1º ESO cuadernillo bilingue

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Juan N. Puebla RodrguezLola Reguera Doblado 1

Matemticas

Material curricular

1 ESO BILINGE


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DEPARTAMENTO MATEMTICAS2

INDEX

UNIT I: NUMBERS.... 3

UNIT II: INTEGERS... 20

UNIT III: POWERS. 33

UNIT IV: DIVISIBILITY.. 44

UNIT V: FRACTIONS.. 55

UNIT VI: WHAT IS MAGNITUDE? .... 72

UNIT VII: ALGEBRA. 87

UNIT VIII:GEOMETRY.... 106


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NUMBERS UNIT I

Exercise 1.Say the numbers:

3 18 34 355 5 01550 406 1,520 36 247 5,000400 2,000,000 234,289 21

480 57 35 68 66212 2,389 756 13 7290.023 6.2 3.8112.99 123 $2.34 12.67

Exercise 2.Write the number that you hear.

Exercise 3.Say the operations:

a) 2 + 3 = b) 5 8 = c) 12 8 = d) 18 9 =

e) 10 + 15 = f) 3 12 = g) 21 7 = h) 81 9 =

i) 23 + 7 = j) 102 17 = k) 12 6 = l) 125 35 =

m) 33 3 = n) 14 6 = o) 111 22 = p) 222 23 =


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PLACE VALUE

In the number 3 147 286 (three million, one hundred and forty seven thousand, two

hundred and eighty six), the figure 2 has a value of 200 (two hundred), and the figure 3

has a value of 3 000 000 (three million).

Exercise 4.What is the value of the figure 8? What is the value of the figure 4?

Exercise 5.

a)Describe the numbers 384 in words.

b) Describe the number 79 in words.

c) Describe the number 348 in words.

d) Describe the numbers 9 139 in words.

e) Describe the numbers 125 .978 in words.

f) Describe the numbers 4. 235 225 in words.

COMPARING NUMBERS

Symbol MeaningExample in

SymbolsExample in Words

> Greater than 7 > 4 7 is greater than 4

< Less than 4 < 7 4 is less than 7

= Equal to 7 = 7 7 is equal to 7


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Exercise 6.Use < (less than), equals (=) or > ( greater than) to compare each set of

numbers.

a) (12 7) y (3 5) b ) (24 18) y 6 c) 23 y (100 77)

ORDER TO OPERATION

Problem: Evaluate the following arithmetic

expression: 3 + 4 2

It seems that each student interpreted the problem differently, resulting in two different

answers. Student 1 performed the operation of addition first, then multiplication;

whereas student 2 performed multiplication first, then addition. When performing

arithmetic operations there can be only one correct answer. We need a set of rules in

order to avoid this kind of confusion. Mathematicians have devised astandard order

of operations for calculations involving more than one arithmetic operation.

Rule 1: First perform any calculations inside parentheses.Rule 2: Next perform all multiplications and divisions, working from left to right.

Rule 3: Lastly, perform all additions and subtractions, working from left to right.

The above problem was solved correctly by Student 2 since she followed Rules 2 and

3.

Let's look at some Exercises of solving arithmetic expressions using these rules.

Student 1 Student 2

3 + 4 2= 3 + 4 2=

= 7 2 = 3 + 8

= 14 = 11
http://x2826022670%28%27arithmetic_operation%27%29/http://x2826022670%28%27arithmetic_operation%27%29/


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Exercise7.Evaluate each expression using the rules for order of operations.

Order of Operations

Expression Evaluation Operation

6 + 7 8 = 6 + 7 8 Multiplication

= Addition

=

16 82 = Division

= Subtraction

=

(25 11) 3 = Parentheses

= Multiplication

=

In Exercise 7, each problem involved only 2 operations. Let's look at some Exercises

that involve more than two operations.

Exercise 8.Evaluate 3 + 6 (5 + 4) 3 7 using the order of operations.

Exercise 9. Evaluate 9 5 (8 3) 2 + 6 using the order of operations.

Solution: Step 1: . = .. Parentheses

Step 2: = Division

Step 3: = Multiplication

Step 4: = Subtraction

Step 5: = 13 Addition

Solution: Step 1: 3 + 6 (5 + 4) 3 7 = . Parentheses

Step 2: = Multiplication

Step 3: = Division

Step 4: = Addition

Step 5: = 14 Subtraction


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In Exercise 8 and 9, you will notice that multiplication and division were evaluated from

left to right according to Rule 2. Similarly, addition and subtraction were evaluated from

left to right, according to Rule 3.

When two or more operations occur inside a set of parentheses, these operations

should be evaluated according to Rules 2 and 3.

Exercise 10.Evaluate 150 (6 + 3 8) 5 using the order of operations.

Solution: Step 1: = Multiplication inside Parentheses

Step 2: = Addition inside Parentheses

Step 3: = Division

Step 4: = Subtraction

Exercise 11. Write an arithmetic expression for this problem. Then evaluate the

expression using the order of operations.

Mr. Smith charged Jill 32 for parts and 15 per hour for labour to repair her bicycle. If

he spent 3 hours repairing her bike, how much does Jill owe him?Solution:

Jill owes Mr. Smith:

Summary: When evaluating arithmetic expressions, the order of operations is:

Simplify all operations inside parentheses.

Perform all multiplications and divisions, working from left to right.

Perform all additions and subtractions, working from left to right.


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Exercise 12. Evaluate 3 + 6 (5 + 4) 3 7 using the order of operations.

(fill up the gaps)

Step 1: 3 + 6 (5 + 4) 3 7 = (Bracket)

Step 2: 3 + 6 ___ 3 7 = (Multiplication)

Step 3: 3 + ___ 3 7 = (Division)

Step 4 : 3 + ____7 = ( Addition and subtraction)

Exercise 13. Evaluate 9 5 (8 3) 2 + 6 using the order of operations.

In Exercise 8 and 9, you will notice that multiplication and division were evaluated from

left to right.

Exercise 14.Evaluate 150 (6 + 3 8) 5 using the order of operations.

Exercise 15. There are 365 days in a year, and each day has 24 hours in it. Howmany hours are there in a week?

Exercise 16.After a school trip, there's 532 left over. The school decides to share

this out between the 38 students on the trip. How much does each student receive?

Exercise 17.A farmer has 630 eggs. They are to be placed in trays. Each tray holds48 eggs. How many trays can be filled?

DIVISION

Multiplication:

In multiplication two numbers are given and we find their product as:

5 15 = ?

5 15 = 75


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Division:

We can rewrite the above multiplication as a division problem as:

What number times 15 equals 75?

? 15 = 75

This is a division problem.

- Division is also called the inverse of multiplication. To find the

answer we have to divide as:

? = 75 15

Here we have to divide 75 into groups of 15

75 15 = 5 75 divided by 15 equals 5

- Else we could subtract 15 from 75 five times.

Multiplication is repeated addition and division is repeated subtraction.

15 is called the divisor.

75 is called the dividend. It is the number being divided.

4 is called the quotient.

In a division problem:

1. The number that is divided is called the dividend.

2. The number that divides the dividend is called divisor.

3. The number of times the dividend is divisible by the divisor is called the quotient.4. If the dividend is completely not divisible by the divisor it leaves behind a

remainder.

Exercise 18. How do you check if the division is right?

Dividend = Quotient Divisor + Remainder

Here the remainder is 0.

Hence 8 = 4 2

Answer: Multiply the ..............and the ...................

8 2 = 4

Quotient

DivisorDividend

Quotient

DividendDivisor4

82


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Question 1: To check if a division problem was solved correctly,

multiply the quotient and the divisor and add the remainder.

multiply the quotient and the dividend and subtract the remainder.

multiply the remainder and the divisor and add the quotient.

Question 2: In the problem 30 5 = 6, what is the divisor?

6

5

30

Question 3: If a division problem has no remainder,

quotient = dividend divisor

divisor = dividend quotient

dividend = quotien divisor

Question 4: In the problem 30 5 = 6, what is the quotient?

6

30

5

Question 5: In the problem 30 5 = 6, what is the dividend?

5

630

Question 6: The number you are dividing by is called:

quotient

dividend

divisor

remainder

Question 7: The number that is remaining after division problem is called:

quotient

remainder

dividend

divisor

Question 8: In the problem 30 5 = 6, what is the answer?

5

30

6


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ROUNDING NUMBERS Rounding makes numbers that are easier to work with in your head.

Rounded numbers are only approximate.

An exact answer generally can not be obtained using rounded numbers.

Use rounding to get an answer that is close but that does not have to be exact.

How to round numbers

1.Make the numbers that end in 1 through 4 into the next lower number that ends in 0.

2.Numbers that end in a digit of 5 or more should be rounded up to the next even ten.

For example 74 rounded to the nearest ten would be 70.

The number 88 rounded to the nearest ten would be 90.

Exercise 19.Fill in the gaps.

424 rounded to the nearest hundred is ..........

988 rounded to the nearest hundred is ..........

6424 rounded to the nearest thousand is . .........

8788 rounded to the nearest thousand is ...........

DECIMALS ORDER

We use a decimal point to separate units from parts of a whole (tenths, hundredths,

thousandths etc).

A tenth is 1/10 of a unit

A hundredth is 1/100 of a unit

A thousandth is 1/1000 of a unit

In the number 34,27, the value of the figure 2 is a tenth, and the value of the figure 7 is

a hundredth.

Ordering decimals

Hundreds are greater than tens, tens are greater than units, units are greater than

tenths, and tenths are greater than hundredths!


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When ordering numbers, we should always compare the digits on the left first.

For example, which is greater: 2,701 or 2,71?

Units Tenths Hundredths Thousandths

2 7 0 1

2 7 1 0

Both numbers have two units and seven tenths, but 2.701 has no hundredths, whereas

2.71 has one hundredth. Therefore, 2.71 is greater than 2.701.

Another way to look at it is to add a zero to the end of 2.71 (this does not change its

value, because it is after the decimal point!).

The two numbers are now 2.710 and 2.701. It is quite easy to see that 2.710 is bigger

(just as 2710 is bigger than 2701).

Exercise 20. Write decimal numbers:

a) Describe the number 0,2 in words.

b) Describe the number 2,48 in words.c) Describe the numbers 0.345 in words.

d) Describe the numbers 1,8 in words.

e) Describe the numbers 2,005 in words.

ADDING AND SUBTRACTING DECIMALS

Adding and subtracting decimals is easy. Simply add or subtract as normal, but make

sure that you keep the decimal points aligned.

Exercise 21.Solve

a) Four and twenty seven hundredths plus two and three tenths.

b) Fifty two and seven hundredths minus twenty seven and three tenths.

c) Twenty times one hundred.

d) Eight times two hundred and twenty four.


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MULTIPLYING AND DIVIDING DECIMALS

Multiplying and dividing by 10, 100, 1000 .You need to learn the following rules:

Action Rule Example

Multiply by 10 Move the decimal point one place to the right 2.45 10 = 24.5

Multiply by 100 Move the decimal point two places to the right 2.45 100 = 245

Multiply by 1000 Move the decimal point three places to the right 2.45 1000= 2450

Divide by 10 Move the decimal point one place to the left 46.7 10 = 4.67

Divide by 100 Move the decimal point two places to the left 46.7 100 = 0.467

Divide by 1000 Move the decimal point three places to the left 46.7 1000=0.0467

Exercise 22.Products and division. Solve:

a) 6.23 1000

b) 1,2 100

c) 115.1 1000

d) 125 100

e) 1000 22,2

Ejercicio 23.En cada uno de los casos, escribe el resultado con letras en ingls:

1) 1.1 + 2.4 = 5) 3.07 10 =

2) 3.25 + 1.04 = 6) 2.5 100 =

3) 5.26 1.18 = 7) 2.31 2 =

4) 3.75 + 1.23 2.41 = 8) 0.8 1.2 =

5) 4.2 0.07 = 9) 6.074 2 =


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Exercise 24.Write down the missing numbers.

a) 15 = 45 f) 50 200 =

b) 297 = 2,97 g) 70 = 4900

c) 741 1000 = h) 140 30 =

d) 3,5 = 3500 i) 60 = 3600

e) 14,2 10 = j) 0,12 = 0,0012

Exercise 25.A builder requires 65 pieces of expensive timber. The Timber costs

$126 a piece What is the total cost of the timber?

Exercise 26.Use the following answer to work out the questions below

5300 120 = 636000

a) 53 1,2 = f) 0,53 1,2 =

b) 53 120 = g) 530 0,12 =

c) 1200 0,53 = h) 53 120000 =

d) 53 12 = i) 0,053 12 =

Exercise 27.Each of these expressions can work out to 20 with brackets in the right

places. Write the expressions with bracket.

a) 30 6 4 = b) 26 3 + 3 = c) 10 7 17 =

d) 4 + 6 6 2 = e) 15 + 25 9 7 = f) 5 24 6 =

Exercise 28.Work out

a) 20 5 (2 + 3) 2 + 6 = b) 15 2 ( 7 4 ) 7 =

c) ( 2 + 6 ) 4 5 2 = d) 3 12 4 ( 10 8 ) =

e) 2 0 15 3 = f) 2 + 5 7 =

g) 3,2 100 0,2 5 = h) 1,22 0,022 10 =


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The numbers at the Start of the maze can only survive if they manage to find a route tothe Exit which has transformed them into the numbers shown.

Start numbers are 10, 4, 6 and 3

Exit numbers are 9, 3.5, 12,20,8 and 14

Which numbers manage to survive the maze?

START

EXIT


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ORDINAL NUMBERS

Ordinal numbers refer to a position in a series. Common ordinals include:

10th tenth

1st first 11th eleventh

2nd second 12th twelfth (note " f " , not "v" )20th twentieth

3rd third 13th thirteenth 30th thirtieth

4th fourth 14th fourteenth 40th fortieth

5th fifth 15th fifteenth 50th fiftieth

6th sixth 16th sixteenth 60th sixtieth

7th seventh 17th seventeenth 70th seventieth

8th eighth (on ly one " t " )18th eighteenth 80th eightieth

9th ninth (no " e" ) 19th nineteenth 90th ninetieth

Ordinal numbers such as 21st, 33rd, etc., are formed by combining a cardinalten with

an ordinalunit.

21st twentyfirst 58th fiftyeighth 83rd eightythird

25th twentyfifth 64th sixtyfourth 99th ninetyninth

32nd thirtysecond 79th seventyninth
http://en.wikipedia.org/wiki/Ordinal_number_(linguistics)http://en.wikipedia.org/wiki/Tenthhttp://en.wikipedia.org/wiki/Firsthttp://en.wikipedia.org/wiki/Eleventhhttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Twelfthhttp://en.wikipedia.org/wiki/Twentiethhttp://en.wikipedia.org/wiki/Thirdhttp://en.wikipedia.org/wiki/Thirteenthhttp://en.wikipedia.org/w/index.php?title=Thirtieth&action=edit&redlink=1http://en.wikipedia.org/wiki/Fourthhttp://en.wikipedia.org/w/index.php?title=Fourteenth&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Fortieth&action=edit&redlink=1http://en.wikipedia.org/wiki/Fifthhttp://en.wikipedia.org/wiki/Fifteenthhttp://en.wikipedia.org/w/index.php?title=Fiftieth&action=edit&redlink=1http://en.wikipedia.org/wiki/Sixthhttp://en.wikipedia.org/wiki/Sixteenthhttp://en.wikipedia.org/w/index.php?title=Sixtieth&action=edit&redlink=1http://en.wikipedia.org/wiki/Seventhhttp://en.wikipedia.org/wiki/Seventeenthhttp://en.wikipedia.org/w/index.php?title=Seventieth&action=edit&redlink=1http://en.wikipedia.org/wiki/Eighthhttp://en.wikipedia.org/w/index.php?title=Eighteenth&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Eightieth&action=edit&redlink=1http://en.wikipedia.org/wiki/Ninthhttp://en.wikipedia.org/w/index.php?title=Nineteenth&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Ninetieth&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Ninetieth&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Nineteenth&action=edit&redlink=1http://en.wikipedia.org/wiki/Ninthhttp://en.wikipedia.org/w/index.php?title=Eightieth&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Eighteenth&action=edit&redlink=1http://en.wikipedia.org/wiki/Eighthhttp://en.wikipedia.org/w/index.php?title=Seventieth&action=edit&redlink=1http://en.wikipedia.org/wiki/Seventeenthhttp://en.wikipedia.org/wiki/Seventhhttp://en.wikipedia.org/w/index.php?title=Sixtieth&action=edit&redlink=1http://en.wikipedia.org/wiki/Sixteenthhttp://en.wikipedia.org/wiki/Sixthhttp://en.wikipedia.org/w/index.php?title=Fiftieth&action=edit&redlink=1http://en.wikipedia.org/wiki/Fifteenthhttp://en.wikipedia.org/wiki/Fifthhttp://en.wikipedia.org/w/index.php?title=Fortieth&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Fourteenth&action=edit&redlink=1http://en.wikipedia.org/wiki/Fourthhttp://en.wikipedia.org/w/index.php?title=Thirtieth&action=edit&redlink=1http://en.wikipedia.org/wiki/Thirteenthhttp://en.wikipedia.org/wiki/Thirdhttp://en.wikipedia.org/wiki/Twentiethhttp://en.wikipedia.org/wiki/Twelfthhttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Eleventhhttp://en.wikipedia.org/wiki/Firsthttp://en.wikipedia.org/wiki/Tenthhttp://en.wikipedia.org/wiki/Ordinal_number_(linguistics)


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AUTOEVALUACIN Matemticas UNIT I

Ejercicio 1. Explica cmo se aproximan por redondeo los nmeros naturales y

decimales. Aydate de ejemplos.

Exercise 2.Write the following numbers:

a) Twelve and fifteen thousandths.

b) Seven hundredths.

c) Eight and nine thousandths.

Ejercicio 3. En una recta numrica ,explica la forma de situar los siguientes puntos:

1,2 ; 1,45 ; 1,03.

Exercise 4.If olive oil cost 3.15 per litre How much would you pay for 0.75 litres?.

Exercise 5.Alfredo and Vanessa buy in the store:

- five cartons of milk at 1,05 per carton

- a 3,5 Kg of cod at 13.25 / Kg.

- A package of cookies for 2.85

What was the total amount paid?

Exercise 6.Manuel has bought 1.25 Kg of apples and 0.750 Kg. of strawberries. If

the strawberries are 1,45 /Kg and the apples are 1.30 /Kg How much change will he

receive if he paid with a 50 bill?

Ejercicio 7.Solve:

a) 6.4 2 (127)= b) 21:(3+4)+6= c) 8,01 2 4,11=

d) 5 ( 0,8 + 0,6)= e) 1,9 + 2 ( 1,3 0,7) = f) (11,210,05)220,21=

Exercise 8.A shopowner buys one hundred twenty five dresses at thirteen point

twenty euros per dress. At what price should the shopowner sell the dresses if five

dresses will not be sold and he wants to make two hundred fifty euros profit?

Ejercicio 9.En una granja, entre caballos y ovejas, hay 847 cabezas. Sabiendo que

hay 31 caballos y que el nmero de vacas supera al de caballos en 108 unidades

cul es el nmero de ovejas?


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REFUERZO Matemticas UNIDAD I

Nombre y Apellidos 1 ___

Fecha de entrega:

Atencin: todos los ejercicios deben realizarse en DIN A4 con bolgrafo azul o

negro.

Copia todos los enunciados y en los problemas explica el procedimiento

seguido. Utiliza una hoja, que debes entregar, para realizar todas las

operaciones. (No debes utilizar calculadora)

Ejercicio 1.Calcular:

a) 20,23 100 = b) 12,04 1000 = c) 13,345 100 = d) 0,234 100 =

e) 0,23 : 10 = f) 12,3 : 100 = g) 234 : 100 = h) 89,2 : 10 =

Ejercicio 2. Explica cmo se redondean nmeros naturales y decimales. Utiliza

ejemplos.

Ejercicio 3. Trece amigos ganan un premio de 12831 Cunto dinero le

corresponden a cada uno si todos apostaron igual cantidad?

Ejercicio 4.En una recta numrica ,explica la forma de situar los siguientes puntos:

2,3 ; 3,05 ; 3,99.

Ejercicio 5.Si el litro de aceite de oliva virgen est a 3,42 /litro cunto costarn

0,25 litros?

Ejercicio 6.Vanesa y Kevin compran en el sper:

- cinco cartones de leche a 0,97 el cartn

- 2 Kg. de pechuga de pollo a 7,99 el Kg.

- dos cajas de cola 0,45 e la lata, sabiendo que cada caja contiene doce latas

- 2 Kg. de garbanzos a 1,43 / Kg.

Cunto pagan a la cajera del supermercado?


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Ejercicio 7. Jos ha comprado 1,5 Kg. de manzanas y 0,85 Kg. de peras, 2,5 Kg. de

cerezas. Teniendo en cuenta que las cerezas estn a 2,35 /Kg., las manzanas a

1,25/Kg. y las peras 1,25/Kg Cunto le devolvern si paga con un billete de

20?Sabras expresar las operaciones como una nica operacin combinada?

Ejercicio 8.Calcular detallando el proceso seguido:

a) 3 4 2 (21 17) = b) 20 2 (5 4) =

c) 3( 23 12) + 2 ( 21 18) = d) 3 2,2 2 ,2 (2,1 1,7) =

e) 2,3 2, 01 1,2 (1, 5 1, 23 ) = f) 3 ( 2,3 1,2) =

Ejercicio 9. Propiedad de la divisin Al multiplicar el dividendo y el divisor por el

mismo nmero, el cociente no vara Cundo la aplicamos y para qu? Aydate de

ejemplos.

Ejercicio 10. Luis ha comprado una camiseta y unos pantalones por 52 . Si los

pantalones le han costado el triple de la camiseta Cunto le ha costado cada prenda?

Ejercicio 11.Calcula:

a) 0,004 0, 7 = b) 0,23 0, 3 = c) 0,008 0,5 = d) 1,2 0, 4 =

e) 0,008: 0,002 = f) 12 : 0,3 = g) 12 : 0,5 = h) 60 : 0,2 =

Ejercicio 12.Escribe los siguientes nmeros:

a) Dos unidades y cinco milsimas. b) Doce centsimas

c) Diez unidades y ocho centsimas d) 1,023

e) 0,07 g) 23, 4

Ejercicio 13.Un meln est a 59 cntimos el kg. Cunto pagars por un meln de

2,45 Kg.?

Ejercicio 14.Calcula el cociente con dos cifras decimales:

a) 12 : 0,9 = b) 4 : 0,25 c) 15 : 18,3 = d) 4,6 : 1,23 f) 2,385 : 6,9 =

Ejercicio 15.Calcular detallando el proceso seguido:

a) 314 3 (11 7) + 23= b) 20 3 (7 4) + 4= c) (23 12)2+ 23=

d)32 (2,1 1,7) = e) 1,32, 05 1,5(1,58 1,23) = f) 5( 24 10) 5 2

=


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INTEGERS UNIT II

RECOGNISING NEGATIVE NUMBERS

You have to be able to use negative (or directed) numbers in many everyday contexts,for example with temperatures, bank balances etc.

A temperature of 6 degrees below zero is 6.

If your bank account is 100 overdrawn, then your balance is 100 .

In your exam, you might be asked to put some numbers in order. You have to

remember, for instance, that 10 is less than 5.

Sometimes it helps to see negative numbers on a number line.

9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8

Exercise 1.Order the numbers 5, 2, 3, 9, 2.5, 0.5.

The lowest number in this list is .............

The highest number is ...........................

From lowest to highest, the order is..............

Negative numbers and temperature

You need to be able to work out rises and falls in temperature using negative numbers.

9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8

A number line can help us to understand changes in temperature. For example, if the

overnight temperature fell to 4C, and by midday it had risen to 7C, then, counting

from 4 up to 7 on the number line,

Negative Positive


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9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8

the temperature has gone up by 11C

Exercise 2.At 6 pm, the temperature was 3C. By midnight, it had dropped to5C.

How great was the fall in temperature?

Exercise 3.At 7 am, Joe recorded the temperature in his garden as being 4C.

He went back out outside at 1 pm and found that the temperature had increased by

12C. What was the temperature at 1 pm?

Exercise 4. a.At 6 pm, the outside temperature was 7C. By 6 am, the temperature

had dropped by 13C. What was the temperature at 6am?

b. At 10am the temperature was 2C. How much did the temperature

increase by?

c. At 5 pm the temperature increased by 5C. What was the

temperature at 5 pm?

d. At 8 pm the temperature dropped by 5C. What was the temperature

at 8 pm?

Exercise 5.Here is a table showing temperatures in cities worldwide:

City Barcelona London New York Moscow Oslo Paris Sydney

Temp 17C 5 C 7 C 9 C 5C 12C 23C

What is the difference in temperature between Moscow and Oslo? And between

Sydney and Oslo?

What was the difference in temperature between Barcelona and Moscow? How much

higher was the temperature in London than the temperature in Oslo?


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ADDING POSITIVE AND NEGATIVE INTEGERS.

If one of the integers is negative, subtract the absolute value of it from the other

number.

Example: 14 + ( 6) = 14 6 = 8

If both of the integers are negative, add their absolute values and prefix the number

with a negative sign.

SUBTRACTING NEGATIVE AND POSITIVE INTEGERS.

To subtract integers, change the sign on the integer that is to be subtracted.

If both signs are positive, the answer will be positive.

Example: 14 (6) = 14 + 6 = 20

If both signs are negative, the answer will be negative.

Example: 14 (+6) = 14 6 = 20

If the signs are different subtract the smaller absolute value from the larger

absolute value. The sign will be the sign of the integer that produced the larger

absolute value.

Example: 14 (+6) = 14 6 = 8

Example: 14 (6) = 14 + 6 = 8

MULTIPLICATION OF TWO AND THREE INTEGERS

Multiplication of Integers is similar to multiplication of whole numbers (both positive)

except the sign of the product needs to be determined.

If both factors are positive, the product will be positive.

If both factors are negative, the product will be positive.

If only one of the factors is negative, the product will be negative.

In other words, if the signs are the same the product will be positive, if they are

different, the product will be negative.


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DIVISION OF NEGATIVE AND POSITIVE INTEGERS

Division of Integers is similar to division of whole numbers (both positive) except the

sign of the quotient needs to be determined.

If both the dividend and divisor are positive, the quotient will be positive.

If both the dividend and divisor are negative, the quotient will be positive.

If only one of the dividend or divisor is negative, the quotient will be negative.

In other words, if the signs are the same the quotient will be positive, if they are

different, the quotient will be negative.

Exercise 6.Work out:

a) 1000 (2) = b) 1000 9000 = c) 12 12 =

d) 12 + 12 = e) (200) ( 4) = f) (144) 12 =

g) (125) (5) = h) 1 1 = i) 1 2 2 =

j) (12 ) (12) = k) 40 (4) = l) 10 + (10) =

ABSOLUTE VALUEThe absolute value of a number is just the value of the numeral, ignoring the sign, that

is, the distance the number is from zero on the number line.

Symbol .... Written as 44 and 44

Exercise 7.Use < (less than), equals(=) or > ( greater than) to compare each set of

numbers.

10..........10a) C10C........25b)

0...........12c) 75.........57d)

112............112e) 57............75f)


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Exercise 8.Write down the following operations and solve:

a) Negative four times, open bracket seven minus eight closed brackets, plus eighteen

divided by three equals.

b) Four times open brackets twelve minus sixteen closed brackets, minus twenty five

divided by five equals.

c) Negative twenty divided by four plus three times open brackets eight minus

seventeen closed brackets equals.

d) Open brackets, negative two, close bracket times open square bracket, open

bracket, negative five, close bracket minus open bracket negative two close bracket,

close square bracket equal.

Exercise 9. A painter is on the middle step of a ladder.

This means that there are an equal number of steps above and below the painter. If the

painter goes down 4 steps, then up 7 steps, and then down 13 steps, he will be on the

first step of the ladder. How

many steps does the ladder

have all together?

Exercise 10.A group of 68 students went to the amusement park.

Some travelled by car while others rode the school bus .On the way to the amusement

park, 41 students rode in the bus and 3 students rode in each car. During the trip home4 students rode in each car. How many students came home by bus?

Exercise 11. Look at the dates of birth and deaths of the following people:

Pythagoras de Samos (582 507 BCE.), Cleopatra (69 30 BCE), Alejandro Magno

(356 323 BCE), Herodes Antipas (20 BCE, 39), Miguel ngel (1475 1564), San

Francisco de Ass (1182 1226) y Gustavo Adolfo Bcquer (1836 1870)

a) Order them by least amount of years lived to most.b) Chronologically
http://es.wikipedia.org/wiki/A%C3%B1os_580_a._C.http://es.wikipedia.org/wiki/A%C3%B1os_500_a._C.http://es.wikipedia.org/wiki/A%C3%B1os_500_a._C.http://es.wikipedia.org/wiki/A%C3%B1os_580_a._C.


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Ejercicio 12. Calcular, paso a paso:

983)(a) 111842b)

4119c) 10791d)

64284)e) 462752e)

94102)(831f)

DIRECTED NUMBERS (1)

1) 4 1 = 21) 3 9 = 41) 6 + (9) =

2) 6 4 = 22) 8 ( 8) = 42) 5 (8) =

3)1 6 = 23)1 2 = 43) 4 + (11) =

4) 4 + 1 = 24) 2 + 2 = 44)7 + (12) =

5) 4 1 = 25) 8 + 4 = 45)2 (4) =

6)1 ( 2) = 26) 6 11 = 46)1 + (8) =

7) 5 12 = 27) 9 + 12 = 47)6 (19) =

8) 4 7 = 28) 3 (6) = 48)7 + (8) =

9)10 + 15 = 29) 2 9 = 49)4 (5) =

10) 17 17 = 30) 8 7 = 50) 3 + (11) =

11) 815 = 3) 4 6 = 51) 4 + (6) =

12) 5 (5) = 32) 9 + 7 = 52) 6 (18) =

13) 14 12 = 33)13 9 = 53) 4 (3) =

14)10 18 = 34)11 + 13 = 54)8 (5) =

15) 9 5 = 35) 10 (9) = 55)3 + (9) =

16) 2 6 = 36) 14 18 = 56)5 + (18) =

17)3 ( 7) = 37)14 + 18 = 57) 3 + (2) =

18)17 7 = 38)14 ( 18) = 58) 5 (4) =

19) 10 17 = 39) 14 18 = 59)3 + (17) =

20)13 + 15 = 40)18 14 = 60)8 (9) =


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DIRECTED NUMBERS (2)

1) 5 2 = 21) 2 7 = 41) 8 (9) =

2) 9 ( 4) = 22) 1( 5) = 42) 1 (+8) =

3) 10 ( 6) = 23)18 + 21 = 43) 4 (11) =

4) (4) (+10) = 24) (12) ( 2) = 44) 10 + (2) =

5) 4 1 = 25) 8 + 4 = 45)2 (24) =

6)10 ( 7) = 26) ( 6) ( 11) = 46) (2) (11) =

7) 5 12 = 27) 3 + 1 = 47) (6) (9) =

8) 3( 80) = 28) (3) ( 8) = 48)3 + (3) =

9)1 15 = 29)1 9 = 49) 4 (8) =

10) 17 (17) = 30) 8 ( 7) = 50) 3 (11) =

11)8 (5) = 31)4 (+ 6) = 51) (4) (6) =

12) 5 (5) = 32) ( 9) (+ 7) = 52)6 (8) =

13)1 (12) = 33)13 9 = 53) (4) (3) =

14) (10) (18) = 34)11 + 13 = 54) (8) (5) =

15)9 500 = 35) 1000 ( 9) = 55) (3) (9) =

16) ( 2 ) (+600) = 36) 14 18 = 56) (5) (8) =

17)3 ( 7) = 37)14 + 18 = 57) 3 + (2) =

18)17 7 = 38)14 ( 18) = 58) 5 (4) =

19) 10 17 = 39) 14 18 = 59)3 + (17) =

20)13 + 15 = 40)18 14 = 60) 8 (9) =


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AUTOEVALUACIN Matemticas UNIT II

Ejercicio 1. Contesta razonadamente:

a) Define valor absoluto de un nmero. Qu smbolos utilizamos? Aydate de

ejemplos.

b) Qu smbolo utilizamos para designar el conjunto de los nmeros enteros? Qu

entiendes por opuesto de un nmero entero?

c) Escribe dos nmeros enteros distintos que tengan el mismo valor absoluto.

Exercise 2. There are two opposite integers that are 10 units apart on the number

line What are these two number?

Exercise 3.Order the following integers from least to greatest (>


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a) +10 ( 8) = e) ( 2 + 4 ) ( +1 3) =

b) ( +10) ( 8) = f) (10 21) ( 2 + 3 ) =

c) (10) + ( 8 ) = g) + ( 21 + 8 ) ( +10 30) =

d) ( 8 + 9) ( +2) = h) (111 ) ( +1+1+1 ) =

Ejercicio 8. Calcular

a) ( + 800 ) ( 2) = b) ( 2000) (+100) = c) ( 24 ) ( + 3) =

d) ( 800 ) ( 2) = f) ( +2000) ( 100) = g) ( 24 ) +( +3) =

h) ( 1000 ) ( 4) = i ) ( 25) ( + 5) = j) ( 56 ) ( 8) =

k) ( 800 ) ( 2) = l) ( 2000) + ( + 100) = m) ( 24 ) ( + 3) =

Ejercicio 9. Calcular, paso a paso:

252)(a) 61236b)

2108c) 106411d)

64184)e) 483754f)

9222)(1032f)

Exercise 10.The thermometer in a walkin refrigerator reads 5 C. The thermostat is

changed in order to lower temperature decreases 2 C every 5 minutes, what will the

temperature be in one hour?

Ejercicio 11. Hemos recibido un mensaje en clave formado por los siguientes

nmeros {1, +2, 5, +7, 10, 4, 8, 1, +5}. Sabemos que el significado de dicho

mensaje es el resultado de calcular el valor absoluto de la suma de dichos nmeros.

Cul es el significado del mensaje?


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REFUERZO Matemticas UNIDAD II


Fecha de entrega:


negro.




Ejercicio 1.Completa.

El conjunto de los nmeros enteros se representa por la letra ___ , y est formado

por:

Los nmeros, _________________ que son los positivos +1, +2, +3, +4, ..

El cero 0

Los nmeros ______________ _______

Ejercicio 2. Asocia un nmero, positivo o negativo segn corresponda a cada uno de

los enunciados:a) La peluquera est en el 4 piso ______

b) Mi coche est en el stano n 2 ______

c) Debo en el banco 123 Euros __________

d) Un termmetro marca 4 bajo cero______

e) Hoy han cado 8 litros de agua por m2 ______

f) Tengo 4 euros en la cartera y 2 euros en el bolsillo_________________

g) He perdido 2 euros __________

h) El ascensor sube 6 plantas _________

Ejercicio 3. Representa en una recta numrica los siguientes nmeros enteros:

a) 3, 0, +4, 6, +6, 7, +1, 1, +3

b) Ordena todos esos nmeros anteriores de menor a mayor (recuerda que tienes que

colocar el signo).

Ejercicio 4. Define que es el valor absoluto de un nmero y pon ejemplos (no te

olvides en los ejemplos de poner las rayas verticales).


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+ + = ++ =

+ = = +

Ejercicio 5.Define quien es el opuesto de un nmero entero. Pon ejemplos.

Ejercicio 6.Escribe cinco elementos ms en las siguientes series numricas:

a) 0, 11, 2, 2, .........

b) 6, 4, 2, 0, 2, ........

c) 8, 4, 0, .........


a) +5 + 8 = b)3 9 = c) + 12 + 14 = d) +2 + 1 + 7 =

e) + 3 + 9 = f) + 6 + 1 + 14 = g) 4 6 3 2 = h) 46 50 =

i) 5 8 = j) + 4 + 6 + 3 + 2 = k) 25 30 5 = l) 6 1 14 =

m) 2 1 7 = n) 4 9 = ) 12 14 = o) 4 + 3 5 =

Ejercicio 8.Efecta las siguientes operaciones:

a) 7 + 5 = d) 5 9 = g) 5 + 6 8 =

b) 3 6 = e) + 9 + 2 = h) + 6 + 4 12 =

c) 8 + 12 = f) + 6 9 = i) 20 40 60 =

Ejercicio 9.Quita parntesis y calcula:

a) ( + 12 ) + ( + 15 ) = e) ( +8 ) ( +6 ) =

b) ( 14 ) + ( + 4 ) = f) ( 18 ) ( +10 ) =

c) (14 ) + ( 2 ) = g) ( 50) ( 10) =

d) ( +30 ) + ( 45 ) = h) ( +8 ) ( 12 ) =

Ejercicio 10.Quita parntesis y calcula:

a) ( 8) ( 4) + ( 6) ( + 2) ( 9) =

b) ( + 7 ) ( + 5 ) + (11 ) + ( +4 ) =

c) ( + 15 ) + ( 13 ) ( +12 ) ( 10) =d) ( 2) ( 8) + ( 4) ( 6) + ( 7) =

e) ( +12 ) (14 ) ( +16 ) (20 ) =

f) ( 4) + ( 8) ( +7 )+ ( +16 ) =

Recuerda que para multiplicar y/o dividir nmeros enteros,

se multiplican y/o dividen como los naturales y se aplica la

regla de los signos:


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Ejercicio 11.Calcula los siguientes productos:

a) (+4)(+5) = b) (+4)( 5) = c) ( 4)(+5) = d) (4)(5 ) =

e) (+12)(+3) = f) (+12)( 3) = g) (12)(+3) = h) ( 12)( 3) =

Ejercicio 12.Calcula los siguientes cocientes:

a) (+24) : (+3) = b) (+24) : ( 3) = c) ( 24) : (+3) = d) ( 24) : ( 3) =

e) (+120) : (+10) = f) (+120) : (10) = g) (120) : (+10) = h) (120) : (10) =


a) (7) (3) = b) (+2) (5) = c) (+4) (+3) =

d) (1) (8) = e) (2) (7) (1) = f) (+3) ( 2) ( 4) =

g) (+28) : (+4) = h) (21) : (3) = i) (3) : (3) =

j) (+35) : (7) = k) 30 : (6) = l) ( 48) : 6 =

Operaciones combinadas

A la hora de resolveroperaciones combinadas, ten en cuenta:

1 Resolver los parntesis o corchetes.

2 Hacer las multiplicaciones y las divisiones.

3 Hacer las sumas y las restas.

Ejercicio 14.Realiza:

a) 5( 4 9 ) = d) 48 : ( 8 4 ) = g) ( 3)[ ( 5 ) + ( 4 )]=

b) 6( 12 + 8 ) = e) 9[ ( 7) ( 3) + (+1 )]= h)50: [(5)+(5)] =

c) 12 : ( 6 8) = f) )2(:10)(4)(6)( =

Ejercicio 15. Expresa ayudndote de una recta numrica las siguientes situaciones:

a) Tengo en mi cuenta 12 euros, pero me llega una factura de 15 euros. En qu

situacin estoy?

b) El ascensor est en el tercer stano y ha subido cinco plantas. Dnde se

encuentra?

c) Ayer, la temperatura a las nueve de la maana era de 4 C. A medioda haba

subido 6 C ms, a las cinco de la tarde marcaba 5 C ms, a las nueve de la noche

haba bajado 7 C y a las doce de la noche an haba bajado otros 4 C.

Qu temperatura marcaba el termmetro a las doce de la noche?


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Ejercicio 16. Carlomagno naci en el ao 742, Pitgoras en el ao 580 a.n.e.,

Sneca en el ao 3 a.n.e. y Tiberio en el ao 42 a.n.e. Ordnalos por antigedad, del

ms antiguo al ms moderno.

Ejercicio 17.Euclides naci en el ao 300 a.n.e. y Pitgoras en el ao 580 a.n.e.

Cul de ellos naci antes? Cuntos aos antes?

Ejercicio 18.Ayer a las 8 h de la tarde el termmetro marcaba 2C. A las 12 h de la

noche la temperatura descendi 5C. Qu temperatura marc el termmetro a las 12h

de la noche?

Ejercicio 19 La pirmide de Keops se termina de construir aproximadamente hacia el

ao 2.600 a.n.e. Cuntos aos han transcurridos desde su terminacin?

Ejercicio 20. Pitgoras naci en el ao 580 a.n.e y muri en el ao 501 a.n.e.

Cuntos aos vivi Pitgoras?


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POWERS UNIT III

9 is a square number.

3 3 = 93 3 can also be written as 32 (3 squared)

8 is a cube number.

2 2 2 = 8

2 2 2 can also be written as 23 (2 cubed)

The notation 32 and 23 is known as index form. The small digit is called the index

numberorpower.We have already seen that 32 = 3 3 = 9, and that 23 = 2 2 2 = 8.

Similarly, 54 (five to the power of 4) = 5 5 5 5 = 625, and

35 (three to the power of 5) = 3 3 3 3 3 = 243.

In each case, the index number tells us how many times we should multiply the

numbers together.

When the index number is two, we say 'squared'.

When the index number is three, we say 'cubed'.

When the index number is greater than three we say 'to the power of'.

For example:

72 is 'seven squared',

33 is 'three cubed',

37 is 'three to the power of seven',

45 is 'four to the power of five'.

All scientific calculators have a 'power' button [xy]. This is particularly useful when the

index number is large. (After all, you're very likely to make a mistake if you try to

calculate 410 by typing 4 4 4 4 4 4 4 4 4 4 into your calculator!).

Instead, type

4 shift xy

10 =


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Exercise 1.Copy and complete the following table then check your answers.

The first has been done for you!

Exercise 2.Use your calculator to find the values of the following:

a) 211 b) 58 c) 26 35

INDEX LAWS FOR MULTIPLICATION AND DIVISION.MULTIPLICATION

23 = 2 2 2

so 23 25 = 2 2 2 2 2 2 2 2 = 28

25 = 2 2 2 2 2

Can you see what happened? We had 3 twos from 23 and 5 twos from 25, so altogether

we had 8 twos.

In general

Four cube

43

4 4 4

64

Two to the power of seven 128

104

5555

122

2222

1010

64

27

1000

2m 2n =2(m + n)


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Exercise 3.

a) 25 24 = b) 27 23 = c) 34 32 = d) 256 254=

DIVISION

If we divide 25 by 23:

So 25 23 = 22

In general

Exercise 4.

a) 25 22 = b) 27 23 = c) 510 53 = d) 459 454 =

Exercise 5.Fill in the gaps.

powers base exponent Expanded Form value

1)Three squared

Three to the power of two32 3 2 33 9

2)2 4

3) 2 64

4)444

5) Ten to the power of four.

6)1000

7) 3 27

8)3 1

9)(2) 5

10)3 81

11)121

2m 2n =2(m - n)


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12)1 5

13)2 144

14) Negative four to the

power of five

15)(3) 243

16)2 6

17)10000

18) ( 3)2

19)7 343

20)8000

21)125

22) Negative ten to thepower of six

23)6 1296

24)(2)4 16

SQUARE ROOT

We already know that 3 3 can be written as 32 (3 squared).

The opposite of squaring a number is called finding the

square root.

Examples

The square root of 16 is 4 (because 4 4 = 16).




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Exercise 6.

a) What is the square root of 4?

b) What is the square root of 121?

c) What is the square root of 49?

d) What is the square root of 100?

e) What is the square root of 144?

f) What is the square root of 25?

g) What is the square root of 81?

h) What is the square root of 9?

i) What is the square root of 16?

j) What is the square root of 64?

k) What is the square root of 36?

(Read the instructions carefully and choose the correct answer.)

Exercise 7. What is 25 ?

Possible answers: a) 5 b) 5 c) 625 d)2.5

Exercise 8. What is 289 ?

Possible answers: a) 17 b) 17 c) 2.89 d) 83521

Exercise 9. What is 2 rounded to two decimal places?

Possible answers: a) 2.00 b) 1.41 c) 1.40 d) 0.5


Possible answers: a) 10 b) 30 c) 31.62 d) 31.00


Possible answers: a) 5 b) 3 c) 3.1 d) 4


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Exercise 12. What is 68 rounded to one decimal places?

Possible answers: a) 8 b) 9.1 c) 34.1 d) 8.2

Exercise 13. What is 0.0016 ?Possible answers: a) 0.04 b) 0.04 c) 0.40 d) 0.004

Exercise 14. What is 0.01 ?

Exercise 15.Calculate the value of the following square roots. If it is not a perfect

square, then indicate which two numbers the answer is between.

a)

144

b)65

c)32

d) 225 e) 97 f) 256

SCIENTIFIC NOTATION

For very large numbers, it is sometimes simpler to use "scientific notation" (so called,

because scientists often deal with very large numbers).

The format for writing a number in scientific notation is fairly simple:

First digit of the number followed by the decimal point and then all the rest of the

digits of the number, times (10 to an appropriate power). The conversion is fairly

simple.

Example: Write 12400 in scientific notation

This is not a very large number, but it will work nicely for an example. To convert this to

scientific notation, I first write "1.24". This is not the same number, but (1.24)(10000) =

12400 is, and 10000 = 104. Then, in scientific notation, 124 is written as 1.24 102.


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AUTOEVALUACIN Matemticas UNIT III

Exercise 1.Write in exponential form and indicate how you would read it:

a) 7 7 7 7 b) 9 9 9 c) 6 6

d) 3 e) 6 6 6 6 6 f) 5 5 5 5 5 5

Exercise 2. Express the followingexponents in product form:

36 Dos al cuadrado 71

Cinco al cubo 90 (6)4

Exercise 3.Mentally calculate the value of the following:

a) 122 b) 52 102 c) 92 d) 53 e) 23 103 f) 62

Exercise 4.Calculate:

a) ( 6 2 )3 b) ( 24 : 3 )3 c) ( 15 6 )2

d) ( m n )3 e) ( m : n )3 f) ( 3 + 2 )3

Exercise 5.Simplify to one exponent:

a) (m3)4 b) 43 45 c) n8 : n3 d) (42)3

e) 33

35

3 f) a6

: a4

g) (43

)2

h) x5

x2

i) 85: 82 j) (63)3 k) x3 x4 l) x6 : x5

Exercise 6. Calculate the value of the following square roots. If it is not a perfect

square, then indicate which two numbers the answer is between.

a) 25 b) 121 c) 32

d) 50 e) 20 f) 2

g) 1 h) 36 i) 140

Exercise 7.Calcula el valor de:

a43 b34 c126 d 23 e24 f (3 )4

g1121 h 21 i) (7)0 j) 22 k) ( 100)2 l)10 4

Exercise 8.Calcula con todas sus cifras el valor de los siguientes nmeros e indica

cmo se leen:a) 5 107 b) 8,7 1010 c) 3,12 1014


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Exercise 9. The spaceship Intrepid VII, piloted by the glorious crew of captain

Bermudez was stationed on planet Alfa in a far away galaxy. While there they receive a

call for help from planet Beta, which is situated 1.810 6 km away.

a) Write out the distance between Alpha and Beta in expanded form.

b) Write out how you would readthe distance.

Ejercicio 10.Expresa en notacin cientfica

a) 5000 b) 6.560.000.000 c) 12.000.000

Exercise 11.In a train station in a certain city it is very busy:

a) A train will leave from track 1 with four cars. Each car has four sections, each

section has four compartments, and each compartment has four seats. Express in

exponential orm and calculate the number of passengers that can go in a car and

the number of people who can travel in the train.

b) A train will leave from track 2 with 6 cars, and it is known that there will be 2 4 33

passengers on the train, equally distribute among the 6 cars. Calculate the total

numbers of passengers on the train and the number of passengers in each car.

c) A train departed 2 hours ago from the station. It made 4 stops before arriving at its

destination. The amount of passengers for each part of the trip is described below.

Departure: Departure with 26 3 people.

STATION A: 42 people got on, 23 got off

STATION B: 22 42 people got off

STATION C: 25 people got on, 27 got off

STATION D: 34 people got on, 52 got off

DESTINATION: 23 22 3 people got off

Complete this table:

Stations Got on Got off Number of people who stayed on the trainDeparture (S) 26 3 0 192

A 42 23 192 + 42 23 = 192 + 16 8 =

B

C

D

Destination(F)

Did any passengers stay on the train?


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REFUERZO Matemticas UNIDAD III


Fecha de entrega:


negro.




Recordemos que es una potencia: Es el resultado que se obtiene al multiplicar

repetidas veces un mismo nmero

Ejemplo: 24 = 2 2 2 2 = 16

En esta potencia ( 24) el 2 es la base, y el 4 es el exponente.

OJO! NO CONFUNDAS 24 QUE ES 16 CON 24 QUE SERIA 2 + 2 + 2 + 2 = 8

Ejercicio 1.

a) Escribe en forma de potencia:

3 3 3 3 = 2 2 2 2 2 = 5 5 =

b) Escribe como producto:

3 + 3 + 3 + 3 = 2 + 2 + 2 + 2 + 2 = 5 + 5 =

Ejercicio 2. Calcula el resultado de las siguientes potencias:

a) 34= b) 53= c) 62= d) ( 8)3= e) 83=

f) 83= g) 107= h) 107= i)(10)7= j) 24=

k) ( 2)4= l) (2)4= m) (+3)3= n) ( 3)3= o) 33=

Ejercicio 3.Calcula el valor de los siguientes productos:

a) 24 3 = 16 3 = 48 b) 32 5 = c) 23 3 5 = d) 52 22 3 =

e) 33 2 = e) 7 52 = f) 105 = g) 103 5 =


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Ejercicio 4.Escribe en forma de potencia de base 10:

a) 1000 = b) 100

c) 10.000= d) 1.000.000 =

Ejercicio 5.Haz la descomposicin polinmica de los nmeros siguientes:

a) 26.479

b) 5.807.245

Ejercicio 6. Escribe el nmero que corresponde a estas descomposiciones

polinmicas:

a) 5104 + 3 102 + 6 10 +4 =

b) 3106 + 8104 + 7103 + 2102 + 9 =

Ejercicio 7.Escribe con notacin cientfica:

a) 430000000000

b) 0000000000025

OPERACIONES CON POTENCIAS: (Repasa en el libro las operaciones con

potencias y realiza estos ejercicios)

Ejercicio 8 .Escribe como una sola potencia:

a) 22 23 24 2 = b) 35334= c) 5352=

d) 10310410= e ( 6)3( 6)4( 6)= f) ( 7)2( 7)3(7 )0=

g) ( 5)4( 5)5( 5)3= h) (82)3= i) (104)5=

j) (3

3

)

3

= k) ( 9

2

)

4

= l) 3

2

: 3

2

=m) (5)8 : (5)3= n)27:23= o) 77:78=

Ejercicio 9.Realiza como potencia de un producto:

a) (352)2= b) (432)3= c) (abc)n=

Ejercicio 10.Reduce a una sola potencia:

a) x8: x3 = b) x3 x2 x = c) (x2)3 =


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Ejercicio 11. Define que es raz cuadrada y pon el nombre a cada una de sus

elementos:

ba

n

Ejercicio 12.Calcula las siguientes races cuadradas enteras buscando mentalmente

el nmero cuyo cuadrado nos d el radicando:

4 = 81 = 100 = 49 = 64 =

Ejercicio 13.Un paquete tiene 12 cajas, cada caja tiene 12 estuches. Cada estuche,

12 rotuladores. Cuntos rotuladores hay en un paquete? Y en 12 paquetes?

(expresa primero el resultado en forma de potencia y despus calcula)

Ejercicio 14. Un jardinero tiene que plantar 121 rboles formando un cuadrado.

Cuntos rboles tendr cada lado?

Ejercicio 15.Calcula en cm la longitud del lado de un cuadrado que tiene 529 m 2 de

rea. ( rea del cuadrado: A = L2 )


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DIVISIBILITY UNIT IV

MULTIPLES

Debbie works for a food and drink company. She decides how items are packagedtogether so that customers can buy in bulk.

Debbie decides that tins of tea should be packaged in pairs.

How many tins are there in 4 packages?...........

The numbers of tins, .......................... are called multiples of two.

The multiples of two are the even numbers.

The others numbers are odd.

Exercise 1.Write the multiples of the following: a) 3 b) 7 c) 10

FACTORS

Debbie thinks that small bottles of Juice will sell well in packages of 12.

She designs a package. It has 3 rows of 4

3 4 = 12.

We say that 3 and 4 are factors of 12

Exercise 2.Draw other ways of arranging 12 bottles in a rectangular package.

The different arrangements tell you all the factors of 12.

Another way of saying this is that 12 is divisible by .......................

PRIMES

Next month there is going to be a special promotion on Juice.When you buy a pack of 12 you get one bottle free.

Exercise 3.How can Debbie arrange 13 bottles in a rectangular package?

A prime number has just two different factors, 1 and itself

The prime number are,............................................................ (up to ten numbers)


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PRIMES FACTORIZATION

Exercise 4.Which of these are prime numbers? 19, 20, 43, 84

If a number is not itself prime then can be written as a product.

E.g.: 18 = 2 3 3 ; 75 = 3 5 5

each of these is a prime number. These called prime factorization

Exercise 5.What is the prime factorization of:

a) 20 b) 84 c) 12 d) 45 e) 36

f) 100 g) 56 h) 32 i) 360 j) 128

Exercise 6.Here is a tray of chocolate cakes:

How many chocolate cakes are there in

a) 2 trays b) 3 trays c) 4 trays

Exercise 7.Lucy has 18 cork tiles

She wants to use them to make a rectangular notice board in her room.

She sketches this arrangement.

a) Draw all the other ways of making a rectangular notice board using exactly 18 tiles.

b) Now list all the factors the of 18

Exercise 8.List all the factors of

15 8 20 7 16 35

45 36 60 100 72 144

Exercise 9.List all the primes numbers between 20 and 40 .


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Let's look at some tests for divisibility and examples of each.

Divisibility Tests Example

A number is divisible by 2 if the last digit is0, 2, 4, 6 or 8.

168 is divisible by 2 since the lastdigit is 8.

A number is divisible by 3 if the sum of the

digits is divisible by 3.

168 is divisible by 3 since the sum

of the digits is 15 (1+6+8=15), and

15 is divisible by 3.

A number is divisible by 5 if the last digit is

either 0 or 5.

195 is divisible by 5 since the last

digit is 5.

A number is divisible by 10 if the last digit is

0.

1,470 is divisible by 10 since the

last digit is 0.

Example 1: Determine whether 150 is divisible by 2, 3, 5, and 10.

150 is divisible by 2 since the last digit is 0.

150 is divisible by 3 since the sum of the digits is 6 (1+5+0 = 6), and 6 is divisible by 3.



Solution: 150 is divisible by 2, 3, 5, and 10.

Example 2: Determine whether 2047 is divisible by 2, 3, 5, and 10.

2047 is not divisible by 2 because the last digit is not 0, 2, 4, 6, 8.

2047 is not divisible by 3 because the sum of the digits is 13, and 13 is not divisible by

3.

2047 is not divisible by 5 because the last digit is not 0 nor 5.

2047 is not divisible by 10 because the last digit is not 0.

Solution: 2047 is not divisible by 2, 3, 5, nor 10.

Exercise 10. Determine whether 225 is divisible by 2, 3, 5, and 10 (complete thisexercise in the same manner as example 1 and 2 ).


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Exercise 11.Determine whether 7,168 is divisible by 2, 3, 5, and 10 (complete this

exercise in the same miner as example 1 and 2 ).

Exercise 12. Determine whether 2433 is divisible by 2, 3, 5, and 10 (complete this

exercise in the same manner as example 1 and 2 ).

LCM (LEAST COMMON MULTIPLE) AND GCF (GREATEST COMMON

FACTOR)

To find either the Least Common Multiple (LCM) or Greatest Common Factor (GCF) of

two numbers, you always start out the same way: you find the prime factorizations of

the two numbers. Then you put the factors into a nice neat grid of rows and columns,

and compare and contrast and take what you need.

Example: Find the GCF and LCM of 2940 and 3150.

First, I need to factor each value:

My prime factorizations are:

2940 = 2 2 3 5 7 7

3150 = 2 3 3 5 5 7

I will write these factors out, all nice and neat, with the factors lined up according to

occurrence.

This orderly listing, with each factor having its own column, will do most of the work for

me.

The Greatest Common Factor, the GCF (MCD), is the biggest number that will divide

into (is a factor of) both 2940 and 3150. In other words, it's the number that contains all

the factors common to both numbers. In this case, the GCF is the product of all the

factors that 2940 and 3150 have in common.


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Looking at the nice neat listing, I can see that the numbers both have a factor of 2;

2940 has a second copy of the factor 2, but 3150 does not, so I can only count the one

copy toward my GCF. The numbers also share one copy of 3, one copy of 5, and one

copy of 7.

2940 2 2 3 5 7 7

3150 2 3 3 5 5 7

GCF 2 3 5 7

Then the GCF is 2 3 5 7 = 210

On the other hand, the Least Common Multiple (mcm), the LCM, it is the smallest

number that contains both 2940 and 3150 as factors, the smallest number that is a

multiple of both these values. Then it will be the smallest number that contains one of

every factor in these two numbers.

2940 2 2 3 5 7 7

3150 2 3 3 5 5 7

LCM 2 2 3 3 5 5 7 7

Then the LCM is 2 2 3 3 5 5 7 7 = 44,100

By using this "factor" method of listing the prime factors neatly in a table, you can

always easily find the LCM and GCF. Completely factor the numbers you are given, listthe factors neatly with only one factor for each column (you can have 2s columns, 3s

columns, etc, but a 3 would never go in a 2s column), and then carry the needed

factors down to the bottom row.

For the GCF, you carry down only those factors that all the listings share; for the LCM,

you carry down all the factors, regardless of how many or few values contained that

factor in their listings.


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Exercise 12. Find the LCM and GCF of 27, 90, and 84.

a) I need to find the prime factorizations:

b) Then I will list these factorizations neatly:

Then the GCF (being the product of the shared factors) and the LCM (being the

product of all factors) are given by:

Exercise 13.Find the GCF and LCM of :

a) 10 and 15 b) 8 and 24 c) 225 and 90 d) 60 and 135 e) 135 and 90

f) 225 and 100 g) 150 and 36 h) 54 and 225 i) 60 and 36 j) 24, 27 and 8


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SELFEVALUATION Mathematic UNIT IV

Exercise 1. a) Complete the table with the first ten multiples of the each number:

Multiple of 2

Multiple of 5

Multiple of 4

Multiple of 6

b) What is the least common multiple of two and five?

c) What is the least common multiple of four and six?

Exercise 2. Put the following number in the table:

39 9 10 8 50 14 16 104 32 18 91 98

65 55 33 115 51 77 88 25 27 49 119

Multiple of 3

Multiple of 4

Multiple of 5

Multiple of 7

Exercise 3. Copy and complete the following definition: The numbers you obtain by

multiplying a number by the natural numbers are called.of that

number.

Exercise 4.A television channel shows a nature documentary every six hours, andanother channel shows the documentary every four hours. Every how many hours do

the documentaries coincide on the two channels?

Exercise 5.Ismael is going to prepare hot dogs and want to buy the same number of

sausages as buns (special kind of bread). The sausages are sold in package of six and

the buns in packages of tour. What is the least number of packages that he has to buy

for each?


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Exercise 6.Define following terms:

a) Multiples b) Common multiples c) Lowest common multiple

Exercise 7. Which of the following numbers are factors of 114?:

2 3 4 5 6

Write the criteria of divisibility that you have used:

Exercise 8.Complete these tables:

Divisores de 40

Divisores de 24

Divisores comunes a 40 y 24

Mximo comn divisor de 40 y 24

Divisores de 20

Divisores de 16

Divisores comunes a 20 y 16

Mximo comn divisor de 20 y 16

Exercise 9.Classify the numbers between 10 and 20 as prime or composite.

Exercise 10 Write T or F if the statements are true o false. Justify your answers.

a) 24.583 es divisible por 2

b) 16.666 es divisible por 3

c) 130.425 es divisible por 5

d) 194.680 es divisible por 10

Exercise 11.Find the multiple of 24 whose digits sum to 9 and is between 300 and

400.

Exercise 12.Copia y completa las siguientes definiciones:

a) Los nmeros que dividen a otro nmero y dan de resto 0, se llaman ....de

ese nmero.

b) Un nmero primo es un nmero que slo tiene. el 1 y l mismo.

c) Los nmeros que tienen ms de dos divisores se llaman..


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a) m.c.m. (18, 30) = b) m.c.m. (48, 36) = c) m.c.m. (15, 30) =

d) M.C.D. (18, 30) = e) m.c.m.(240, 600, 960) = f) M.C.D.(48,36) =

Exercise 14.Three cousins, John , Ana and Tom frequently visit their grandparents

John every four days, Ana every six days, and Tom every eight days. If they just visited

them all together, when will they all coincide again? How many times will they coincide

in a year?

Ejercicio 15.Calcula

a) M.C.D.(390, 900) b) M.C.D.(504, 792) c) M.C.D.(180, 276, 444)

Ejercicio 16.Calcula

a) m.c.m.(10, 20, ) b) m.c.m.(4, 18, 16) c) m.c.m.(25, 35, 45)

Exercise 17.To build a fence 35m long and 28 m wide you must place the post the

same distance apart along the whole fence. What is the maximum distance that one

could place the post apart from each others? How many posts will there be?

Exercise 18.In a hostel there are three tour groups with 40, 56 and 72 people. The

waiter wants to organize the tables in the restaurant so that there is an equal number of

people at each table and the most people possible without missing the groups. How

should people sit at each table?

Ejercicio 19. Un cometa es visible desde la tierra cada 16 aos, y otro, cada 24

aos. El ltimo ao que fueron visibles conjuntamente fue en 1968. En qu ao

volvern a coincidir?

Ejercicio 20.Un carpintero dispone de tres listones de madera de 40, 60 y 90 cm de

longitud, respectivamente. Desea dividirlos en trozos iguales y de la mayor medida

posible, sin que sobre madera. Qu longitud deben tener esos trozos?

Ejercicio 21. Un electricista tiene tres rollos de cable de 96, 120 y 144 metros de

longitud. Desea cortarlos en trozos iguales de la mayor longitud posible, sin que quede

ningn trozo sobrante. Qu longitud tendr cada trozo?


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REFUERZO Matemticas UNIDAD IV


Fecha de entrega:

Atencin: todos los ejercicios deben realizarse en DIN A4 con bolgrafo azul.


seguido.

Se valorar la presentacin y limpieza. Las pginas deben ir numeradas.

Ejercicio 1.Explica el significado de:

a) Mltiplo de un nmero b)Divisor de un nmero

Ejercicio 2.Calcula los siete primeros mltiplos de los siguientes nmeros (utiliza la

notacin adecuada:

a) 7 b) 9 c) 10 d) 25

Ejercicio 3. Halla todos los divisores de los siguientes nmeros (utiliza la notacin

adecuada):

a) 30 b) 60 c) 25 d) 27 e) 26 f) 7

Ejercicio 4. Qu es un nmero primo?Encuentra todos los nmeros primos

menores que 30.

Ejercicio 5.Escribe los criterios de divisibilidad del: a) 2 b) 3 c) 5 d) 10

Para qu sirven los criterios de divisibilidad?

Ejercicio 6.Escribe la palabra divisible, divisor o mltiplo segn corresponda:a) 4 es..de 28

b) 30 es..entre 6

c) 5 es ....de 10

d)10 es.....de 5

Ejercicio 7.Realiza la descomposicin en factores primos de los siguientes nmeros:

a) 36 b) 62 c) 120 d) 450 e) 242 f) 39


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Ejercicio 8. Escribe los nmeros a los que corresponden estas descomposiciones

factoriales:

a) 23 3 b) 52 7 11 c) 3 22 72 d) 23 53

Ejercicio 9. Calcula el M.C.D y el m.c.m de los siguientes nmeros por el mtodo

ptimo:

a) 12 y 18 b) 24 y 36 c) 50, 30 y 20 d) 21168 y 462

Ejercicio 10.Por la Avenida de la Diputacin pasa el autobs de la Barrosa cada 30

minutos y el autobs de Sancti Petri cada 45 minutos. Si a las 9 de la maana han

coincidido. A qu hora volvern a coincidir?

Ejercicio 11.Se puede llenar un nmero exacto de garrafas de 15 litros con un bidn

que contiene 170 litros? Y con un bidn de 180 litros?

Ejercicio 12.En un albergue coinciden tres grupos de excursionistas de 40, 56 y 72

personas cada grupo. El camarero quiere organizar el comedor de forma que en cada

mesa haya igual nmero de comensales y se rena el mayor nmero de personas

posible sin mezclar los grupos. Cuntos comensales sentarn en cada mesa?

Ejercicio 13. Un granjero ha recogido de sus gallinas 30 huevos morenos y 80

huevos blancos. Quiere envasarlos en recipientes con la mayor capacidad posible y

con el mismo nmero de huevos (sin mezclar los blancos con los morenos). Cuntos

huevos debe poner en cada recipiente?

Ejercicio 14. Un cometa es visible desde la tierra cada 16 aos, y otro, cada 24

aos. El ltimo ao que fueron visibles conjuntamente fue en 1968. En qu ao

volvern a coincidir?

Ejercicio 15.Halla un nmero mayor que 100 y menor que 200, que sea mltiplo de

dos, de tres y de cinco.

Ejercicio 16.Halla un nmero mltiplo de siete y de once que sea mayor que 100.


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FRACTIONS UNIT V

WHAT IS A COMMON FRACTION?

Well, there is a special collection of numbers called fractions, which are usually

denoted byb

a, where "a" and "b" are integers numbers and "b"is not equal to 0".

Numerator

Denominator

There are two distinct meanings of fractions: partwhole and quotient.

THE PARTWHOLE: The partwhole explanation of a fraction is where a number like1/5 indicates that a whole has been separated into five equal parts and one of those

parts are being considered.

This table is a great help to get a feel of how a fractional part compares to the

whole

The Whole

1/2 1/2

1/3 1/3 1/3

1/4 1/4 1/4 1/4

1/5 1/5 1/5 1/5 1/5

1/6 1/6 1/6 1/6 1/6 1/6

1/7 1/7 1/7 1/7 1/7 1/7 1/7

1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8

The Division Symbol ("/" or "__") used in a fraction tells you that everything above the

division symbol is the numeratorand must be treated as if it were one number, and

everything below the division symbol is the denominatorand also must be treated as

if it were one number.


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Basically, the numerator tells you how many parts we are talking about and the

denominator tell you how many parts the whole is divided into. So a fraction like 6/7

tells you that we are looking at six (6) parts of a whole that is divided into seven (7)

equal parts.

A QUOTIENT: The fraction3

2may be considered as a quotient, 2 3. This

explanation also arises from a dividing up situation.

Division by Zero: The denominator of a fraction cannot have the value zero. If the

denominator of a fraction is zero, this is not a legal fraction because it's overall value

is undefined.

Zero in the Numerator: The numerator of a fraction can have a value of zero. Any

legal fraction (denominator not equal to zero) with a numerator equal to zero has an

overall value of "zero."

Any Integer Can Be Written as a Fraction: You can express any integer as a fraction

by simply dividing by 1, or you can express any integer as a fraction by simply

choosing a numerator and denominator so that the overall value is equal to the integer.

Exercise 1.Write two examples of any Integer which Can Be Written as a Fraction.

Exercise 2. The number we write as 1 over 2 :2

1is ".........................................." .

The number we write as 2 over 3 :3

2is called ...................................

The number we write as 3 over 4 : 4

3

is called ..................................."

Exercise 3. Copy and fill in the gaps

a) The number write 5 over 88

5is called .....................................

b) The number write....................................."Sixteenthirds

c) ...................................."Ninehalves


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This number 1/4 is called "onequarter" or "onefourth," because the numerator

is one quarter of the denominator and 1/4 itself is one quarter of 1.

Particular cases:

When the denominator is a power of 10, however, we always say the decimal name.

56 thousandths1000

56

WHAT IS A PROPER FRACTION?

A fraction that is less than 1: ,...5

1,

3

2

WHAT IS AN IMPROPER FRACTION?

A fraction greater than or equal to 1: ....5

11,

3

13

How can we recognize a proper fraction? The numerator is greater than the

denominator.

Exercise 4.Answer with a whole number and a remainder, which ever makes sense.

a) How many basketball teams 5 on a team can you make from 23 students?b) You are going on a trip of 23 miles, and you have gone a fifth of the distance. How

far have you gone?


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FRACTIONS

1)2

1 Onehalf

2)3

2 Twothirds

3)4

3 Threefourths

4)5

4 Fourfifths

5)6

4 Foursixths

6)7

5 Fivesevenths

7)8

6 Sixeighths

8)9

7 Sevenninths

9)10

7 Seventenths

10)12

5 Fivetwelfths

11)23

15 Fifteentwenty thirds

12)43

21 Twenty one forty thirds

13)72

22 Twenty two seventy seconds

14)24

21 Twenty one twenty fourths

15)35

12 Twelve thirty fifths

16)22

34 Thirty four twenty seconds

for #s 1116 you can also say the number over the number ex. Fifteen over twenty

three.


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Exercise 5.There are ten trucks and three minivans parked in front of the store. What

fraction of the vehicles are minivans?

Exercise 6. Maria has five red beach balls and seven orange beach balls. What

fraction of Marias beach balls is orange?

Exercise 7. Michael has ten quarters and seven dimes. What fraction of Michael's

coins are dimes?

Exercise8. Ruth has three gold rings and ten silver rings. What fraction of Ruths

rings is silver?

Exercise 9. Thomas bought thirty sheets of paper. He has used twothirds of the

paper. How many sheets of paper are left?

Exercise10.Ashley has fortyfive notebooks. Twofifths of them are for school. How

many of the notebooks are for school?

Exercise 11. Miss Lee has thirty plants. Threefifths of them have flowers. How many

of her plants don't have flowers.

Exercise12.There are forty cows in the field. Half of the cows are black and white.

How many of the cows are not black and white?

Exercise 13. Betty has seven pink ribbons and ten blue ribbons. What fraction of

Bettys ribbons is pink?

Exercise14.Danielle ate fivesevenths of her orange before lunch and oneseventhof her orange after lunch. How much of her orange did she eat in all?

Exercise 15. There are nine pieces of pizza. Cameron ate twoninths of the pizza for

dinner. He ate twothirds of the pizza for a bedtime snack. How much of the pizza has

he eaten in all?


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WHAT ARE EQUIVALENT FRACTIONS?

Equivalent fractions are simply equal fractions. They have the same value. They are at

the same place on the number line.

The following is called the principle of equivalent fractions:

If we mul t ip ly or div ide both the numerator and denom inator by the same

num ber, we w i l l obtain an equivalent f ract ion.

Exercise16.Name three fractions that are equivalent to4

3

Exercise 17.Write the missing term.

123

2

155

4

204

3

549

2

408

7

426

5

8

7

2

15

16

5

24

4

3

303

2

455

3

12

5

3

42

9

7

14

3

2

56

9

8

48

7

6

Exercise18.Reduce each fraction to lowest terms.

a) 12

8b)

6

3c)

15

5d)

24

16

e) 63

21f)

35

20g)

54

12h)

48

6

i) 81

63j)

60

15k)

36

16l)

72

24

m) 40

30n)

900

600o)

400

50p)

12000

1800
http://www.themathpage.com/Arith/fractions.htm#numlinehttp://www.themathpage.com/Arith/fractions.htm#numlinehttp://www.themathpage.com/Arith/fractions.htm#numline


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HOW DO WE ADD OR SUBTRACT FRACTIONS?

The denominators must be the same. Add or subtract only the numerators, and keep

that same denominator.

Exercise 19.Work out:

a) 5 eighths + 2 eighths =

b) 4 tenths 5 tenths =

c) 3 seventh + 2 seventh =

Exercise 20. Calculate the following.

a) 5

2

5

3

b) 7

2

7

3

c) 165

16

15

d) 5

4

5

1

5

3e)

4

3

4

1

4

2f)

15

4

15

8

15

10

HOW DO WE ADD OR SUBTRACT FRACTIONS THAT DO NOT HAVE THE

SAME DENOMINATOR?

Make the denominators the same by changing to equivalent fractions.

What number should we choose as the common denominator?

Choose a common multiple of the original denominators. Choose their L.C.M.

Example: 8

3

2

1

LC.M. of 2 and 8, is 8 itself. We will change2

1to a fraction whose denominator is 8

8

4

2

1 Therefore,

8

7

8

3

8

4

8

3

2

1

In practice, it is necessary to write the common denominator only once:

8

7

8

34

8

3

2

1


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Exercise 21.Choose the LCM of denominators to calculate the following.

a) 6

1

3

2b)

4

1

12

5c)

32

5

2

1

d) 15

2

5

3e)

32

7

8

3f)

4

3

16

15

g) 10

7

15

11h)

12

5

8

7i)

12

5

9

8

Exercise22. Calculate the following.

a) 3

1

2

1b)

5

2

8

3c)

10

7

9

7

d) 51

72 e) 3

243 f) 9

587

Exercise 23. In a recent exam, one eighth of the students got A, two fifths got B, and

the rest got C. What fraction got C?

HOW CAN WE COMPARE FRACTIONS WHEN THE NUMERATORS ANDDENOMINATORS ARE DIFFERENT?

Change them to equivalent fractions with the same denominator. Then, compare the

numerators.

Exercise24. Compare. Writegreater than (>), less than (


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Now we are going to do the exercise 4

Answers:

a) We can make 4 teams from 23 students 3 we will be left out.

b) A fifth of 23 miles is5

34 we consider 1 mile to have any at all part.

Exercise 25. Which of these fractions are less than 1, equal to 1, or greater than 1?

9

10

9

9

9

8

8

8

5

8

2

3

3

2,,,,,,

WHAT IS A MIXED NUMBER?

A whole number plus aproperfraction3

12

3

12

The andin "2 and onethird" means plus.

1 23

12 3

HOW DO WE CHANGE AN IMPROPER FRACTION TO A MIXED NUMBER

OR A WHOLE NUMBER?

Divide the numerator by the denominator. Write the quotient (4), and write the

remainder (1) as the numerator of the fraction; do not change the denominator.

2

14

2

9

Exercise 26.Write each improper fraction as a mixed number in simplest form:

2

17;

2

7;

5

13;

4

28;

9

32;

5

43

9

1

2

4


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HOW DO WE CHANGE A MIXED NUMBER TO AN IMPROPER FRACTION?

Multiply the whole number (4) by the denominator (2), and add the numerator (1). Write

that sum (9) as the numerator of the improper fraction. Keep the same denominator

4

9

2

142

2

14

2

14

Exercise 27.Write each mixed number as an improper fraction in simplest form.

5

27;

2

110;

3

25;

4

17;

2

13

Exercise 28.Fill in the missing number.

a)3

13

5 b)

2

13

7

c)2

47

30 d)

6

5

6

71

HOW DO WE MULTIPLY A FRACTION BY A FRACTION?

Multiply the numerators and multiply the denominators.

db

ca

d

c

b

a

Exercise 29.Cancel before multiplying, if possible.

a) 8

7

5

3 b)

9

7

8

5 c)

9

4

8

3

d) 4

3

3

2 f)

8

9

51

2 g)

9

4

4

9

h) 18

512 h)

9

712 i)

7

342

j) 7

5

3

2

3

1 k)

4

1

5

3

3

2 l)

2

5

3

4

20

6


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HOW DO WE DIVIDE FRACTIONS?

Step 1: Turn the second fraction (the one you want to divide by) upsidedown (this is

now a reciprocal).

Step 2: Multiply the first fraction by the reciprocal of the second

Step 3: Simplify the fraction (if needed)

Exercise 30.Cancel before dividing, if possible.

a) 5

7

4

3b)

3

2

5

3c)

11

3

9

2

d) 15

3

5

4f)

9

8

6

5g)

10

9

15

4

h) 12

11

8

5h)

2

1

4

3i) 2

3

1

j) 35

2k)

5

23 l) 2

7

2

Exercise 31. A bottle of medicine contains 8 oz. Each dose of the medicine is3

2oz.

How many doses are in the bottle?

Exercise 32.A bottle of medicine contains 15 oz. Each dose of the medicine is 2

oz. How many doses are there in the bottle?

cb

da

c

d

b

a

d

c

b

a


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AUTOEVALUACIN Matmaticas UNIT V

Exercise 1. Represent the following fractions:4

7,

5

2,

3

1

Exercise 2.The shaded part represents what fraction of the whole in the following

images?

a) b) c) d)

e) f) g) h)

Exercise 3. Change each fraction into a decimal:

8

9d)

25

17c)

5

4b)

1000

85a)

Exercise 4. Answer each question and justify your responses.

a) The fraction 3/5 is more or less than one?

b) The fraction 3/4 is more or less than 1/2?

c) Which fraction is greater 2/5 or 2/4?

d) Which fraction is greater 2/4 or 4/8?

Exercise 5. Write three equivalent fractions for each of the following.

7

3d)25

10c)3

1b)10

8a)

Exercise 6. Determine if the following fractions are equivalent.

48

27y

16

9b)

27

12y

9

4a)


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Exercise 7.Simplify the following fractions until they are irreducible:

Exercise 8. Solve the following problems:

a) A family earns 2800 a month and spends 1200 on the mortgage for their

apartment. What fraction of the total earnings do they spend on the mortgage?

b) In a warehouse there are 1500 packages to send; today they have sent threefifths

of the total How many packages have they sent today?

c) James bought a new car for 21000, which used twothirds of his savings. How

much was in his savings originally?

d) A family dedicates half of its monthly earnings to pay the mortgage. If they earnings

are 2800, how much do they pay monthly for the mortgage?

e) A farmer decides to sell 240 cows, if he has 680 cows in his herd, what fraction of

the herd will be sold?

f) For Beatrizs present, Sandra has saved 15 , which is twofitths of the total cost.

How much does the present cost?

Exercise 9. Solve:

6

52e)

74

3d)

6

5

3

2c)

48

1

2

5

4

1b)

6

5

12

2

3

1

4

3a)

2

3

6

42j)

6

4:2i)

53

6h)

6

4

2

6g)

6

4

3

6f)

Exercise 10. Solve:

10

912

5

3

5

2c)

10

11

5

2

2

1b)

3

23

4

36a)

:

:

294

1540

45

15

45

75

2860

2717

945

1575

5082

726


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Exercise 11. Peter spent threetenths of his money in books, one fifth in C.D.s, one

tenth in magazines, and one fourth in other things. What fraction on his money has he

spent? What fraction remains?

Exercise 12. From a roll of 48 m of cable, 2/3 has been used. How many meters of

cable remain?

Exercise 13.Una camioneta transporta 3/5 de tonelada de arena en cada viaje. Cada

da hace cinco viajes. Cuntas toneladas transporta al cabo de seis das?

Exercise 14. Para hacer un disfraz se han utilizado los 3/5 de una pieza de tela de 25

metros. Si el precio del metro de tela es de 3 euros, cunto ha costado la tela del

disfraz?


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REFUERZO Matemticas UNIDAD V

Nombre y Apellidos 1___

Fecha de entrega:

Atencin: todos los ejercicios deben realizarse en DIN A4 con bolgrafo azul.


seguido.

Se valorar la presentacin y limpieza. Las pginas deben ir numeradas.

Ejercicio 1. .Qu fraccin representa la parte sombreada de cada uno de los

siguientes dibujos?

a) b) c) d)

Ejercicio 2.Transforma cada una de estas fracciones en un nmero decimal y cada

decimal en fraccin:

1000

1234h)

7

12g)1.2f)1,03e)0,012d)

2

17c)

5

4b)

100

8a)

Ejercicio 3.Realiza las siguientes operaciones con fracciones:

4

3-3d)2

3

2c)

8

1

2

5b)9

2

3

5

4

5a)

4

Ejercicio 4.Resuelve estos problemas:

a) Una joven ingresa 280 mensuales por trabajos eventuales en un super y le da a

su madre 80 euros todos los meses.Qu fraccin de sus ingresos representa esta

suma?


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b) Lus Alfonso gasta 60 mensuales en la cuota de un gimnasio, lo que supone los

dos tercios de su sueldo.Cunto dinero cobra mensualmente?

c) Una madre de familia tiene 5/9 de una tableta de chocolate y le da a su hija

Elizabeth Pilar 2/9.Cunto le queda?

d) Un campo mide 2000 metros cuadrados. Cuntos metros cuadrados tiene 1/4 del

campo? Y 3/4 del c

1º ESO cuadernillo bilingue

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