudiante de primer año) and sophomore (estudiante de segundo año)

classes at Norwood High School are decorating floats

(carrozas) for homecoming (fiesta de vuelta a casa).

The freshmen have already spent £90 on their float, plus

they need to buy floral sheeting that costs £66 per roll. The

sophomores, who have spent £61 so far on theirs, still need

to purchase vinyl grass at £95 per roll.

Both classes plan to buy the same number of rolls, since they have the same area to cover.

By coincidence, the two floats will have the same total cost in the end. How many rolls will

each class be buying? How much will each class spend in total? [Sol: 1 roll, £156]

17. The admission fee (entrada) at a small fair (feria) is 1.50€ for children and 4.00€ for adults.

On a certain day, 2200 people enter the fair and 5050€ is collected. How many children and

how many adults attended? [Sol: 1500 children and 700 adults attended the fair.]

18. A boat travelled 210 miles downstream (río abajo) and back. The trip downstream took 10

hours. The trip back took 70 hours. What is the speed of the boat in still water (si el agua

estuviese quieta)? What is the speed of the current? (Measure them in miles per hour mph).

(*) In order to set the problem up, think if you have to add or subtract the speed of the boat and the

speed of the current. [Sol: boat: 12 mph, current: 9 mph.]

19. A landscaping company placed two orders with a nursery (vivero). The first order was for

13 bushes (arbustos) and 4 trees, and totalled $487. The second order was for 6 bushes

and 2 trees, and totalled $232. What were the costs of one bush and of one tree? (They do

not know it because the bills do not list the per-item price). [Sol: bush $23; tree: $ 47.]

56

20. Flying to Kampala with a tailwind (volando con viento de cola) a plane averaged 158 km/h.

On the return trip the plane only averaged 112 km/h while flying back into the same wind.

Find the speed of the wind and the speed of the plane in still air.

(*) In order to set the problem up, think if you have to add or subtract the speed of the plane and

the speed of the wind. [Sol. Plane: 135 km/h, Wind: 23 km/h.]

21. Matt and Ming are selling fruit for a school fundraiser. Customers can buy small boxes of

oranges and large boxes of oranges. Matt sold 3 small boxes of oranges and 14 large boxes

of oranges for a total of $203. Ming sold 11 small boxes of oranges and 11 large boxes of

oranges for a total of $220. Find the cost each of one small box of oranges and one large

box of oranges. [Sol: small box of oranges: $7, large box of oranges: $13.]

22. 3 kilos of coffee, which costs €16 per kg, are mixed with 5 kilos of a different type of coffee,

which costs €12 per kg. How much does each kg of the mixture cost?

23. 400 litres of wine costing €9 per litre is watered down with 50 litres of water. How much

does each litre of the watered down wine cost?

24. A passenger jet took three hours to fly 1800 miles in the direction of the jetstream

(corriente de aire). The return trip against the jetstream took four hours. What was the

jet's speed in still air (con el aire quieto) and the jetstream's speed? (Write the answer in

miles per hour mph). [Sol: The jet’s speed in still air was 525mph and the jetstream’s spead 75mph.]

(*) In order to set the problem up, think if you have to add or subtract the speed of the plane and

the speed of the wind before multiplying by the number of hours.

25. Find two positive numbers whose sum is 20 and their product is 96. [Sol: 12 and 8].

26. Last Wednesday, two friends met up after school to read the book they were both assigned

in Literature class. Tony can read 1 page per minute, and he had already read 60 pages.

Belle, who has a reading speed of 2 pages per minute, had read 20 pages. Eventually they

had read the same number of pages. How many pages had each of them read?

27. I spent 16€ on potatoes yesterday. I only remember that the price per kg (which was less

than the kg.) plus the number of kilograms was 10. How many kg did I buy? What was the

price per kg? [Sol: 8kg, and they cost 2€ per kg].

28. The side of my house is 6m largest than the side of my swimming-pool. Their areas sum

90m2. If both are squared, find the measure of their sides. [Sol: House: 9m, Pool: 3m].

57


Bilingual section

EQUATION SYSTEMS.

1. (1 pt.) Fill so the solution to the system is x=3, y=2.

2 (1 pt.) Solve using the graphing method

3 (2 pts.) Solve using the elimination method

4 (2 pts.) Solve

5. (2 pts.) Solve using the substitution method

6 (2 pts.) Pedro and Juan are making presentations for a class project. Pedro's slideshow starts

with a verbal introduction that is 19 seconds long, and then each slide (diapositiva) is left up for 5 seconds. Juan leaves each slide onscreen for 10 seconds, and his introduction lasts 9 seconds. Pedro and Juan notice that their presentations have both the same number of slides and the same duration.

- How many slides are in each presentation?

- How long is each presentation?

(Solve the problem using the substitution method).

59

RECTA, PARÁBOLA E HIPÉRBOLA

FUNCIÓN AFÍN (y=mx+n). Es una recta; m es la pendiente y n la ordenada en el origen.

Pendiente (m): mide la inclinación de la recta. m>0: es creciente; m<0 es decreciente.

- Es el cociente de cuánto se sube (o baja) entre lo que se avanza al pasar de un punto a otro.

Para pasar de ángulo a número se usa la tangente (tan). Al revés, con tan-1.

Ej. Para una recta de inclinación de 45º, m=tan(45º)=1. Por eso tan-1(1)=45º.

FUNCIÓN CUADRÁTICA (y=ax2+bx+c). Es una parábola

- Forma: Es cóncava () si a>0, y convexa () si a<0.

- Corte con eje x (y=0): resolvemos

- Corte con eje y (x=0): punto (0,c).

- Eje de simetría: ecuación

. Corta a la parábola en el vértice.

Cuando avanzamos una unidad desde el vértice, a mide cuánto subimos (o bajamos).

FUNCIÓN DE PROPORCIONALIDAD INVERSA ( y=

). Es una hipérbola equilátera (y =

).

- Asíntotas: horizontal y=b. Vertical x=a.

Son ejes de simetría.

- Centro (de simetría): punto (a,b).

Es la intersección de las asíntotas;

k es el área del rectángulo formado por el centro y un punto cualquiera de la hipérbola.

Ecuaciones de la recta: Ej. m=2 y pasa por P(1,3)

Explícita: y=mx+n y=2x+1

General: ax+by=c -2x+y=1

Implícita: ax+by+c=0 2x-y+1=0

Punto-pendiente: y-b=m(x-a) y-3=2(x-1)

60

STRAIGHT LINE, PARABOLA AND HYPERBOLA

AFFINE FUNCTION (y=mx+n). Straight line: m is the slope and n the ordinate at the origin.

Slope (m): measures the inclination of the line. m>0: it is increasing; m<0 it is decreasing.

- It is the quotient of how much it goes up (or down) by how much it moves forward when

going from one point to another.

To compute the slope corresponding to an angle we use the tangent (tan). And round, we

use tan-1.

E.g. For a line having a 45º inclination, m=tan(45º)=1. Thus, tan-1(1)=45º.

QUADRATIC FUNCTION (y=ax2+bx+c). It is a parabola

- Shape: concave () when a>0, and convex () when a<0.

- x-intercept (y=0): we solve

- y-intercept (x=0): it is the point (0,c).

- Symmetry axis: equation

. It meets the parabola at the vertex.

When we go one unit forward from the vertex, a measures how much we go up (or down).

INVERSE PROPORTION FUNCTION ( y=

). It is a rectangular hyperbola (y =

).

- Asymptotes: horizontal y=b. Vertical x=a.

They are symmetry axes.

- Centre (of symmetry): it is the point (a,b).

It is where the asymptotes meet.

k is the area of the rectangle formed by the centre and any point in the hyperbola.

Equations of a line: E.g. m=2 and passes through P(1,3)

Explicit: y=mx+n y=2x+1

General: ax+by=c -2x+y=1

Implicit: ax+by+c=0 2x-y+1=0

Point-slope: y-b=m(x-a) y-3=2(x-1)

61

ELEMENTARY FUNCTIONS - WORD PROBLEMS

[*] For problems 1 to 10, write angles using degrees, minutes and seconds, and draw a picture for the setup.

1. A ladder is resting against a wall with an angle of 60o. Write an equation for the ladder

(suppose the bottom is at P(0,0)). The top of the ladder touches the wall at a height of

2.5m. Find the distance from the bottom of the ladder to the wall.

2. A bird is flying up in the sky with an angle of 30o. Now, it is in the point (3, 5). Find the

equation of its flight.

3. At 12:00pm, a plane is taking off (despegando), making an angle of 60º with the ground.

Write the equation of its movement (suppose the point where it takes off is P(0,0)).

When the shadow it at 500m, how high is it flying?

When the plane is flying 1200m high, how far is its shadow?

4. At 12:00pm, we are flying a kite with an angle of 80o32’15’’. If the shadow is at 0.5m away

from us, how high is it flying? When the kite is flying at 4m, how far is the shadow?

5. On a shooting range (campo de tiro), Luis shoots his airgun from point P(-2,0) with an angle

of 40º, and Angel from point Q(3,0) with an angle of 50o11’40’’. If the target (el blanco) is at

point T(14’67 , 14), does any of them hit the target?

6. On a road, we find a signal indicating that the slope is 10% (i.e., m=0.10).

What is the angle of the hill? When we have climbed 10 metres, how far

have we reached?

7. A radar detects two ships on the sea. The first one follows the

straight line y=0.84x+2, and the second one, the line y=0.364x-

3. What is the angle between them? [Sol: 20º1’44’’]

8. Fernanda is watching planes fly on the sky. She

notices that one of them follows the trajectory

y=3.73x+1, and another one y=-5.67x+5. Find the

angle between both trajectories. [Sol: 155o]

9. In a map, there is a road to Mérida following the equation y=2x-1, and another one,

following the equation y=0.1x+0.7. What is the angle between them?

10. We are writing equations for some kites in the sky. For the first one we have y=11x+2, and

for the second one, y= 14x-5. What is the angle between both?

62

11. Ana has been analyzing the light bill for some months. She has found that it follows the

equation y=10x2-60x+130, from the first month to the 6th month.

a) When did she spend the most? And the least? How much were each?

b) When did she spend 50€?

12. The quadratic equation p=2r2-8r+48 models the gross profit made by a shop that sells r

ovens. It is only valid from 1 to 5 ovens. When is the profit greater? And smaller?

13. In a restaurant, they are using a model to predict how many clients they will have, from

1p.m. to 8p.m. The equation is y=-x2+8x+12.

a) When are they expecting to have the most clients? How many will there be?

b) And the least clients? How many?

14. Analysts in a company have predicted that the shares (acciones) price in the next 20 days

can be predicted using the equation y=2x2-20x+1050.

a) When will the price be highest? And lowest? How much will it be?

b) When will the price be 1008€?

c) Would you advice (aconsejar) to wait one more month to someone who wants to buy

shares on the 3rd month? And to someone who wants to buy on the 7th month? Why?

15. We are using the quadratic function y=-3x2+42x-72 to predict the number of telephone

calls that will be received in certain call centre, from 2 pm to 10pm.

a) When will they have the most calls? And the least? How many will there be?

b) When will there be 63 calls? And 27 calls?

c) When is the number of calls expected to increase; at 5pm or at 9pm?

16. When blowing fireworks, Natalia discovers that one of them follows the parabola

y= -t2+24t+2 (in feet), from the instant t=0 to t=10, when it explodes. [*] 1 foot=30.48cm

a) What was the highest point of its trajectory? And the lowest?

b) How high was it on the instant t=3? When was it 130 feet high?

17. At night, the timer (programador) turns the heating off at 1:00. Then, the temperature

(in ºC) falls, until the timer turns the heating back on. Suppose the temperature follows the

equation y=x2-10x+35, from 1:00am to 8:00am.

a) When did the timer turn the heating back on? What was the temperature then?

b) When was the temperature equal to 26ºC?

c) What was the temperature at 1:30am?

63

18. A ramp has a 30º slope and passes through A(1,1). Write its equation. [y=0.58x+0.42]

19. Find the equation and the angle of the lines passing through “A(-2,1) ; B(2,3)” and “A(-2,1)

; C(-1,4)”. What’s the angle between them? [y=0.5x+2; 26o33’54’’ and y=3x+7; 71

o33’54’’. 45

o]

20. Find the equation of the line passing through C(2,1) having a 45º slope. [y=x-1]

21. A baseball is hit from home base with an upward speed of 80feet per second. The gain in

height of the ball from the point it was struck is modelled by the equation h=-16t2+80t,

where t is the time in seconds.

a) How far has the ball risen after 2 seconds? [Sol: 96 feet]

b) After how many seconds is the ball 64 feet above its start height= [1 and 4 seconds]

c) The ball is caught at the same height that it was struck (h=0). After how many seconds is

the ball caught? [t=5 seconds]

22. Frank opened a savings account with $500 last year. He believes that the amount of

money in the savings account (S) can be modeled by the formula S=-2t2+60t+500, where

t is the number of years, from 1 to 20.

a) What is the maximum amount? [After 15 years, $950]

b) And the minimum? [The first year, $558]

23. The amount of money, in tens of thousands of dollars, deposited at the local bank in the

last 10 years, can be modeled by the formula y=-7x2+280x+12000, where x represents

the number of the year (x from 1 to 10). [Hint: be careful: x=20 is not allowed!]

a) In what years will the maximum deposits be reached? [Last year (x=10). $14,10010,000]

b) And the minimum? [Ten years ago (x=1). $12,27310,000]

24. Mrs Franklin uses the formula A=-10m2+60m+300 to predict how the amount an

average-customer (cliente medio) spends at her store will change over time, where A is

the amount in $, and m (up to 9 months) is the number of months after the customer’s

initial purchase.

a) What is the maximum the average customer will spend according to this model and

after how many months will it occur? [3 months. $390]

b) The model predicts that the amount the average-customer spends will eventually fall to

$140.00. After how many months is this predict to happen? [8 months]

c) What is the minimum the average-customer will spend? [9 months. ·$30]

64

FINDING A QUADRATIC EQUATION – WORD PROBLEMS

25. Pepa has been tracking the number of pages she reads for 12 days (days 1st to 12th ). She

read the least on the seventh day (110 pages). The next day she read 113.

a) Write a quadratic equation to represent the amount of pages she reads every day.

[Sol: y=3(x-7)2+110=3x2-42x+257]

b) On what day did she read the most pages? [Sol: 218p] How many? [Sol: The 1st

day]

c) On what days did she read 122 pages? [Sol: On days 5th and 9th]

26. Manolo is carefully tracking the number of gas cylinders (bombonas) he is buying for his

restaurant. He finds that the minimum amount of gas cylinders was 1, on the 2nd month.

The next month he bought 2. Also, from months 1 to 4, they follow a quadratic formula.

a) Write the formula representing this situation. [Sol: y=1·(x-2)2+1]

b) How many gas cylinders did he buy on the fourth month? [Sol: 17 cylinders]

27. We have found that the amount of kg. harvested (cosechados) on a field can be expressed

using a quadratic formula, from days 1 to 16th. The day they harvested the most was the

6th, with 500kg. The next day, they harvested 496kg.

a) Write the formula representing this situation. [y=-4(x-6)2+500]

b) When did they harvest the least? [Sol: The 16th day; 100kgs]

c) When did they harvest 436kgs? [Sol: On days 2nd and 10th.]

28. We are studying the relationship between net income (beneficios netos) and the extra

working hours in a company. We think that the most net incomes will be €320, working 2

extra-hours, whereas (mientras que) working 3 extra hours will be only €240.

a) Write a quadratic equation to calculate the net income. [Sol: y=-80(x-2)2+320]

b) What is the net income for no extra hours? [Sol: No incomes (0€)]

c) Is there any moment where extra-hours will mean to lose money? [Sol: For more than 4h]

29. The city council (ayuntamiento) has ordered a sculpture. It will measure between 20 and 30

dm high. The price will depend on its size (raw material pieces, working hours, etc.). The

sculptor will set the cheapest price at 2,000€, for a 23dm sculpture. Also, a 24 dm sculpture

would cost 2,100€. Lastly, he is going to set the prices using a quadratic function.

a) Write the price as a quadratic function of the size. [Sol: y=100(x-23)2+2000]

b) What is the highest price? [Sol: 6,900€ for a 30dm high sculpture]

c) How much would a 25dm sculpture cost? [Sol: 2,400€]

d) What sculpture would cost 4500€? [Sol: Both 18m and 28m high sculptures]

65

ELEMENTARY FUNCTIONS – REVISION EXERCISES

Analyze the following functions. Make a chart with the information and plot their graphs.

1) y=2x2-8x+6

2) y=-x2+2x+15

3) y=x2+8x+12

4) y=-3x2-6x+9

5)

6)

[1] [2] [3]

[4] [5] [6]

SHAPE

Symmetry

Vertex (Max./Minimum.)

Intercepts y-intercept

x-intercepts

Monotony increasing

decreasing

-3 -2 -1 0 1 2 3 4

66

Analyze the following functions..

1) y = 2

2) y = 1

3) y =

- 1 4) y =

5)

6) y =

[1] [2]

[3] [4]

[5] [6]

SHAPE

Symmetry centre

Asymptotes horizontal

vertical

Intervals

Monotony increasing

decreasing

Curvature concave ()

convex ()

67


Bilingual section

STRAIGHT LINE, PARABOLA AND HYPERBOLA.

1. (3 pts.) Fill the chart (you may round the results of tan and tan-1 to tenths)

2 (4 pts.) Analyze: (a) y = x2 - 4x - 5, (b)

. (PLEASE TURN THE PAGE OVER)

3 (1 pt.) On a map, a hill has a 5º42’38’’ inclination, and passes through point (5, 2.5). Find the equation of the hill.

[En un plano, una cuesta tiene una inclinación de 5o42’38’’ y pasa por el punto (5, 2’5). Escribe la ecuación de la cuesta.]

4 (2 pts) The daily revenue, in thousands of dollars, for a car manufacturer is modeled by the equation y=-3x2+60x+1060, where x is the price of each car in thousands of dollars; for prices between $3000 (x=3) and $35000 (x=35).

[ Los ingresos diarios de un fabricante de coches, en miles de dólares, son y=-3x2+60x+1060, donde x es el precio del coche en miles de dólares; para precios entre 3000 (x=3) y 35000 (x=35) ]

a) At what price must the cars be sold to receive maximum revenue (and how much is it)? (¿A qué precio hay que vender los coches para obtener el mayor beneficio (y cuánto es)?)

b) And the lowest? Would they lose money? How much? (¿Y el menor? ¿Perderían dinero? ¿cuánto?)

c) What should be the price of cars manufactured to have $685000 revenue (y=685)? (¿Cuál debería ser el precio de los coches fabricados para obtener un beneficio de $685000 (y=685)?)

Passes through Slope Equation Angle

(2 , 3) and (5 , 18) º ‘ ‘’

(-1 , 8) and ( 5 , -4) º ‘ ‘’

(1 , 10) and (0, ) y = 5.7x+4.3 º ‘ ‘’

(0 , 5) 29º

68

2 (4 pts.) Analyze the following functions. Plot the graph along with its main elements.

Table of values:

EQUATION y = x2 - 4x - 5

SHAPE

Symmetry

Vertex (Max./Minimum.)

Intercepts y-intercept

x-intercepts

Monotony increasing

decreasing

EQUATION

SHAPE

Symmetry centre

Asymptotes horizontal

vertical

Intervals

Monotony increasing

decreasing

Curvature concave ()

convex ()

x y

3

4

5

69

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

FUNCIONES

FUNCIONES

- Una función es una relación entre dos conjuntos (inicial y final), de manera que a cada

elemento del primero le corresponde uno solo del segundo.

- Variables: independiente (x): conjunto inicial. Dependiente (y): conjunto final.

Formas de expresar una función:

- Lenguaje natural: una frase que describe la relación. Ejemplo: “el doble de cada número”.

- Tabla de valores: en una fila (o columna) colocamos la variable independiente, y en otra la

dependiente. Ejemplo:

- Una gráfica: en el eje de abscisas colocamos la variable

independiente (x) y en el de ordenadas la dependiente (y).

Ejemplo:

- Una fórmula: una ecuación describe cómo están relacionadas las variables. Substituyendo el

valor de “x” en la fórmula, se obtiene el de “y”. Ejemplo: y 2x. Para x=3 resulta y=2·3=6.

Características de una función:

- Dominio: valores para los que está definida.

- Continua: puede dibujarse de un solo trazo.

- Discontinua: si la gráfica da algún salto.

- Puntos de corte con los ejes: los que están sobre los ejes de coordenadas.

- Creciente: cuando al aumentar “x”, aumenta “y”.

- Decreciente: cuando al aumentar “x”, disminuye “y”.

- Curvatura: si se curva hacia abajo es cóncava (). Si se curva hacia arriba, es convexa () .

Punto de inflexión: punto de la gráfica donde cambia el tipo de curvatura.

- Máximo: punto de la gráfica donde la función toma el valor más alto de su entorno.

- Mínimo: punto de la gráfica donde la función toma el valor más bajo de su entorno.

- Periódica: cuando la gráfica se repite a intervalos iguales. Periodo: la amplitud del intervalo.

- Simétrica: puede ser respecto un eje (axial) o respecto un punto (central).

- Asíntota: recta (horizontal, vertical u oblicua) hacia la que tiende la gráfica “hacia el infinito”.

[*] Para fracciones algebraicas, pueden calcularse efectuando la división.

x 0 1 2 3 4 5

y 0 2 4 6 8 10

70

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

FUNCTIONS

FUNCTIONS

- A function is a relationship between two sets (initial/input and final/output), such that each

input is associated to one and only one output.

- Independent variable (x): the input to a function. - Dependent variable (y): the output.

Ways to give out a function:

- Natural language: a sentence describing the relationship. Example: “the double of each

number”.

- Table of values: we put the independent variable in a row (or column) and the dependent in

another. Example:

- A graph: we put the independent variable (x) in the

abscissa axis and the dependent (y) in the ordinate.

Example:

- A formula: and equation describing how the variables are related. By substituting the value

of “x” in the formula, one gets “y” value. Example: y 2x. For x=3 results y=2·3=6.

Characteristics of a function:

- Domain: values for which it is defined.

- Continuous: it can be drawn without lifting the pencil from the paper (no gaps).

- Discontinuous: its graph jumps somewhere (there are gaps).

- X and Y Intercepts: the points where the graph meets the axis (X and Y).

- Increasing: the more value of “x”, the more value of “y”.

- Decreasing: the more value of “x”, the less value of “y”.

- Curvature: if it is bent downwards, it is concave (). If it is bent upwards, it is convex ().

Inflexion point: point of the graph where the type of curvature changes.

- Maximum: point where the function takes the largest value within a neighbourhood.

- Minimum: point where the function takes the smallest value within a neighbourhood.

- Periodic: when the graph is repeated at equal intervals. Period: the length of the interval.

- Symmetric: can be about a line (axial) or about a point (central).

- Asymptote: line (horizontal, vertical or oblique) the graph approaches to “at the infinity”.

[*] For algebraic fractions, they can be computed by performing the division.

x 0 1 2 3 4 5

y 0 2 4 6 8 10

71

Simetría respecto un punto / symmetry about a point.

Un polinomio es simétrico respecto al origen si su expresión es del tipo p(x)=c5x5+c3x3+c1x.

Para averiguar si es simétrico respecto algún punto (a,b), hacemos una traslación que lleve

(a,b) al origen, y después comprobamos si entonces es simétrico respecto al origen.

[*] En particular, todos los polinomios de grado 3 son simétricos respecto algún punto.

Cálculo del centro de simetría (en general):

Obtenemos p(x+a) y Ordenamos el resultado por las potencias de x.

Vemos si para algún valor de “a”, los coeficientes de grado par se anulan.

(El término independiente no es necesario).

Ejemplos: (Cálculos con el programa wiris).

72

Simetría respecto un eje / symmetry about a line (axis).

Un polinomio es simétrico respecto el eje “x 0” si su expresión es del tipo

p(x)=c6x6+c4x4+c2x2+c0. Para averiguar si es simétrico respecto otro eje x=a, hacemos una

traslación que lo lleve al origen, y después comprobamos si entonces es simétrico respecto el

eje “x 0”.

[*] En particular, los polinomios de grado 2 (parábolas) son simétricos respecto un eje.

Cálculo del centro de simetría (en general):

Obtenemos p(x+a) y Ordenamos el resultado por las potencias de x.

Vemos si para algún valor de “a”, los coeficientes de grado impar se anulan.

Ejemplos: (Cálculos con el programa wiris).

73

FUNCTIONS – REVISION EXERCISES

For the following functions, use the computer (“Wiris”, “Geogebra” or “Graph” programs, for

e.g.) to plot them. Use the plot to describe the function by eye (to eyeball the description).

1. Compute the asymptotes of the following algebraic fractions:

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

2. Use the computer (Wiris program for e.g.) to find out if the following functions have any

symmetry centre and to write their reduced equation.

a) -5x3+15x2-11x+3 b) 2x3+12x2+28x+25 c) 2x3+12x2+28x+25 d) x5-5x4+7x3-x2-x-1

e) x5-7x4+12x3 f) x5-10x4+34x3-44x2+16x+1. g) 2x3-6x2-x+4 h) –x5+5x4-3x3-11x2+9x+1

3. Use the computer (Wiris program for e.g.) to find out if the following functions are symmetric

about any line, and to plot the function.

a) 2x4-8x3+7x2+2x b) x6-6x5+11x4-4x3-6x2+4x+2. c) 3x4-14x3+15x2 d) x4-8x3+20x2-16x+1

e) -x4-4x3+8x+4 f) –x6+6x5-11x4+4x3+4x2+1 g) x4+12x3+50x2+84x+44 h) 3x6-7x4+3x2+2.

Solutions:

1. Asymptotes:

[a] x=2, x=-2, y=x+1. [b] x=-3, x=3, y=1. [c] x=1, x=-2, y=2x-1 [d] x=3, x=2, y=x-4

[e] x=-1, x=1, y=-2. [f] x=-5, x=3, y=3 [g] x=-2, x=3, y=1 [h] x=-2, x=1, y=-x+3.

[i] x=0, x=-2, x=4, y=2x-1 [j] x=1, x=-3, x=-1, y=x+2 [k] x=2, x=1, x=-3, y=-x+3 [l] x=-5, x=2, x=3, y=3x

2. [a] C(1,2). P=-5(x-1)3+4(x-1)+2 [b] C(-2,1). P=2(x+2)

3+4(x+2)+1 [c] C(1,-1). P=2(x-1)

3-3(x-1)-1.

[d] C(1,0). P=(x-1)5-3(x-1)3+3(x-1). [e] No symmetry. [f] C(2,1). P=(x-2)5-6(x-2)3+8(x-2)+1

[g] C(1,-1). P=2(x-1)3-7(x-1)-1 [h] C(1,0). P=-(x-1)5+7 (x-1)3-7(x-1).

3. [a] x=1; 2(x-1)4-5(x-1)

2+3 [b] x=1; (x-1)

6-4(x-1)

4+3(x-1)

2+2 [c] No symmetry.

[d] x=2; (x-2)4-4(x-2)2+1 [e]x=-1; -(x+1)4+6(x+1)2-1. [f] x=1; -(x-1)6+4(x-1)4-5(x-1)2+3.

[g] x=-3; (x+3)4-4(x+3)2-1 [h]x=0; 3x6-7x4+3x2+2.

74

4. Plot a function having the following characteristics:

.

5. Describe the following functions:

a) Maximum: (-1,4). Inflexion (2,2). Asymptotes: x=-6, y=-1, y=1 Negative: (- ,-4). Positive (-4,+ )

Increasing: (-6,-1). Decreasing: (- ,-6)(-1,+ ).

Convex: (- ,-6)(-6,2) . Concave: (2,+ ).

b) Minimum: (1,2). Maximum: (3, -2) Asymptotes: x=0, x=2, x=4. Positive: (0,2). Negative: (2,4)

Decreasing: (0,1)(3,4). Increasing: (1,2)(2,3). Concave: (0,2). Convex: (2,4). Periodic. Period=4.

d) Domain: (- ,6). Maximum (2,5). Minimum (4,1). Inflexion: origin (0,0) and (3,3). Asymptote: x=6. Negative: (- ,0). Positive (0,6).

Increasing: (- ,2)(4,6). Decreasing: (2,4).

Concave: (- ,0)(3,6). Convex: (0,3).

c) Discontinuities x=-4 and x=4. Minimum (0,-4). Asymptotes: x=-4, x=4.

Negative: (0,2'5)(4,5). Positive: (2'5,4)(5,+ )

Increasing: (0,4)(4,+ ). Concave: (0,4). Convex: (4,+ ). “Even”: Symmetric about the line x=0.

a) b)

INTERVALS (a) and (b)

Domain

Signs Positive

Negative

Monotony Increasing

Decreasing

Curvature Convex

Concave

POINTS (a) and (b)

Discontinuity

Maxima

Minima

Inflexion

Intercepts

SYMMETRY:

ASYMPTOTES: No asymptotes.

75


Bilingual section

FUNCTIONS.

1. (1 pt.) Say, if the following graphs correspond to any function. Justify your answer.

a) b)

2. (0.5 pts.) Compute the domain of the following function:

3. (4.5 pts.) Describe this function “by eye”, considering its graph. Compute the equation of its

asymptotes and plot them.

x=Pupil Pedro Ana Manuel María

f(x)=Mark 7 8 5 4

(1.25 pts.) INTERVALS

Domain

Signs Positive

Negative

Monotony Increasing

Decreasing

Curvature Convex

Concave

(1 pt.) POINTS

Discontinuity

Maxima

Minima

Inflexion

Intercepts

(2 pts) ASYMPTOTES

Type Equation

(0.25 pts) SYMMETRY:

-40 -20 0 20 40 -40 -20 0 20 40

76

4. (1 pt). Consider the following wiris code. Say, if the function y=x4-8x3+16x2-16 has any symmetry (explain why). If so, write the reduced equation:

Answer:

5. (3 pts.) Plot a function satisfying the following conditions:

- Maximum: (2,5). Minimum (6,-5). Inflexion: (4,0).

- x-Intercepts: (0,0), (4,0), (9,0).

- Positive: (0,4) (9, ). Negative (4,9).

- Increasing in (0,2)U(6, ) and decreasing in (2,6).

- Convex in (0,4) and concave in (4, ).

- It is “even” (symmetric about the origin).

77

ESTADÍSTICA

- Estadística: la ciencia que se ocupa de recoger, resumir y analizar datos. También de sacar

conclusiones, hacer predicciones y tomar decisiones.

- Población: los elementos de los que queremos obtener información.

- Muestra: parte de la población que usamos en el estudio. Debe elegirse aleatoriamente.

Muestreo: puede ser aleatorio, sistemático (elegir uno cada k) o estratificado (agrupando).

En el estratificado, al elegir el tamaño de la muestra (afijación), puede tomarse

igual en cada estrato, o proporcional a su tamaño.

Muestra sesgada: Si algún elemento tiene más probabilidad de ser elegido.

- Variable/carácter estadístico: característica de la población en la que estamos interesados.

Cuantitativas: se miden con números.

Discreta: puede tomar pocos valores. Ejemplo: “nota de la 1ª evaluación”.

Continua: puede tomar muchos valores. Ejemplo: “altura”.

Cualitativas: medimos cualidades (y no números). Ejemplo: “color favorito”.

RESÚMENES DE DATOS

Tabla de Frecuencias

- Frecuencia absoluta: número de veces que se repite el dato.

- Frecuencia relativa: indica qué parte del total es ese dato. Podemos medirlo en porcentaje.

- Frecuencia acumulada: el número de datos que hay menores o iguales que él.

Si hay muchos datos puede convenir agruparlos por intervalos, indicando la marca de clase.

Gráficos Estadísticos

- Diagrama de barras: en el eje horizontal se coloca la variable estadística y en el vertical la

frecuencia. Si los datos están agrupados, usamos un histograma (las barras se dibujan juntas).

- Polígono de frecuencias: unimos las barras del gráfico con segmentos.

- Diagrama de sectores: dividimos un círculo en partes proporcionales a la frecuencia.

- Caja y bigotes: ponemos los valores mínimo, máximo y cuartiles en la recta numérica.

- Diagrama de tallo-hojas: ordenamos los datos en una tabla. El tallo es la primera parte del

número y la hoja la última cifra. Hay una hoja por cada dato (y puede haber hojas iguales).

- Diagrama de dispersión: se comparan los valores de dos variables, en un plano cartesiano.

Línea de tendencia: recta que describe “aproximadamente” la relación entre dos variables.

Puede usarse para hacer predicciones a partir del diagrama de dispersión.

78

Medidas de Centralización

- Media ( ): suma de los datos dividida entre el número de datos.

Media ponderada: cuando no todos los datos tienen la misma importancia, los

multiplicamos por unos “pesos”, antes de hacer la media (i.e. ponderamos los datos).

Truncada/recortada (y winsorizada): eliminando (o sustituyendo) un porcentaje de datos

en los extremos. Suele escogerse entre el 5% y 25% de los datos.

[*] Media Geométrica: se multiplican los datos y se hace la raíz. Suele usarse al resumir

magnitudes diferentes sin que influyan los rangos. Ej: para 2, 12, 9; G=

=6.

- Moda: el dato que tenga la mayor frecuencia (puede haber varias modas).

- Mediana: tras ordenar los datos, el que queda justo en la mitad.

Medidas de de posición

Cuartiles: son los que quedan en el 25%, 50% (la mitad), y el 75% de los datos.

Percentil (p): el dato tal que % de la población es menor que él.

Ej. Para el percentil p10, el 10% de los datos son menores que él, y el 90% mayores.

Parámetros de dispersión

- Desviación media absoluta: promedio de (los valores absolutos de) la diferencia de cada

dato con la media.

- Desviación típica (): medida de cómo de separados están los datos entre sí:

Para calcularlo usamos la Varianza (2): “media de cuadrados” – “cuadrado de la media”.

- Coeficiente de variación (CV): mide la dispersión en porcentaje, comparando con la media.

CV=

. (*) Si es mayor del 30%, la dispersión es grande.

79

STATISTICS

- Statistics is the science that gathers, summarizes and analyses data. It's also dedicated to

find conclusions, make predictions and make decisions.

- Population: is a collection of elements whose properties are analyzed.

- Sample: is a part of the population of interest. It must be chosen at random.

Sampling: it can be random, systematic (pick every kth element) or stratified (grouping first).

In stratified sampling, when chosing the sample’s size (allocation), it can be

equal on each stratum, or proportional to their sizes.

Biased sample: some people or items are more likely to be chosen than others.

- Variable/characteristic: feature of the population we are interested in. It can be

Quantitative: anything that can be expressed as a number.

Discrete: it can only take a few values. Example: “mark in the 1st term”.

Continuous: it can only take many values. Example: “height”.

Qualitative: when we measure qualities (and not numbers). Example: “favourite colour”.

DATA SUMMARIES

Frequency tables/charts

- Absolute frequency: the number of times the data value occurs.

- Relative frequency: what part of the total is that data. It can be measured in percentage.

- Cumulative frequency: the number of data which are less or equal than a given data.

If the variable is continuous, we group data into intervals, indicating the class mark.

Statistic Graphs

- Bar graph: we put the statistic variable on the horizontal axis and the frequency on the

vertical. If data is organized into groups, we use a histogram (bars are drawn together).

- Frequency polygon: join the bars by line segments.

- Circle graph (pie chart): a circular chart divided into sectors, proportional to the frequency.

- Box and whisker: we place the least data, the greatest data and quartiles on the number line.

- Stem-and-leaf plot: the stem tells the first part of the number and the leaf tells the last digit

of the number. There is one leaf for each datum (and leaves can be repeated).

- Scatter plot: it is used to compare the values of two variables in a Cartesian plane.

Trend line roughly describes the relationship between two variables in a set of data. You can

use a trend line to make predictions from a scatter plot.

80

Central Tendency measures

- Mean ( ): the sum of the numbers divided by the number of data.

Weighted mean: when not all data are equally important, we multiply them by some

“weights” before computing the mean (i.e. by weighting the data).

Trimmed/Truncated (and winsorized) mean: discarding (or replacing) parts of the data at

the high and low ends. Often 5 to 25 percent of the ends are replaced.

[*] Geometric Mean: is the root of their product. It is often used when comparing different

magnitudes without the influence of their ranges. E.g.: for 2, 12, 9; G=

=6.

- Mode: the number that appears most often. (There can be several modes.)

- Median: when the numbers are arranged from least to greatest, it is in the middle.

Position measures

Quartiles split the data into four sections. Each section contains 25% of the data.

Percentile (centile): the value of a variable below which a certain percent of observations fall.

E.g. for the 10th percentile, a 10% of the data are less than p10, and a 90% are greater.

Dispersion parameters

- Mean absolute deviation (MAD): Average of (the absolute values of) the difference between

the data set and the mean.

- Standard deviation (): measure of how far a set of numbers are spread out from each other

To compute it we use: Variance (2): “mean of squares” – “mean squared”.

- Coefficient of variation (CV): measures the dispersion in percentage, related to the mean.

CV=

. (*) If it is greater than 30%, the dispersion is big.

81

STATISTICS - WORD PROBLEMS

1. Describe the following statistics and say if the samples are biased.

a) A sample of 1000 people are telephoned on a Saturday to find out how people at Mérida

spend their leisure time. It’s found people spend time watching TV.

b) Isaac would like to know if people prefer comic books or novels. He waits outside of a

bookstore and asks 20 people exiting whether they purchased comics or novels.

c) A town wants to find out if people are happy with a proposal to tear down (desmantelar)

a section of a park and replace it with a parking lot. To learn about the citizens’ opinion,

the town council sends a person to the location in the park to ask all passersby whether

they would like to see a parking lot built at the location.

d) 3000 girls were surveyed for a study to determine the number of hours-of-television

teens watch per day.

e) To gather opinion about a concert, the first 25 people leaving the hall were interviewed.

f) An experiment into how children react to a certain vitamin supplement was conducted

using 5000 boys from all over Europe.

g) A study is carried out to discover whether residents of a .town think a road should be

widened. A questionnaire was mailed to every household (hogar) in the town. Only one

tenth of the residents replied, but a 95% of the replies were against the road widening.

2. Describe the following surveys

a) In a given neighbourhood (vecindario), the owners of 50 dogs were asked about the

height and breed (raza) of their pets, and the number of times they walked them each

week. We picked neighbours from each street, proportionally to its population.

b) Of all the people who are members at a sports centre, 30 people are asked how much

time they spend exercising.

c) Of all the people entering in an airport, two of every 15 people are asked which

destination they have chosen for their holidays.

d) Twenty of a school’s students are chosen at random and are asked about the number

of times they go to the cinema in a month. We picked 2 from each of the 10 classes.

e) We ask one out of every 15 customers on a video shop about the type of film he has

hired (alquilar).

Sol: a) Pop: the dogs in the neighbourhood. Sample: 50 dogs. Var: nº. of times (discrete), height (continuous),

breed (qualitative). Stratified sampling (proportional allocation).

b) Pop: members of the sport centre. Sample: 30 people. Var: time spent exercising (continuous). Random sampling.

c) Pop: people in the airport. Sample: 2 of every 15. Var: destination (quatitative). Systematic sampling.

d) Pop: all the students at the school. Sample: 20 students. Var: no. times gone to the cinema (discrete). Stratified (equal)

e) Pop: customers of the video-club. Sample: 1 out of 15. Var: type of film hired (qualitative). Systematic

82

3. We are using stratified sampling in the following surveys (estudio). Determine the

allocation for each stratum (estrato) using both the equal and the proportional allocation.

a) We are gathering information about the marks obtained in 3ºESO by pupils in certain

Town. We want a 24 people sample. Strata (estratos) will be their three High-Schools.

High A: 96 “3ºESO-pupils”, High B: 72, High C: 120.

b) We are asked to take a sample of 40 staff out of a company which has the following:

male, full time: 90 male, part time: 18 female, full time: 9 female, part time: 63

c) There are 240 workers in a factory. The table shows the number of each type of worker in

the factory. A sample of 40 is required.

d) An inspector visits a large company to check their vehicles.

The company has 10 large-load vehicles, 120 light vans and 20 cars. The inspector decides

to sample 10% of the vehicles. Each type of vehicle is to be represented in the sample.

e) A small village has a population of 400. The population is classified by age as shown in the

table below. A sample of 50 is planned (12.5% of the population).

f) We are taking a 3 kg meat sample at the butcher shop, to survey its quality. There, they

have 130kg pork; 70kg beef and 100kg chicken.

g) To check the school bus safety, we inspect a 6 buses sample. The companies’ fleets (flota)

are: Company-A: 5 buses, Company-B: 10 buses, Company-C: 15 buses.

h) We take a 61 people sample in Extremadura living in a capital. Consider its population.

Badajoz: 150,000 inhabitants; Cáceres: 95,000; Mérida: 60,000.

i) The number of students in each year group at a high-school is shown in the table.

Suppose that a sample of 10% is to be chosen.

Sol: Equal allocation Proportional allocation

[a] 8 pupils each. High A: 8 pupils, B: 6 pupils; C: 10 pupils.

[b] 10 people each. Male, full time: 20. Male, part time: 4. Female, full time 2. Female, part time: 14.

[c] 10 workers of each type. Managers:4. Craftsmen: 20. Labourers: 9. Administrators: 7.

[d] 5 of each type of vehicle. 1 Large-load; 12 light-vans and 2 cars.

[e] 10 people each. [0,12], 4 people. [13,24], 7 p. [25-40], 16 p. [41-60], 13 p. [61+], 10 p.

[f] 3kg of each meat. Pork: 1.95kg, beef: 1.05kg, chicken: 1.5kg

[g] 2 buses from each company. Company A: 1 bus. Company B: 2 buses. Company C: 3 buses.

[h] 20 people from each city. Badajoz: 30 people. Cáceres: 19 people. Mérida: 12 people.

[i] 18 students each. 1ºESO: 18; 2ºESO: 20, 3ºESO: 17; 4ºESO: 19; Bach: 16.

Managers Craftsmen (artesano) Labourers (peon) Administrators

24 120 54 42

Age (years) 0-12 13-24 25-40 41-60 61+

# of people 32 56 128 104 80

Group 1ºESO 2ºESO 3ºESO 4ºESO Bach

Students 180 200 170 190 160

83

4. The English teacher uses a weighted mean to calculate the final mark of her pupils. The

chart shows the marks of 4 pupils (out of 10) in each section, and the weights in

percentage. Calculate their final marks.

5. In a school, 85 boys and 35 girls appeared in a public examination. The mean mark of boys

was found to be 40% whereas (mientras que) the mean marks of girls was 60%. Determine

the average marks percentage of the school. [Hint: Use a weighted mean] [Sol: 45.83%]

6. A class of 25 students took a science test. 10 students had an average (arithmetic mean)

score of 80. The other students had an average score of 60. What is the average score of

the whole class? [Hint: Use a weighted mean] [Sol: 68]

7. In a figure skating (patinaje artístico) contest, the scoring method consists of a trimmed

mean, discarding (desechar) the lowest and the highest scores. Find the marks:

8. Rafael and Luisa ask their Maths teacher to set their final marks using the 25% trimmed

mean of their 16 exams. Find their final mark using both the mean and the trimmed mean.

9. For comparing three companies, they are rated at 0 to 5 for their environmental

sustainability, and are rated at 0 to 100 for their financial viability. We don’t want the

financial viability to be given more weight, so we decide to use a geometric mean.

10. We are measuring three computers performance by running a couple of programs, and

then finding the geometric average of their execution time:

Pupil Writing

(30%)

Listening

(25%)

Reading

(15%)

Exercises

(20%)

Behaviour

(10%)

Final

Mark

Sergio 8 7 7 10 10 =8.20

Elena 10 9 10 5 5 =8.25

Isabel 6 7 5 9 10 =7.10

Alonso 5 5,5 6,5 4 2 =4.85

Company Enviromental Sus. Financial Via. G. Mean

A 1 90 =9,49

B 5 30 =12,24

C 3 60 =13,42

Computer Program 1. Program 2. G. Mean

A 1 second 1000 seconds =31.62

B 10 seconds 100 seconds =31.62

C 20 seconds 20 seconds =20

Contestant Judge#1 Judge#2 Judge#3 Judge#4 Judge#5 Mean

Pedro 6 10 6.5 4 7 =6.5

Roberto 8 1 7 8 9 =7.67

Pupil Marks Mean 25% t. mean

Rafael 0, 2, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 10, 10, 10. =6.86 =7,5

Luisa 4, 5, 6, 6, 6, 7, 7, 8, 8, 8, 8,8, 9, 9, 10, 10 =7.44 =7.5

84

11. Charles counted the number of jump rope turns each student jumped in P.E. before

messing up (Col: fallar). Fill the chart using the six data given:

12. The following diagram shows the marks obtained by 30 students in a class:

a) Organise the data using a table of frequencies.

b) Calculate the quartiles, and percentile p60.

c) Draw the box-and-whisker plot.

d) Calculate mean, standard deviation and Coefficient of variation

e) There are two modes. Which are they?

13. An oral surgeon (cirujano) is tracking how many patients were referred (remitido) to him by

various local dentists. In the past year, 7 dentists made the following number of referrals:

14. Logan used coupons 5 times at the store last month. Fill the chart, considering his savings:

15. Juan played a word game in which points are awarded depending on the letters that are

used. He played 7 words and earned the following scores: Fill the chart

16. Preston volunteered at the city park.

He planted 7 flower beds (parterre) with: Fill the chart

17. Ana bought 6 toys for her puppy (cachorro), Pipo. Fill the chart, considering their prices:

Mean CV Q1 Median Q3

5.36 points 2.47 beads 46% 3 5 8

3 patients 5 patients 4 patients 5 patients

7 patients 8 patients 3 patients

Mean MAD CV

5 patients 1.43 patients 1.8 patients 36%

62€ 73 € 24€

95€ 89€

Mean MAD CV

68.60€ 20.48€ 25.20€ 36.7%

9€ 9€ 2€

1€ 4€ 8€

Mean MAD Mode CV

5.50€ 3.17€ 9€ 3.30€ 60.1%

Mean MAD CV

53 points 17.1 points 23.9points 45.2%

98 points 15 points 31 points 57 points

59 points 56 plants 55 points

Mean MAD CV

53 plants 19.1 plants 23.7 plants 44.8%

17 plants 53 plants 83 plants 80 plants

60 plants 56 plants 22 plants

57 turns 65 turns 85 turns

69 turns 43 turns 35 turns

Mean MAD CV

85

Butterscotch candies per bag

Stem Leaf

2 5 6 7 9

3 3 4 7 7 9

4 2 2 5 8

5 4 6 8

6 2 8

7 6

8 3 3 8 8

18. Beatriz volunteered to stamp hands at the entrance of a museum 6 times this year.

The number of visitors each time was: Fill the chart.

19. The following plot shows the number of butterscotch (sirope de azúcar moreno y mantequilla) candies found in some sweets-bags:

a) How many bags did we survey? [Sol: 23]

b) How many bags had at least 45 butterscotch candies but fewer than 79 butterscotch candies?

c) Compute the quartiles and draw the corresponding box-and-whiskers plot.

d) Calculate percentiles: p40= , p85=

e) Calculate the frequency table, grouping by tens. And draw the histogram.

20. For the following statistical surveys, answer the following questions:

a) What type of variable are we surveying?

b) To represent the data set graphically, would you construct a bar graph or a histogram?

Why is that choice better than the other? (construct the graph).

c) Fill in a chart with tendency measures:

Survey#1: In the past year, you have recorded the number of tickets that a movie theatre

has sold, grouped by months. [For variable “Number of tickets”]

Survey#2. For a recent science project, you collected data regarding the distribution of fish

and aquatic life in a nearby pond (estanque-pozo). Your data consists of the amount of

living creatures found in each 1 meter depth increment in the pond. [For variable “# of creatures”]

Survey#3. A shoe store in your local mall has recorded the number of each type of shoes

that it has sold in the past month. [flip-flop=chancleta; high heel=tacones de aguja, loafer=mocasín]

4 visitors 6 visitors 1 visitor

4 visitors 5 visitors 7 visitors

Mean MAD Mode CV

4.5

visitors

1.5

visitors 4 visitors

1.89

visitors 42.1%

Month Jan Feb March April May Jun July Aug. Sept. Oct. Nov. Dec

Tickets 25 30 15 20 30 35 40 20 25 15 20 30

(c) Solutions Mean CV Q1 Median Q3

#1) =25.4 =7.49 =29.5% =20 =25 =30

#2) =30.5 =19.2 =63% =15 =22 =47

#3) =37.4 =9.38 =25.1% =31 =34.5 =39.5

Depth 0-1 m. 1-2 m. 2-3 m. 3-4 m. 4-5 m. 5-6 m. 6-7 m. 7-8 m. 8-9 m. 9-10 m.

#Creatures. 10 19 23 47 68 51 43 21 15 8

Type flip-flops tennis shoes sandals high heels boots walking shoes running shoes loafers

#Pairs. 35 60 42 37 29 32 30 34

Mean CV Q1 Median Q3

86

21. We are surveying how many people live in each of the High-School pupil’s homes. So we

ask two students from each class. These are the results: [*] Suppose there are 10 classes.

a) Describe this statistical study.

b) Fill in the table of frequencies.

c) Fill the chart

d) Calculate the quartiles, and draw the box-and-whiskers plot.

22. The following table shows the distribution of the area, in square metres, of some flats

randomly picked in Mérida.

a) Describe this survey.

b) Fill using class marks

c) Draw the corresponding statistical graphs.

23. We are measuring the heights, in centimetres, of children arriving at a youth hostel

(albergue juvenil). So, we measure one out of every 30 children who arrive during the Holy

Week. The results are the following.

a) Describe this statistical survey.

b) Fill in the table of frequencies. Group by tens ( [130,140), [140,150), …).

c) Fill using class marks

d) Draw the corresponding statistical graphs.

24. We are surveying the weight of football players. So, we measure 22 football players from

two teams playing against each other in a match. The results are the following:


b) Fill in the table of frequencies. Group by tens ( [60,70), [70,80), …).

c) Fill using class marks


4 3 6 5 4 5 6 5 4 6

5 3 3 4 4 7 4 4 5 4

Mean Mode CV

Mean Mode CV

143 153 164 173 143 168 156 154 161 148

177 161 179 165 157 174 152 139 144 155

Mean Mode CV

75 74 79 82 65 73 78 76 85 82 95

89 78 81 68 74 73 69 77 89 92 83

Mean Mode CV

=85 60-80 =17.75 =20.88%

Surface area No. of flats

60-80 50

80-100 30

100-120 15

120-140 5

Totals 100

87

25. We are surveying the traffic accidents on the roads in Extremadura. So, we pick 30 roads

randomly and count the numbers of accidents in a weekend. The results are the following:


b) Fill in the table of frequencies.

c) Fill


[a] Population: roads in Extremadura, variable: number of accidents (quantitative, discrete). Sampling: 30 roads (random).

26. The following table displays the time worked per day by some employees in a multinational

company. Employees were chosen from 5 countries, proportionally to the number of

employees on each country.


b) Fill using class marks

c) Draw the corresponding statistical graphs.

[a] Pop: employees in the multinational, var: hours worked (continuous). Sampling: 200 emp. (stratified), proportional alloc.

27. Consider the following scatter plots. Say if they show a trend (positive or negative) or not.

If there is a trend, draw “by eye” the line of best fit (trend line).

a) b) c)

d) e) f)

Mean Mode CV

=7,95 hours [8,9) hours =0, 86h =10.88%

Time (hours) #employees

[5,6) 10

[6,7) 20

[7,8) 40

[8,9) 130

Totals

Mean Mode CV

=2 accidents =2 accidents =1.18 accidents =59%

1 3 0 4 3 0 4 2 2 3 3 2 1 1 2

2 2 2 2 1 1 2 0 2 1 1 2 3 5 3

88

28. Look at the scatter plot. Write the equation of the trend line, and use it to estimate the

image of the unkwnown “x”

a) b) c)

d) e) f)

29. A factory orders an audit (auditoría-revisión financiera) to find out if their expenses in

advertising are effective. The data collected is (in thousands of €):

Expenses in advertising (X) 1 2 3 3 4 5 6 7 8

Sales (ventas) (Y) 15 16 14 16 17 20 18 18 19

Frequency 1 3 2 1 2 2 3 2 1

a) Make the scatter plot and identify possible trends.

b) Make the frequency table for each variable (expenses and sales)

c) Expenses: Sales:

30. We are surveying the relationship between the number of days a pupil is going out during a

festival, and the money spent (in total). After asking 20 pupils, we get:

a) Make the scatter plot and identify possible trends.

b) Make the frequency table for each variable (days and money)

c) Days: Money:

Mean Mode CV

Mean Mode CV

Mean Mode CV

3,75days 4 days 1,09 days 29,06%

Mean Mode CV

0 x 2 4 6 8 10 0 4 8 x 12 16 20 0 5 10 x 20 25

0 20 40 x 60 80 100 0 5 x 10 15 20 25 0 5 10 x 15 20 25

days (X) 1 2 3 3 4 4 4 5 5

money (Y) 20 32 45 50 55 65 70 70 75

no pupils 1 2 1 2 2 4 3 3 2

89


Bilingual section

STATISTICS.

1. (1.5 pts.) In a funfair (parque de atracciones) they want to know their visitors’ opinion. They

want all types of people to be represented. The table below shows the information about

people entering today. They will survey 44 people.

a) (0.5 pts.) What type of sampling should they use?

b) (1 pt.) Fill the chart:

2. (1 pt.) The Ph. E. teacher will set the final mark using a weighted mean of the skills shown in

the chart below (it includes how many points they are worth). Fill it with the final marks.

3. (1 pt.) We want to know if there is any relationship between the number of hours per day a

shop is open, and its net income (ingreso neto). We have the following data:

Make the scatter plot and identify possible trends. Use it to predict the income for 9 hours.

4. (6.5 pts.) We are surveying the height of girls studying 3ºESO in our high-school, measuring 5

girls in each of the 4 groups (20 in total). The results are shown in the table below:

a. (2 pts.) Describe this survey and make the box-and-whisker plot.

b. (1 pt.) Fill the table of frequencies.

c. (0.5 pts.) Is it better to use a bar graph or a histogram? Why? (construct it).

d. (3 pts.) Fill

# of people and type Man Woman Boy Girl Totals

Total 112 126 210 168

______________ allocation

______________ allocation

Skill and

mark

Written exam

(3 points.)

Handstand

(1p)

Basketball

(2p)

Cooper

test (4p)

Final

mark

Pedro 8 4 9 6

Ana 6 10 5 8

Height #Girls

[150,152) 1

[152,154) 2

[154,156) 3

[156,158) 3

[158,160) 5

[160,162) 4

[162,164) 2

Mean Mode CV p10

Hours opened (X) 4 5 5 6 6 6 7 7 7 8 9

Net income (Y) 80 90 105 120 130 140 130 150 180 190

91

PROBABILIDAD

Experimento: suceso que podemos repetir bajo condiciones semejantes. Puede ser:

Deterministas: sólo hay un resultado posible.

Aleatorio: si hay varios resultados posibles. Cada uno de estos resultados se llama

suceso elemental. El conjunto de todos los resultados posibles se llama Espacio Muestral (E).

La unión de varios sucesos elementales se llama suceso compuesto.

Experimento compuesto: cuando el experimento puede dividirse en otros más

sencillos. Para calcular el espacio muestral podemos usar un diagrama de árbol.

Ejemplo: “lanzar dos monedas” puede descomponerse en: 1ª moneda y 2ª moneda.

- Si no necesitamos todo el espacio muestral, podemos podar el árbol: no dibujar las ramas

que no nos interesan.

Probabilidad: es el grado de creencia que tenemos de que ocurra un suceso. La medimos en

porcentaje, fracción o número (entre 0 y 1).

Un suceso de probabilidad 100%, se llama suceso seguro, y el de probabilidad 0% se llama

suceso imposible.

Cálculo de probabilidades:

- Regla de LaPlace:

, pero sólo puede usarse cuando podemos afirmar

que todos los casos tienen la misma probabilidad (son equiprobables).

- Diagramas de árbol: la probabilidad de cada suceso se calcula multiplicando la de cada rama.

- Ley de los grandes números: la probabilidad es la frecuencia relativa, siempre que se haya

repetido el experimento muchas veces.

Técnicas para contar (recuento):

- Permutaciones: las posibles colocaciones de todos los elementos del conjunto. Se escribe

Pn=n! = n·(n-1)·…·1. Por ejemplo, para 5 elementos, P5=5!=5·4·3·2·1=120 posibilidades.

Si tomamos sólo algunos elementos, (variaciones); por ejemplo elegir 3 elementos de los 5,

entonces hay 5P3= 5·4·3 = 60 posibilidades.

Si hay elementos repetidos, por ejemplo 2 iguales y 3 iguales,

=10 posibilidades.

- Combinaciones. Tomar varios elementos, sin que importe el orden. Por ejemplo, elegir 3

alumnos de 5 voluntarios; 5C3=

= 10 posibilidades. (Dividimos entre las 6

posibles formas de recolocar esos 3 alumnos).

El o

rden

SÍ i

mp

ort

a

92

PROBABILITY

Experiment: test we can repeat under similar conditions. It can be:

Deterministic: there is only one possible result.

Random: There is more than one possible result. Each of them is called simple event l.

The set of all possible outcomes is called the Sample Space (S).

If an event has more than one elements then it is called a compound event.

Compound Experiment: when the experiment can be break into some more

simple ones. We can use a tree diagram to calculate the sample space.

Example: “tossing two coins” may be broke into: 1st coin and 2nd coin.

- If we do not need the whole sample space we can prune the tree: do not draw the branches

we are not interested in.

Probability: degree of likelihood that something will happen. We can measure it in

percentage, fraction or number (between 0 and 1).

A 100% probable event is called sure event, and a 0% probable event is called impossible

event.

Finding probabilities:

- LaPlace’s rule (rule of Succession):

, it can only be used when we

can state that all outcomes are equally likely (are equiprobable).

- Tree diagrams: the probability of each event is computed multiplying the probability of each

branch.

- Law of Large Numbers: the relative frequency is the probability, as long as we repeat the

experiment many times.

Counting techniques:

- Permutations: all possible arrangements of the elements in a set. It is written as Pn.

For instance, for 5 elements, P5=5!=5·4·3·2·1=120 possibilities.

If we pick only some elements, (variations); for e.g. choose 3 elements out of 5, there are

5P3= 5·4·3 = 60 possibilities.

If there are repeated elements, for e.g. 2 equals and 3 equals,

=10 possibilities.

- Combinations. Pick some elements, when order does not matter. For e.g., pick 3 pupils out

of 5 volunteers; 5C3=

= 10 possibilities. (We divide by the 6 possible ways of

arranging those 3 pupils).

Ord

er D

OES

mat

ter

93

PROBABILITY - WORD PROBLEMS

1. Mary builds a wooden birdhouse that is shaped like a cube. She paints 2 sides red, 1 side

green, and 3 sides black. If she picks a side at random for the front, what is the probability

that she will not pick a red side?

2. You roll a 6-sided die (dado). What is P(divisor of 16)?

3. In February, 85% of an airline’s flights arrived on time. What is the probability that one of its

flights arrived late in February?

4. This table that shows the results of a survey that asked students in a classroom to choose

their favourite fruit:

a) Suppose a student in the classroom is picked at random. Find the probability that the

student’s favourite fruit is strawberry.

b) Suppose a student in the classroom is picked at random. Find the probability that the

student’s favourite fruit is not an apple or a banana.

5. Consider the following spinner.

a) Find the probability of spinning a number that is greater than or equal to 4.

b) What is the probability of spinning a number that is not a 2 or a 3?

c) What is P(prime number)?

6. Boats from all along the Atlantic coast dock at a busy marina. Of the first 16 boats to dock at

the marina one day, 4 were from Massachusetts. What is the (experimental) probability that

the next boat to dock will be from Massachusetts?

7. “Santa Eulalia” Bakery recently sold 12 desserts (postre), including 4 biscuits. What is the

(experimental) probability that the next dessert sold will be a biscuit?

8. Of the last 12 contestants on a game show, 8 won a prize. What is the (experimental)

probability that the next contestant will win a prize?

9. You flip a coin twice. What is P(heads, tails)? Write your answer as a percentage.

10. You pick a card at random, and then pick another card at random.

Find P(They sum 6) and P(They sum more or equal than 8)

a) With replacement.

b) Without replacement.

Fruit Orange Apple Banana Strawberry Other

# of students 3 8 11 6 4

1 2 3 4 5

94

11. Counting outcomes:

a) An Italian restaurant offers mozzarella cheese, swiss cheese, sausage, ham, onions, and

mushrooms for pizza toppings. For this week’s special, you must choose one cheese, one

meat, and one vegetable topping. Find the number of possible outcomes.

b) Alicia has a black and a white teddy bear. Clara has a black, a white, a brown, and a pink

teddy bear. Each girl picks a teddy bear at random to bring to a sleepover (quedarse a dormir)

party. How many different combinations can the girls bring?

c) A candy maker offers milk, dark, or white chocolates with solid, cream, jelly,

nut, fruit, or caramel centres. How many different chocolates can she make?

d) In a lottery game, balls numbered 0 to 9 are placed in each of four

chambers (cámara) of a drawing machine. One ball is drawn from each

chamber. How many four-number combinations are possible?

12. In a game of checkers (damas), there are 12 red game pieces and 12 black game pieces. Julio

is setting up the board to begin playing. [Hint: use a tree, and remember to prune it!]

a) What is the probability that the first two checkers he pulls from the box at random will be

two red checkers?

b) What is the probability that the first two pieces are a red followed by a black?

13. Each of the spinners at the right is spun once to determine how a player’s piece is moved in

a board game. [Hint: use a tree, and remember to prune it!]

a) Francisco needs to spin a red and a blue to move to the last square

and win the game. What is the probability that Francisco will win?

b) If Francisco spins a green or a white on either spinner, he will land on a “take an extra turn”

square. What is the probability that Francisco will get an extra turn?

14. Irene keeps her white and black chess pieces in separate bags. For each colour, there are 8

pawns, 2 rooks, 2 bishops, 2 knights, 1 queen, and 1 king.

a) Find the probability of drawing a knight from the bag of white pieces and drawing a pawn

from the bag of black pieces.

- Find the probability of drawing a bishop from the bag of white pieces and then drawing the

queen from the same bag.

b) With replacement. c) Without replacement.

15. We are playing with a 40 cards deck. Find the probability of having four aces (ases) when we

draw 5 cards.

a) With replacement. b) Without replacement.

95

Video Game Playing Time Per Week

Hours # participants

0 18

1-3 43

3-6 35

more than 6 24

Favorite Spectator Sport

Sport Number

football 42

baseball 27

basketball 21

soccer 12

16. A blackjack hand of 2 cards is randomly dealt (repartido) from a standard deck of 52 cards.

What is the probability that the first card is an ace and the second card is a face card?

17. Use the results in the table at the right. In a survey, 102 people were asked to pick their

favourite spectator sport in England.

a) What is the probability that a person’s favourite

spectator sport is baseball?

b) Out of 10,000 people, how many would you expect

to say that baseball is their favourite spectator sport? Round to the nearest person.

18. Use the results of a survey of 120 eighth grade

students shown at the right to find probabilities:

a) That a student plays video games more than 6 hours

per week.

b) Out of 400 students, how many would you expect to play more than 6 hours per week?

19. Only 6 out of 100 English say they leave a tip of more than 20% for satisfactory service in a

restaurant. What is the probability that an English will leave a tip of more than 20%? Out of

1,500 restaurant customers, how many would you expect to leave a tip of more than 20%?

20. Jason has a packet of tomato seeds left over from last year. He plants 36 of the seeds and

only 8 sprout (brotar). What is the experimental probability that a tomato seed from this

packet will sprout?

21. There are 10 boys and 14 girls in a class.

a) If we pick two people at random, what is the probability of choosing a boy and a girl?

b) If we pick five people, what is the probability of picking any boys (at least one)?

c) If we pick four people, what is the probability of choosing two boys and two girls?

Hint: There are

possible cases; all of them having a probability of

=6.42%.

22. We are drawing 5 cards from a deck of 40 cards. Find the probability of drawing:

a) Exactly one face card. b) Any face card (at least one):

c) Four of a kind/quads (poker) (contains all four cards with of one rank).

Hint: For each of the 10 numbers, there are 9cards·5places=45 favourable. 450 in all, out of 40·...·36 outcomes.

d) Exactly two face cards.

Hint: There are

possible cases; all of them having a probability of

3%.

e) Straight (escalera) flush (color) (5 cards in sequence, of the same suit; ace can play high or

low). Hint: There are 7straights·4suits·5!combinations favourable-cases out of 40·...·36 possible outcomes.

96

23. I’ve got a bag containing 5 black marbles, 3 white marbles and 2 pink marbles. I’m picking

three marbles.

a) Draw a tree showing all possibilities for the event “all the same colour”. [Sol: 3 events]

b) Draw a tree showing how many outcomes consist of all three colours. [Sol: 3!=6 events]

c) Fill with probabilities:

24. We are picking two cards at random from the cards shown on the right.

a) If we do it with replacement, how many outcomes are possible?

b) Fill the chart with probabilities

25. Cecilia’s got a bag containing 3 green marbles, 2 blue marbles, 2 purple

marbles and one red marble. Fill the charts with probabilities

a) If she draws 2 marbles.

b) If she draws 4 marbles.

Event (colours) With replacement Without replacement

Red and green =

=

Both green =

=

Two different colours =

=

Event (colours) With replacement Without replacement (Hint)

All green =

=1.98% Impossible There are not 4 green marbles

All the same colour =

+2·

+

=2.78% Impossible

At least two different colours =100 – 2.78 = 97.22% Sure event (P=100%)

It’s the contrary to “all the

same colour”

All four colours =24·

= 7.03% =24··

=17.14%

Each of the 4!=24 favourable outcomes are equally likely

Event (colours) With replacement Without replacement (Hint)

All white

All the same colour

Two or more colours

It’s the contrary to “all

three the same colour”

All three colours

Event (cards) With replacement Without replacement

Circle and cross

Both equal

Both different

97

Shipping List (one each)

Aquarius Purple Tiger Candy Apple Roundelay Desert Dawn Scarlet Knight

Fragrant Plum Shining Hour Golden Girl

Sonia Supreme Linda Ann Sundowner

Mount Shasta Viceroy Pink Parfait

COUNTING TECHNIQUES

26. [Permutations] At lunchtime recess, 12 students race each other across the playground. In

how many ways can students finish in first, second, third, and fourth places?

27. [Permutations] A plumber has 8 jobs to schedule in the next week. One of the jobs is high

priority and must be done first. In how many ways can the next 4 jobs be scheduled?

28. [Permutations] A music festival features 5 jazz bands, 9 rock bands, and 11 school bands.

The bands play at various times over a long holiday weekend.

a) In how many ways can the first 4 rock bands be selected to play?

b) In how many ways can the first 3 school bands be selected to play?

29. [Permutations] Natalia buys a small box of 12 different assorted (surtido) chocolates. She

lets her sister have her 2 favourite chocolates, and then she has just enough left to give one

chocolate to each girl attending basketball practice. In how many ways can Natalia give out

the chocolates to the basketball players?

30. [Combinations] During one month, a movie theatre is planning to show a collection of 9

different Science-Fiction movies. How many different double features (two-film showings)

can they choose to show from this collection?

31. [Combinations] For a history test, students are asked to write essays (redacción) on 4 topics.

They must choose from a list of 10 topics about the European countries they have been

studying. Is this situation a permutation or a combination? Explain. How many ways can a

student choose 4 topics?

32. [Combinations] A taste test of 11 different soft drinks is held at a shopping mall. Each taster

is randomly given 5 of the drinks to taste. How many combinations of soft drinks are

possible?

33. [Combinations] A school book fair is offering a package deal on the opening day. For a

special price, students may purchase any 6 different paperback books from a list of 30 books

that have won the “Felipe Trigo” prize. How many packages are possible?

34. Use the shipping list at the right that shows the rosebushes (rosal) Mrs. Lawson ordered for

her front yard. She wants to plant 9 of them along the walkway from her driveway to her

front porch. How many ways can she plant the rosebushes along the walkway?

a) If order is not important.

b) If order is important.

98


Bilingual section

PROBABILITY.

1. (1.5 pts.) We’ve got a bag containing 3 black marbles (B), 2 white (W) and 5 blue (U). We’ve got

another one with 2 black marbles (B) and 3 white (W). We are drawing one marble out of each bag

and writing their colours.

a. (0.5 pts.) Is this a random experiment? Yes / NO Why?

b. (0.5 pts.) Write the sample space. E={ }

c. (0.5 pts.) Write any two simple events and their probabilities.

2. (2 pts.) Four people are playing a “Cuatrola” game, using a deck which only has face cards (jack-sota,

queen-caballo, king-rey), threes and aces (20 cards in all). In this round, gold coins are trumps (pintan

oros). Each player is dealt 5 cards.

Work out the following probabilities.

- (1 pt.) All 5 cards that I have are trumps “pintes” (gold coins).

- (1 pt.) I have any (at least one) trumps.

3. (1 pt.) A cream advertisement says that in a survey carried out with 50 people, 43 of them liked the

cream a lot. If we try that cream, what is the (experimental) probability that we will like it a lot?

4. (1 pt.) The table below shows information about pupils in a school:

We know that Luis’ brother is in 5ºA. If we pick a child from the school,

find the probability that:

a) He is in the same class as Luis’ brother:

b) He is in 6th grade.

5. (4.5 pts.) We have a bag containing 10 red marbles (R), 4 yellow (Y) and 6 black (B).

a. (1.5 pts.) If we draw two marbles (with replacement). Find the probability of drawing at least

two marbles of the same colour.

b. (1.5 pts.) If we draw two marbles (without replacement). Find the probability that each is of a

different colour.

c. (1.5 pts.) If we draw four marbles (with replacement). Find the probability of drawing at least

two different colours. [Hint: They are not all of the same colour].

Group No. of children

Up to 4th 160

5th A 20

5 th B 20

6 th A 26

6 th B 24

99

apotema

a

b

c

ÁREAS Y VOLÚMENES

- Teorema de Pitágoras: en todo triángulo rectángulo, el cuadrado de la

hipotenusa es igual a la suma de los cuadrados de los catetos.

Terna Pitagórica: si a, b, c son enteros. Ej: (3,4,5); (5,12,13); (7,24,25); (8,15,17),…

ÁREAS

Paralelogramos: base por altura A=b·h.

- Para el rombo y la cometa, A =

(D y d son las diagonales).

Triángulo: A =

. Trapecio: A =

Polígono regular: A =

. (apotema: distancia del centro al lado).

- En general, hay que descomponer la figura en otras de áreas conocidas.

- Longitud de la circunferencia de radio r: L= 2r. Para un arco de a grados: L =

· 2r.

- Área del círculo: A=r2. Para un sector circular de a grados: A =

r2.

VOLÚMENES

Prisma: tiene dos caras iguales y paralelas “bases”, y las otras son paralelogramos.

Cuando la base es un paralelogramo, se llama paralelepípedo. Si además sus ángulos son

rectos, se llama ortoedro (como un rectángulo, con volumen).

Pirámide: una cara es un polígono y el resto triángulos que concurren en un punto.

Cuerpo de revolución: el obtenido haciendo girar una curva plana (generatriz) alrededor de

una recta (eje). Ejemplos: cilindro, cono y esfera.

Principio de Cavalieri: si dos cuerpos de la misma altura tienen secciones de

áreas iguales, entonces tienen el mismo volumen.

- Ejemplos: Volumen del cilindro: V=Abase·h. Volumen del cono:

Abase·h.

Esfera: A=4r2. V=

r3. Para una zona o un casquete, A=2r·h.

Tronco: V=

·h. Volumen: V=

Abase·h.

Volumen: V=Abase·h.

a2 = b2 + c2

D

d

B

b

h

100

a

b

c

apothem

AREAS AND VOLUMES

- Pythagorean Theorem: In a right triangle the square of the hypotenuse

is equal to the sum of the squares of the legs (the other two sides).

Pythagorean Triple: when a, b, c are integers. Ex: (3,4,5); (5,12,13); (7,24,25); (8,15,17),…

AREAS

Parallelograms: base times height A=b·h.

- For rhombus (diamond) and kite, A =

(D and d are the diagonals).

Triangle: A =

. Trapezium: A =

Regular Polygon: A =

. (apothem: distance from centre to sides).

- Usually, one has to break the figure into others of known areas.

- Circumference of a circle of radius r: L= 2r. For an arc of a degrees: L =

· 2r.

- Area of a circle: A=r2. For a circular sector of a degrees: A =

r2.

VOLUMES

Prism: has two equal parallel faces “bases”, and the others are parallelograms.

When the base is a parallelogram, it is called parallelepiped. Moreover if it is

right-angled, it is called cuboid (like a rectangle, with volume).

Pyramid: one face is a polygon and the others triangles that meet at one point

Solid of revolution: it is obtained by rotating a plane curve (generatrix) around a straight

line (axis). Examples: cylinder, cone y sphere.

Cavalieri’s principle: if two bodies have the same height and their sections

have the same areas, then they have the same volume.

- Examples: Volume of the cylinder: V=Abase·h. Volume of the cone:

Abase·h.

Sphere: A=4r2. V=

r3. For any zone or cap, A=2r·h.

Frustum: V=

·h. Volume: V=

Abase·h.

Volume: V=Abase·h.

a2 = b2 + c2

D

d

B

b

h

101

AREA AND VOLUMES – PRACTICE

1. Use the nets to find the surface of each figure. Then, calculate their volume:

a) b) c) d)

e) f)

2. Find the surface of each figure:

a) b) c) d)

e) f)

3. Find the lateral area and surface area of each figure.

a) b) c) d)

102

4. Find the volume and surface of these figures:

a) b) c)

d) e) f)

5. The cone from the previous exercise (section f ) is cut 5cm above its base by a plane that is

parallel to that base. Calculate the volume of the truncated cone created.

[Hint: use proportions to calculate the radius of the smaller base]

6. Do the same for the pyramid in section b).

7. Find the volume of each figure.

a) b) c) d)

e) f) g) h)

8. Find the volume of the following frustums:

a) b) c) d)

Sol: V=17493cm3 V=989.1cm

3 V=693.3cm

3 V=2478.5cm

3.

4m

2m

10m

2m

4m

2m

3cm

11cm

5cm 8cm 8cm

6cm

x 3cm

3cm

12 cm

103

9. Find the volume and the surface area:

a) b) c)

V=140.6m3; S=3081.9mm

2 V=254.8yd3; S=270.8yd

2 V=140ft

3, S=226ft

2

10. Find the volume of the following bodies:

a) b) c)

V=729+234=963cm3. V=52.3+314=366.3cm3. V=5024+133.97=5157.97cm3.

11. Work out the volume (remember to use proportions to find the missing measures):

a) b) c) d)

V=54.5m3. V=56dm3. V=170.2dm3. V=96.25cm3.

12. Find the volume of this polyhedron, and its surface using its net.

3cm

3cm

6cm

1.5cm

1.68cm

6cm

3cm

3cm

5cm

10cm

5cm

3cm

3cm

3cm

4m

2m

6m

6cm 3dm

8dm

4dm

2dm 8dm

2.5dm 7.5dm

104

2.5m

Areas and Volumes – Word Problems

1. A “trunk multi-purpose” measures 40x29x19cm. What is its capacity in litres? [Sol: 22 litres]

2. Trailers that travel on the road behind trucks are rectangular prisms. A typical height for

the inside of these trailers is 2.74m. If the trailer is 2.5m wide and 6m long, what is the

volume of the trailer? And its capacity? [Sol: 41.1m3=41,100litres]

3. A cologne (colonia) is sold in a box measuring 8cm by 5cm by 4cm. How many colognes

may be packed in a carton measuring 64cm by 40cm by 32cm?

4. How much water will it take to fill a swimming pool that measures 2.5m x 10m x 6m? How

long will it take to fill this pool at a rate of 5L every 30 seconds?

5. How many litres of water are required to fill a rectangular swimming pool 18m long and

12m wide which is 2m deep throughout? [Sol 432,000litres]

6. A factory is making jewellery boxes that are going to be chrome plated (chapado). The

chrome plating costs $0.05 for every cm2 covered in chrome. How much would it cost to

chrome plate a rectangular box with dimensions of 18cm x30cm x6 cm?

7. A cylindrical can is just big enough to hold three tennis balls. The radius of a

tennis ball is 5 cm. What is the volume of air that surrounds the tennis balls?

8. A tent used for camping is shown below.

a) Find the volume of the tent.

b) How many m2 of fabric (tela) does it have? (including the floor)

9. Consider the tank on the right.

a) How many litres of water can it hold?

b) Painting it costs 12€ per m2. How much is it for the whole tank?

10. Farmer Jones owns a citrus tree farm in Florida. During some parts of the year the amount

of rain in Florida is not sufficient to maintain maximum growth of citrus so farmer Jones is

going to buy a water tank. Find how much water it will hold if it is a right circular cylinder

with a height of 10 feet and a radius of radius of 5 feet. [*] 1ft=30.48cm [Sol: 785ft3].

11. A soup can has a diameter of 10 cm and a height of 15 cm. What is the volume of the soup

in the can if 0.5 cm of space is left at the top of the can? [Sol: 1138.25 cm3].

12. There are 24 cans of soda and they are arranged in 4 rows of 6. Each can has a radius of 4

cm and a height of 15 cm. These cans are to be packed snugly (ajustado) into a case with no

extra space on the top, bottom, or sides. Calculate how much empty space will not be

occupied by a can in the box. [Sol: 1336.32 cm3].

1.5m

2m

1m

105

TRANSFORMACIONES EN EL PLANO. MOSAICOS

Transformación: correspondencia de una figura con otra, que se llama homóloga.

Si un elemento se transforma en sí mismo, se llama invariante o doble.

Si la transformación conserva las distancias se llama movimiento. Si además

conserva la orientación, es movimiento directo. Si la cambia, es inverso.

Vector: segmento orientado. Su longitud se llama módulo.

Traslación: a cada punto se le suma un vector “director”.

Rotación: cada punto se gira un mismo ángulo (argumento), con el mismo centro de giro.

La simetría central (respecto un punto) es un giro de 180º.

Simetría axial (reflexión): a cada punto le corresponde el que está a la misma distancia del

eje (en perpendicular).

Friso: rectángulo decorado al que se le aplica reiteradamente una traslación.

Mosaico (teselado): conjunto de figuras que recubren el plano mediante traslaciones.

Regular: si está formado por un polígono regular (sólo pueden hacerse con

triángulo, cuadrado y hexágono)

Semirregular: está formado por varios polígonos regulares.

No regular: formado por polígonos irregulares.

Métodos para teselar:

- Compensación de áreas: deformar un lado de un polígono y aplicarle una isometría para

que siga teselando.

- Malla invisible: haciendo simetrías respecto el centro de los lados de un cuadrilátero.

DIR

ECTO

IN

VER

SO

106

TRANSFORMATIONS OF A PLANE. MOSAICS.

Transformation: correspondence of one figure with another, which is called homologue.

Any element that corresponds to itself is called invariant (or double).

When the transformation preserves lengths, it is called movement. If it also

preserves orientation it is a direct movement. If it changes the

orientation, it is called inverse.

Vector: oriented segment. Its length is called moduli.

Translation: every point is added the same “director” vector.

Rotation: each point is turned around a fixed centre of rotation a certain angle (argument).

Central symmetry (around a point) is a 180º rotation.

Axial symmetry axial (reflection): each point is mapped to the other one which is at the

same distance of the axis (in perpendicular).

Frieze: decorated rectangle to which we apply a translation repeatedly.

Mosaic (tessellation): set of figures which fill the plane using translations.

Regular: formed by one regular polygon (it can be done only with triangle,

square and hexagon)

Semi-regular: formed by several regular polygons.

Non-regular: formed by irregular polygons.

Tessellation methods:

- Area compensation: reshape the side of a polygon and apply an isometry such that it still

tessellates.

- Invisible grid: performing symmetries respect the center of each side of a quadrilateral.

DIR

ECT

INV

ERSE

107

TRANSFORMATIONS AND MOSAICS – EXERCISES

1. Graph the image of the figure using the transformation given:

a) translation: 5 units right, 1 up b) translation: 1 unit left, 2 up c) translation: 2 units right, 2 up

d) reflection across y=-2 e) reflection across the x-axis f) reflection across y=-x

g) reflection across y=-1 h) reflection across y=x i) rotation 180º about the origin

j) rotation 90º clock wise k) rotation 180º about the origin l) rotation 90º counterclockwise about the origin about the origin

108

2. Complete the mosaics using the “area compensation” method:

a) b) c)

d) e)

3. Use the “invisible grid” method to make a tessellation. Include the grid.

a) b) c)

4. Continue this irregular mosaic following the pattern (L-substitution tiling)

109


Bilingual section

AREAS AND VOLUMES (7.5 pts).

1. (3 pts.) Find the volume of the figure. Write each solid’s name.

2. (3 pts.) Use the net to find its surface.

V= S=

3. (1.5 pts.) A tent is shaped as a right-cone. The base radius measures 1.5m and its height is 3m. Fabric (tela) for the ground costs 15€ per m2, and for the rest of the tent, 7€ per m2.

a) Find the volume of the tent.

b) How much does the material cost?

Transformations of a plane. Mosaics (2.5 pts)

4. (1.25 pts.) Graph the image of the figure using a reflexion across y=2-x:

2cm

x

2cm 2cm

4cm x

2cm

2cm 12.57cm

5. (1.25 pts.) Complete the mosaic using

the “area compensation” method:

111

APÉNDICE: LUGARES GEOMÉTRICOS

Lugar geométrico: es el conjunto de puntos que cumplen una condición.

Mediatriz: puntos que están a la misma distancia de otros dos.

Las “cónicas” como lugares geométricos:

Circunferencia: puntos que están a la misma distancia “radio” de uno dado “centro”.

Elipse: puntos tales que la suma de distancias a otros dos “focos” es constante.

Hipérbola: puntos tales que la diferencia de distancias a otros dos ”focos” es constante.

Tienen dos ejes de simetría (mayor y menor), que son perpendiculares. El mayor contiene

los focos. Su intersección es el centro de simetría.

Parábola: puntos que están a igual distancia de un punto “foco” que de una recta “directriz”.

Tiene un eje de simetría que es perpendicular a la directriz y contiene al foco y al “vértice”.

APPENDIX. LOCI.

Locus: set of points which satisfy a certain condition.

Segment bisector: points which are at the same distance of some other two.

“Conics” as loci

Circle: set of points which are at the same distance “radius” from a “centre”.

Ellipse: points such that the sum of the distance to other two “focus” is constant.

Hyperbola: points such that the difference of distance to other two “focus” is constant.

Both have two symmetry axes (major and minor) which are perpendicular. The major axis

contains the focus. They meet at the symmetry centre.

Parabola: points which are at the same distance from a “focus” than from a “directrix” line.

It has a symmetry axis which is perpendicular to the directrix and contains the focus and

the “vertex”.

112

LOCI – EXERCISES

1. Use the webs to draw ellipses. Affinity method.

a) b) c)

d) e) f)

2. Use the webs to draw ellipses. Use the definition.

a) b) c)

d) e) f)

113

3. Use the webs to draw parabolas. Use the definition.

a) b) c)

d) e) f)

4. Use the webs to draw hyperbolas. Use the definition.

a) b)

c) d)

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