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

WORKING ON MATHS IN ENGLISH
UNIT 8
RATIO
AND
Isabel Leo de Blas
PROPORTION
1 ST 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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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/numbers/ratio_rev1.sht
ml EXPLANATION AND EXERCISES
http://www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/numbers/ratio_activity.s
html 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
ratio-to-fraction.pdf
-statement-4.pdf
- reduce.pdf
- proportionWS1.pdf
- statement-1.pdf
- ratio-method.pdf
- write ratios.pdf
- Proportions WSAnswer Key 1.pdf - proportional.pdf -singledigit.pdf
- ratio-two-same-units.pdf
- EVALUATION testUNIT 8 RATIO.pdf
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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
UNIT 8
1 ST LEVEL
RATIO AND PROPORTION
"Life is 10% what happens to
us and 90% how we respond to it."
Charles Swindoll
Ratio = 1: 9
Ratio 1 : 7
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,
tuna fish is twice heavier than the small one.
So 2 is the ratio what means that the large
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 =
1
one part of sugar for 2 parts of flour.
2
3
200
400
WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
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
5
=
=5
60
1
So the length is 5 times greater than the width.
In metres is the same result:
3
= 5, so we get the same ratio
0,60
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
4
and 4 / 9
WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
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?
biscuits
2.79 4.45
biscuits
for 3for
pounds
5 pounds
biscuits 7.49 for 7 pounds
Ratio =
1)
2,79
= £ 0,93 per pound
3
2)
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.
cos t
= quotient
weight
4,45
7,49
= £ 0,89 per pound 3)
= £ 1,07 per pound
5
7
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?
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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
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
=4
50gbutter
Recipe 2
100gflour
=4
25gbutter
200 100
Both ratios are equal, so both recipes have the same proportion of ingredients:
=
50
25
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
10
10 : 2 5
and 10 : 14 =
We can see they are equivalent fractions.
7
14
14 : 2 7
We have multiplied the numerator and denominator of the first fraction by 2.
5:7
2) Cross-products Rule:
a c
=
b d
Product of means = Product of Extremes
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 10
=
7 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 x 3.20 = 1.60 x 8
12.8 = 12.8
6
4
8
=
1.60 3.20
WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
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
grams
=
unknown cos t,n
cos t
5
9
=
400
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 m
=
5 10
2)
x 3
=
7 21
3)
6 12
=
4 n
4)
n 30
=
12 24
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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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
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 10
=
7 x
4 h __________ 10 km
x=
7 •10
= 17,5 km
4
7h ___________ x km
Also we can write the proportion like this:
10 x
= and the result is the same. K (constant
4 7
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 C
=
B x
x=
B.C
A
We have 3 elements: A,B,C; and
using the rule of proportion we
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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
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 1
=
450 x
x=
450 •1
= 9 So the car will travel 9 h
50
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?
c) Ten calculators cost 50 b) The bike spends 55 minutes in 7 km
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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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
To solve this problem we have to set a proportion:
5
175 •1
1
x =
=
= 35
175 x
5
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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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
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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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
- 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:
Number of Books
Commission
Book seller
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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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
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
_1,5_
_2 _
2 1,5
=
2,5 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
. So we consider the
100
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
home?
30_____100
students, 30 % have pets at home, How many students have pets at
x
30
=
100 20
x=
30 • 20 600
=
= 6 students have pets at home
100
100
x ______20
2. In a class 3 out 5 students are girls. We can express that ratio as a fraction:
of the class are girls. What is the percentage?
13
3
three fifths
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WORKING ON MATHS IN ENGLISH
3:5 3
60
= 0,6 =
= 60%
5
100
Isabel Leo de Blas
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
3:5
FRACTION
DECIMAL
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
258x100
= 218,64 118
118 %_______258 x=
100 _________ x
without 18% of VAT
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WORKING ON MATHS IN ENGLISH
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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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WORKING ON MATHS IN ENGLISH
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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.
1
The pure of gold is measured in karat. One karat =
weight of the alloy in pure gold.
24
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
Solve:
18
of 36 g: 18 x 36 : 24 = 27 g of pure gold in the ring.
24
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:
10
8
The number of hotels in town B is greater than in town A
10:8 = 1,25 times more hotels.
B
A
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
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WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
- 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
= 13,5 per friend.
4
x ----------- 4 friends
Or with equivalent fractions:
18 3
= and we inverse one of the fractions 4 Then we have:
x
4
3
18 4
=
and by cross product law 18 • 3 = x • 4
x 3
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
students
of
40
45
50
/ students
53
4,55 17
WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
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:
DIRECT PROPORTION
INVERSE PROPORTION
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
= 750 posts
2
Unitary method
We calculate the numbers of posts at a distance of 1 m.
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WORKING ON MATHS IN ENGLISH
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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?
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19
WORKING ON MATHS IN ENGLISH
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20
WORKING ON MATHS IN ENGLISH
A)
Isabel Leo de Blas
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 18 •12
=
x
600
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?
21
WORKING ON MATHS IN ENGLISH
Isabel Leo de Blas
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:
57 133
=
3
x
57x = 3 • 133 x =
3•133
= 7 days
57
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 %.
Bring your money to
CHERRY BANK!! 3% of
interest for 1-year deposit,
don’t miss this great offer.
So the three magnitudes involved are directly related: money, time and interest.
There is a formula to calculate it: profit =
c•r•t
where c = capital or money, r = rent or
100
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
= 58,50 profit in 3 years.
100
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 3 1
650 • 3• 3
where x =
= 58,5 = =
650 x 3
100
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?
22