unit 8: ratio and proportion - Working on Maths in English

WORKING ON MATHS IN ENGLISH Isabel Leo de Blas

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UNIT 8 RATIO AND PROPORTION 1ST

LEVEL

SUMMARY

1. RATIO

1.1 APPLICATIONS OF RATIO

2. PROPORTION

- PROBLEM SOLVING

3. DIRECT PROPORTION

3.1 RULE OF THREE

3.2 APPLICATIONS OF DIRECT PROPORTION

3.2.1 SCALES

- SCALE PRACTICE 1 Reading maps

- SCALE PRACTICE 2 My house map

3.2.2 PROPORTIONAL SHARING-OUT (REPARTO PROPORCIONAL)

3.2.3 SIMILAR FIGURES

3.2.4 PERCENTAGES

3.2.5 PER THOUSAND

3.2.6 MIXTURES AND ALLOYS

3.2.7 PROPORTION AND GEOMETRY

4. INVERSE PROPORTION

- REVISION

EXTENSION: COMPOUND PROPORTION

- SOLVED EXAMPLES

- BANK INTEREST

Webs for more practice:

http://www.bbc.co.uk/skillswise/numbers/wholenumbers/ratioandproportion/ratio/factsheet.shtml RATIO AND PROPORTION WEB


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http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/numbers/ratio_rev1.shtml EXPLANATION AND EXERCISES

http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/numbers/ratio_activity.shtml video ratio

- http://www.bbc.co.uk/schools/ks3bitesize/maths/number/ratio/activity.shtml VIDEO RATIO

- http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/numbers/quiz/ratio/ TEST

- http://www.bbc.co.uk/schools/ks3bitesize/maths/number/ratio/revise1.shtml ratio

- http://www.math.com/school/subject1/lessons/S1U2L1GL.html ratio

- http://www.homeschoolmath.net/teaching/proportions.php

- http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i7/bk8_7i1.htm

- http://www.mathslice.com/actionctl.php?actionid=542 EXERCISES

- http://www.mathleague.com/help/ratio/ratio.htm EXPLAIN

- - http://www.bbc.co.uk/skillswise/numbers/measuring/distance/factsheet.shtml

- http://www.bbc.co.uk/skillswise/numbers/measuring/distance/worksheet.shtml

http://www.bbc.co.uk/skillswise/numbers/measuring/distance/quiz.shtml

http://www.bbc.co.uk/skillswise/numbers/measuring/distance/index.shtml Facts, worksheets and test about SCALES

- http://www.mathslice.com/percent1_ws.php Proportion as %, decimals and fractions

- http://www.gcsemathstutor.com/ratio.php# Sharing proportion

- http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/proportionhirev2.shtml inverse proportion

- http://www.algebralab.org/Word/Word.aspx?file=Algebra_InterestI.xml INTEREST

- http://www.ixl.com/math/grade-7/simple-interest

- http://www.purplemath.com/modules/investmt.htm

NOTE: THERE ARE pdf files: with more activities, problems and test of evaluation.

- statement-2.pdf -statement-3.pdf - reduce.pdf - statement-1.pdf

ratio-to-fraction.pdf - proportionWS1.pdf - ratio-method.pdf - write ratios.pdf

-statement-4.pdf - Proportions WSAnswer Key 1.pdf - proportional.pdf -singledigit.pdf

- ratio-two-same-units.pdf - EVALUATION testUNIT 8 RATIO.pdf


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UNIT 8 RATIO AND PROPORTION 1ST

LEVEL

"Life is 10% what happens to

us and 90% how we respond to it." Charles Swindoll Ratio = 1: 9

Here we have 1 duck taking care of 7 little ducklings. The ratio is 1 to 7 1 : 7 or 1/7

This is a solving problem unit very useful in daily life situations.

1. RATIO

The large tuna fish weighs 5 kg and the small one 2,5 kg. How

much heavier is the large fish? 5 : 2,5 = 2, So 2 is the ratio what means that the large tuna fish is twice heavier than the small one.

The ratio is a quotient between 2 numbers or measurements. Ratio has no units because it indicates the number of times a quantity is bigger than another.

Ratio: a comparison of two quantities by division. Hence, a ratio is used to compare two quantities.

We express ratios as fractions, so we can do the same operations. To find the ratio is better to reduce the fraction, but do not change to a mixed number. Improper fractions are acceptable.

Example: Out of 10 runners, 7 runners finished the race. We are comparing runners to runners by saying 7 out of 10 runners finished.

We can write this three ways: 7 to 10 or 7 : 10 or 7

10

Each ratio is read: 7 to 10

More examples:

a) 6 out of every 8 students in my class watch the Documentary Programme 6 : 8

b) The recipe for cookies contained 200g of sugar for 400 g of flour 200 : 400 200

400

= 1

2 one part of sugar for 2 parts of flour.

Ratio 1 : 7


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c) A rectangle measures 3 m of length and 60 cm of width. What is the ratio between length and width?

- 1st We have to express the measurements in the same unit, cm for example: 300

60 =

5

1 = 5

So the length is 5 times greater than the width.

In metres is the same result: 3

0,60 = 5, so we get the same ratio

Rates are ratios that compare different quantities that cannot be converted to a

common unit

d) I drove 65 km per hour 65 : 1 or 65 km to 1 hour or 65 / 1

Exercises:

1. At school there are 450 students and 45 teachers. Which is the ratio between teachers and students?

2. Marina spends 1 hour per day to study English and 40 minutes to study Maths. Write the ratio.

3. A bus is available for 60 passengers and a car for 5. How many times is greater the

number of rooms in the bus than in the car? 4. Find the ratio of 3 plants for 9

5. A car that runs 55 miles per hour

6. A driver that travelled 180 km in 2 hours

7. 2 books read every 10 days

8. 15 balloons inflated in 10 minutes

9. Eggs cost 1.20 per dozen. How much is 1 egg?

10. Write a statement of comparison about these ratios: 3 : 1 and 4 / 9


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1.1 APPLICATIONS OF RATIO

The value of ratio is the application to real life problem solving.

For example, “which is the best buy?” . Here we compare the cost per unit or weight for each item.

Example: There are 3 sizes of biscuits bags on the grocer’s shelves. Which is the best buy?

Ratio = cos t

weight= quotient

1) 2,79

3 = £ 0,93 per pound 2)

4,45

5 = £ 0,89 per pound 3)

7,49

7 = £ 1,07 per pound

The 5-pound bag is the best buy.

Number of parts

Ratio is used to describe how 2 quantities are related.

For example: The paint mix is 4 parts, with 3 parts blue and 1 part yellow. The order of the ratio is important.

Paint mix ratio: 3 : 1 + 3 + 1= 4 parts in total

Exercises: Solve these problems in groups and invent similar word problems.

1. In an excursion there are 57 students and 3 teachers. What is the ratio? What does it mean?

2. In a class there are 25 students and 10 are boys. Which is the ratio between boys and girls?

3. To prepare a lotion we need 3 parts of alcohol and 1 part of essence. If we want to prepare 1 litre of lotion, how much lotion do we need?

biscuits 2.79

for 3 pounds biscuits 4.45

for 5 pounds biscuits 7.49

for 7 pounds

To find the per-unit cost for each item, calculate the ratio of cost to one unit of weight dividing the numerator by the denominator.


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4. Laura weighs 42 kg and her little brother Lucas 6 kg. Write the ratio that shows how many times is Laura heavier than Lucas

2. PROPORTION

When two ratios are equal we have a proportion.

For example: Look at the ratios between flour and butter in these two recipes for cakes:

Recipe 1 200gflour

50gbutter= 4 Recipe 2

100gflour

25gbutter = 4

200

50=100

25 Both ratios are equal, so both recipes have the same proportion of ingredients:

4 times more flour than butter.

The quotient of any of the ratios in a proportion is called constant of proportion. In this example is 4.

Example: a photographer enlarges a 5 x 7 photo to a 10 x 14

photo.

Can you find out if these ratios form a proportion?

You can use two ways to check this statement:

1) Write each ratio as a fraction in simplest form:

5 : 7 5

7 and 10 : 14

10

14

10 : 2

14 : 2= 5

7 We can see they are equivalent fractions.

We have multiplied the numerator and denominator of the first fraction by 2.

2) Cross-products Rule: Product of means = Product of Extremes

a

b=c

d a . d extremes b . c means a . d = b . c

If the cross products are equal, then the two ratios form a proportion.

Ex.: 5

7 = 10

14 5 . 14 = 7 . 10 70 = 70 So the two photographs form a proportion.

Exercises

1. Write a proportion for each word problem. Use the cross-products rule to check.

a) Four wallet-size photos cost $ 1.60, so 8 photos cost $ 3.20. 4

1.60 =

8

3.20

4 x 3.20 = 1.60 x 8 12.8 = 12.8


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b) In 45 minutes a plane travels 300 km, so in 30 minutes it travels 200 km.

PROBLEM SOLVING

To find a missing term, n, in a proportion we use cross-products rule.

Example: Find the cost of 9 grams of gold if 5 grams of gold cost 400 .

To solve this problem:

- 1st Set up the proportion in a fraction way: grams

cos t =

grams

unknowncos t,n

5

400 =

9

n

- 2nd Cross multiply: The missing value equal to the cross product of the diagonal of the two given numbers divided by the remaining number.

5 .n = 400.9 n = 400•9

5= n = 3600 : 5 = 720

So 9 grams of gold is $720

Notice, the missing term can be at any place in the proportion.

Exercises: Find the missing element symbol in each proportion:

1) 2

5=m

10 2)

x

7=3

21 3)

6

4=12

n 4)

n

12=30

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Can you solve these problems?

1. If 3 bananas cost 75 cents, How much is 20 bananas?

2. A car travels 128 km on 15 litres of gas. Which choice below will tell you how far the car can travel on a full tank of gas that holds 60 litres?

a) 128 x 15 x 60 b) 128x60

15 c)

128x15

60

3. This picture must be enlarge to a width of 10 cm. What will be the height of the enlargement?

6 cm a) 12 cm b) 20 cm c) 15 cm 4 cm Compare your answers in groups and invent similar ones to show to your class.


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3. DIRECT PROPORTION

Two magnitudes are directly proportional: if we multiply one of them by a number,

then the same number multiplies the other.

If 3 pens cost 6 , then 6 cost the double: 12 . The number of pens and their price are

directly proportional magnitudes. :

So we can say that when one quantity is increased, the other quantity is also increased. If we decrease a quantity, the other is also decreased.

Example: Time and distance: more time more distance

If we walk 10 km in 4 hours, How many km can we walk in 7 hours?

4 h __________ 10 km 4

7=10

x x=

7•10

4 = 17,5 km

7h ___________ x km

Also we can write the proportion like this: 10

4=x

7 and the result is the same. K (constant

of proportion) is 2,5 = 10 : 4

Give more similar examples and propose them to the class: 1. A cinema ticket costs 8 , more tickets more money.

Unitary method: you can always find the value of one item and multiply by the numbers of items: 10 : 4 = 2,5. So 2,5 km per hour.

http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i7/bk8_7i2.htm

3.1 RULE OF THREE

Observe that to solve a direct proportion problem we set three elements and we can find a missing one in the direct proportion using cross multiplication. This way of solving direct proportion problems is called simple rule of three or direct rule of three.

A_________B A

B=C

x x =

B.C

A We have 3 elements: A,B,C; and

using the rule of proportion we


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C__________ x can find the fourth element, x.

Solved problem:

How long will it take to a racing car traveling at a rate of 50 mph (miles per hour) to go

450 miles?

50m________1h 50

450=1

x x=

450•1

50 = 9 So the car will travel 9 h

450m _______ x

Exercises: Solve these exercises in groups and propose similar ones to your class.

1. How far will a bus go in 90 minutes if it travels 80 km an hour?

2. How much will you pay for a year if your rent for 3 months is 960 ?

3. What is the rate per hour for traveling 220 km in 80 minutes?

4. If 1 kg of ham is 15 , How much is 125g?

5. An electrician charges 42 per 2 hours of work, how much will he earn for 15 hours?

In groups invent word problems which solution will be (the 1st one is done for you):

a) 3 chocolate boxes are 750 g: Ex: Ten chocolate boxes weigh 2,5 kg, How heavy is 3

chocolate boxes?

b) The bike spends 55 minutes in 7 km c) Ten calculators cost 50

3.2 APPLICATIONS OF DIRECT PROPORTION

Proportion has many applications in problem solving, like cost. In this section we will consider problems involving scales, percent, mixtures, measurements and geometric relationships.

If a problem sets up a relationship between two quantities and then asks you to extend

that same relationship into a new situation, you should consider using a proportion to solve the problem.

We will show here some examples of real life uses of direct proportion.

For example a person in a picture is 5 cm high and in real life he is 1,75 m

tall. Which is the ratio between the picture and the reality?


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To solve this problem we have to set a proportion: 5

175=1

x x =

175•1

5= 35

So the ratio is 1 : 35 and this is called the scale. The picture is 1: 35 scale; hence 1 cm in

the picture is 35 cm in reality.

3.2.1 SCALES

A scale drawing is an accurate picture of something, but different in size.

This is very useful in real life to read maps, for example, and compare distances. If we know the scale we can calculate the real dimensions of the objects.

Scale: the ratio of the picture measure to the actual measure.

Scale ratio= Scale meaure

actual measure

http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/scale.html

LOOK AT THE MAP OF CALIFORNIA

THE SCALE IS 1: 100 km , so 1 cm on the map represents 100 km in reality or in cm

1 : 1 000 000

What’s the distance between Los Angeles and San Diego? It’s 1,5 cm so 1,5 x 100 = 150 km approximately.


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Continue building questions like these ones in your groups

What’s the distance between San Francisco and Sacramento?

How far is Palm Springs from San Bernardino?

SCALE PRACTICE 1 Reading maps

Divide the class in groups. Give them maps of different places (you can get them from a tourist information office or ask the students to bring some to class).

- 1st They have to find the scale

- 2nd Each member of the group must write a question about distances.

- 3rd All members write the questions and answer them individually

- 4th They check the answers together

- 5th Prepare a poster with the project and present the work to the class: explain the map, the scale and some questions. Leave other students to ask questions.

http://www.bbc.co.uk/skillswise/numbers/measuring/distance/quiz.shtml

SCALE PRACTICE 2 My house map

First solve this problem: A house map is a 1:25 scale. What is the measurement of Marina’s room if the dimensions on the map are 16 cm and 12 cm?

A scale 1:25 means that 1 cm on the map is 25 cm in the reality. So 16 cm on the map are 25 x 16 = 400 cm = 4 m; and 12 cm on the plane are 25 x 12 = 300 cm = 3 m in reality.

Then, Marina’s room measures 4 m of length and 3 m of width.

Ask the students to draw scaled maps of their room with furniture for homework.

For example: draw a scale map of your room 1:50.

So 1 cm on the picture is 50 cm in reality.

Take those maps to class and do a dictation

From one student to the rest:


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- DRAW A BED OF 90 X 1,90 So, the bed drawing will be 1,8 x 3,8 cm

One student can do it on the blackboard.

3.2.2 PROPORTIONAL SHARING-OUT (REPARTO PROPORCIONAL)

It’s a proportional distribution of something, for example in prize-giving.

Solved problem:

Tom, Cris and Paula won a lotto prize of 600 . Together they have 30 tickets, so each ticket = 20 . How much will each friend have?

Number of tickets Prize

Tom 10 10x20 = 200

Cris 12 12x20 = 240

Paula 8 8 x 20 = 160

Each friend will be paid according to the number of lottery tickets that have bought.

Exercises:

A bookshop offers a commission of 1000 to share out among the five employees for the 250 books sold on the last book fair. Complete the missing data on the next table:

Book seller

Number of Books Commission

Paul 40

Rose 65

Patricia 50

Angel 55

Lilliam

http://www.gcsemathstutor.com/ratio.php#

Solved problem:

Tom, Palm and Richard have 108 stamps in a collection, which they share out in the ratio

2:3:7. How many stamps does each person receive?


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1st total ratio parts: 2+3+7 = 12

2nd divide the amount by the total parts: 108 : 12 = 9

3rd Multiply each part by its ratio: 9 x 2 = 18 for Tom, 9 x 3 = 27 for Palm and 9 x 7 = 63 for Richard.

Or using rule of three: 108 ____12 x = 108 x 2 : 12 = 18 stamps for Tom

x ______ 2

3.2.3 SIMILAR FIGURES

Corresponding sides of similar figures are in proportion.

2cm 2,5cm 2

2,5=1,5

2

Use grid paper to draw similar figures to these ones in proportion:

3.2.4 PERCENTAGES

Percentages are also ratios whose denominator is100. 50% = 50

100. So we consider the

unit divided by 100.

Percentages compare directly proportional magnitudes, so we can use simple rule of three to solve them or reduce to the unit.

Solved example 1: In a class of 20 students, 30 % have pets at home, How many students have pets at home?

30_____100 30

100=x

20 x =

30•20

100= 600

100 = 6 students have pets at home

x ______20

2. In a class 3 out 5 students are girls. We can express that ratio as a fraction: 3

5 three fifths

of the class are girls. What is the percentage?

_1,5_

_2 _


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3:5 3

5= 0,6 =

60

100= 60% or 3--------- 5 If 3 out of 5 are girls, then x girls are in

x -------- 100 100

If we are 25 students, which percentage are boys?

Complete the table with ratios, fractions, decimals and percentages:

RATIO FRACTION DECIMAL %

3:5 3

5

0,6 60

100 =

60%

= 45 %

0,2

1

2

5 : 3

http://www.mathslice.com/percentgrid_ws.php

http://www.mathslice.com/percent1_ws.php

Percentages have also the application in cost: taxes, discounts, interest of money…

Solved exercise:

What the price of a bike without VAT of 18% that costs 258 ?

258 is the cost of the final product, 100 % + 18 % VAT, so 118% = 258

118 %_______258 x = 258x100

118 = 218,64

100 _________ x without 18% of VAT


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Wok out the following problems and check your results in groups:

1. A television cost 1200 and then we have to add 18% VAT (IVA). Which is the total price?

2. They have 15 % discount on sales. What is final price of a pair of glasses that costs

125 ?

3. A machine takes 30 h to asphalt 10 km of road, How many hours does it take to asphalt 125 km?

4. In a fieldtrip we spend 300 on food for 20 students. If we are going 25 students how much will it be?

5. How much money do I get from a bank deposit of 1000 at 3,5 interest in 1 year and a final discount of 18 % in taxes from the interest got?

6. What is the price with out VAT (18%) of a mobile phone that costs 125 ? 7. A teacher charges 25 per hour of private lessons. After a week of 8.30 h of classes, He has to pay 18% taxes, How much does he earn?

8. In a shop they have two offers: 3 t-shits for 51 or 5 t-shirts for 84 . Which is the best buy?

9. On a recipe for pasta we can read: -200 g of spaghetti, - 50 ml oil and 100 g tomato sauce. Calculate the amount of oil and tomato sauce for 350 g of pasta.

10. Andrew had 12 correct answers of a 16 questions test. Which mark will he have out of 10? A) 8 B) 7 C) 7,5 D) 6,5

3.2.5 PER THOUSAND

Instead of percent % where we divide by 100, we consider now 0 /00

where we divide by 1000.

Here we consider a unit divided in 1000 parts.

Solved example:

How many bins will be placed in a town of 4560 inhabitants if the want to put

5 bins per thousand?

5 0 /00 of 4560 = 5 x 4560 : 1000 = 22,8 rounded 23 bins

Or 5 --------- 1000 This is a direct proportion that can be solved with the rule of three.


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x ---------- 4560

Invent in groups similar word problems and propose them to the class (give the students only 5 minutes to prepare the problems).

3.2.6 MIXTURES AND ALLOYS

Mixtures and alloys are another application of direct proportion.

We want to calculate the price per kilogram of a mixture of 3 kg of red paint at 0,65 /kg, 3 kilograms of green paint at 0,50 /kg and 2 kilograms of blue paint at 0,80 / kg.

3 x 0,65 = 1,95 3 x 0,50 = 1,50 2 x 0,80 = 1,60

1,95+1,50+1,60= 5,05 and Total kg = 3+3+2= 8 So 5,05 : 8 = 0,63 / kg

An alloy is a mixture of metals. In jewellery we use an alloy of gold, copper and silver.

The pure of gold is measured in karat. One karat = 1

24 weight of the alloy in pure gold.

The gold of law must be at least 18 carats. Pure gold has 24 karats (quilates).

Which amount of pure gold does a ring have of 18 karats that weighs 36 g?

18----24

x-----36 or We calculate the 18

24 of 36 g: 18 x 36 : 24 = 27 g of pure gold in the ring.

Solve: What is the weight of 2 earrings of 20 karats that have 24 g of pure gold?

3.2.7 PROPORTION AND GEOMETRY

Expressing different proportional drawings we can transmit information:

We can use geometric figures to express the relationship between two magnitudes:

The number of hotels in town B is greater than in town A 10:8 = 1,25 times more hotels.

The amounts of reading books in different countries A, B and

C: A B C

Create similar proportional drawings to express this information:

- Amount of trees in different countries or towns

- Hours of sun in two countries

8

A

10

B


17

- Expenses in electricity in two buildings, …

http://www.homeschoolmath.net/worksheets/proportions.php

3. INVERSE PROPORTION

On Sunday is John’s birthday. The present price is 54 . We

want to share the cost of the present, how much do we have to

pay if we are 3 friends, and if we are 6?

54 : 3 = 18 each friend

54 : 6 = 9 each friend

Obviously if we are more friends, we will pay less money. Therefore, the price and the payment per friend are related in an inverse proportion. The product between friends and price is constant and always equal to 54.

3 x 18 = 54 6 x 9 = 54

Can you calculate how much will each one pay if we share with 4 friends?

18 ---------- 3 friends 18 • 3 = x • 4 x = 18• 3

4= 13,5 per friend.

x ----------- 4 friends

Or with equivalent fractions: 18

x = 3

4 and we inverse one of the fractions 4

3 Then we have:

18

x= 4

3 and by cross product law 18 • 3 = x • 4

An inverse proportion is when one value increases as the other value decreases.

More explicitly: if two quantities a and b are in inverse proportion, then their product will be constant.

Exercises:

1. Complete the table that represents the number of students and the money they have to pay for a bus that costs 250 for a fieldtrip to Cáceres.

Number of students

40 45 50 53

/ students 4,55


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2. Complete the sentences:

a) When two quantities are related inversely if we double the value of one, the other is reduce to half / double and 1/3 of the value of one quantity correspond to the third / triple of the other.

b) Are these magnitudes related in a direct (D) or an inverse (I) proportion?

1. Price of photocopies and number of photocopies

2. Speed of a car and time

3. Cost of a flight and number of passengers

4. Cost of a rent car shared and number of travelers

5. Portions of a cake and number of guests

6. Litres of gas and km runs

Ask students to imagine more examples of both proportions. We can make a chart showing some of them:

3. Solve these problems

Example:

To make a fence we place 500 posts separated at a distance of 3 m. How many posts do

we need for a distance of 2 m?

3 m ------------ 500 posts This is an inverse proportion: less distance, more posts.

2 m ------------- x posts We have 2 ways of solving it:

Inverse rule of three

3 • 500 = 2 • x x = 3•500

2 = 750 posts

Unitary method

We calculate the numbers of posts at a distance of 1 m.

DIRECT PROPORTION INVERSE PROPORTION


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We need 500 posts to place at every 3 m, so at a distance of 1 m we need the triple: 500 x 3 = 1500. Therefore we will need the half for a distance of 2 m: 750 posts

Continue solving the following problems:

1. 2 people take 3 hours to unload a lorry. If there are 5 people, how long does it take?

2. If a car takes 30 minutes at 60 km / h to run a distance, at what speed must it drive to take 20 minutes?

3. Two workers made a wall in 8 hours, how many hours does it take for 5 workers?

4. If 8 people finish a work in 12 days, then how many days will it take for 3 people?

5. The groceries at home of 4 members are enough for 30 days. If one guest comes and stays with them, how many days will the groceries last?

B

A

A

B

3

3

3

3

3


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A) 1680 B) 100800 C) 420

A) 0,75% B) 0, 15% C) 0,25%

A) 7 km B) 5 km C) 9 km

A) 18 ´ B) 32 ´ C) 33


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A) B) 20 minutes C) 25 minutes

10. If two taps fill a swimming pool in 3 hours, how many taps will we need to fill it in 30´ minutes?

A) 6 B) 8 C) 12

EXTENSION: COMPOUND PROPORTION

The proportion involving more than two quantities is called compound proportion. The quantities could be directly related or inversely related or both.

SOLVED EXAMPLES:

A) 195 men working 10 hours a day can finish a job in 20 days. How many men employed

to finish the job in 15 days if they work 13 hours a day?

Days inverse Hours inverse Men

20 • 10 • 195 = 15 • • x x = 20•10•195

15•13

B) A soap factory makes 600 units in 9 days with help of 20 machines. How many units can

be made in 12 days with the help of 18 machines?

Machines inverse Days direct Units

20 9 600

18 12 x

20•9

600=18•12

x 20 • • • •

C) It is easier to solve some problems by reducing them simple problems of direct or

inverse proportionality.

For example:30 students from 2nd

grade traveled to Paris and

paid 1710 for 3 nights at a hotel. Another group of students

from 4th

grade decides to go also there. They have collected

2660 , how many nights could they pay?


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The number of students and hotel price are directly related, and the number of students is inverse to the number of nights.

Therefore, we can reduce the problem to a simple proportion getting the price per night of each student:

1710 : 30 = 57 for 3 nights.

Each student from 4th grade has 2660 : 20 = 133 to spend X number of nights, so:

57

3=133

x 57x = 3 • 133 x =

3•133

57 = 7 days

D) BANK INTEREST

Calculating bank interest is a problem of direct compound proportion. The bank interest is the profit we get for an amount of money in a bank deposit during a period of time or the interest charged by a bank when we have a loan. It is express in %.

So the three magnitudes involved are directly related: money, time and interest.

There is a formula to calculate it: profit = c • r • t

100 where c = capital or money, r = rent or

interest, t = time

If you invest money on Cherry bank, for 100 you will get 3 for 1 year and for 1 you

will get 0,03

How much will you get for 650 during 3 years?

There are three ways of solving this problem:

1) Formula Profit = 650• 3• 3

100 = 58,50 profit in 3 years.

2) Reduce to the unit: 650 • 0,03 = 19,5 in one year, therefore 19,50 • 3 = 58,5 in 3 years.

3) Set the proportions: 100

650=3

x=1

3 where x =

650• 3• 3

100 = 58,5

Work out this problem and invent similar ones:

1) Andrew invested 2500 on a bank and after a year he got a profit of 56 . How much will he get if he invest 4000 during 3 years?

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unit 8: ratio and proportion - Working on Maths in English

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Transcript of unit 8: ratio and proportion - Working on Maths in English