MODULE 1
INTRODUCTION
Ratios, proportions, and percentages are all about comparing and measuring
stuff. They can help us measure popularity, or find how rare something might be, or
figure out a discount at our favorite store.
A ratio compares any two numbers, like the number of students who can do standing
backflips to the number of students who can't. A proportion compares two equivalent
ratios. A percentage compares any number to a total out of 100, like the number of
corkscrew flips we landed out of 100 tries.
Ratios crop up often in official statistics. The government wants the teacher–
pupil ratio in schools to be increased to one teacher to thirty pupils or less. The birth
rate has fallen: the ratio of children to women of child bearing age has gone down. It
used to be 2.4 to 1, and now it is 1.9 to 1. Predictions for the ratio of working adults to
retired adults is disturbing. Predictions are, that by 2030 the ratio will be two working
adults to every retired person, instead of three to one now, and four to one ten years
ago.
Often ratios are implicit in the language rather than explicitly referred to: one
teacher for 30 pupils; 2.4 children per woman of child bearing age; one retired person
per two working adults. The word ‘per’ often indicates that the concept of ratio is being
used.
Proportion is another way of expressing notions of part and whole. You might
say that the proportion of village inhabitants who are children is a quarter, or that the
proportion of fruit juice in the punch is two thirds, or that the proportion of sand in
the concrete is three quarters.
In a recipe the quantity of each ingredient needed depends upon the number of
portions. As the number of portions increases, the quantity required increases. The
quantity per portion is the same. This is called direct proportion. The quantity is said
to be directly proportional to the number of portions. If 2 potatoes are required for
one portion, 4 will be required for two portions etc. A useful method for direct
proportion problems is to find the quantity for one and multiply by the number you
want.
In Section 2.2 you saw that direct proportion described relationships between
two quantities, where as one increased, so did the other. Sometimes as one quantity
increases the other decreases instead of increasing. This is called indirect proportion.
Team tasks are often an example of this. The time taken to do a job is indirectly
proportional to the number of people in the team.
A difficulty with the real-life context of such problems is that, in many cases, it
is hard to believe that people working in a team will work at the same rate regardless
of the size of the team, unless the team work independently, i.e. ‘in parallel’. The main
idea behind this type of problem is that increasing the number of people working
decreases the time taken to complete the task. (An obvious exception to this is
decision-making in a committee: if two people can reach a decision in an hour, four
people are liable to take twice as long!)
Such problems can be compared with certain problems involving speed: doubling the
number of people working is the same as doubling the speed at which the team work.
In either case the time is halved. It is useful to find out how long it would take one
person to do the whole job, then divide by the number of people sharing the work. This
is a good approach to most indirect proportion problems.
Objectives
At the end of this lesson, you should be able to:
1. Identify Ratio, Proportion and Percent, and Geometry and Measurements;
2. Solve Ratio, Proportion and Percent, and Geometry and Measurements; and
3. Reflect the importance Geometry and measurements in real life;
Lesson: Ratio, Proportion, Percentage, Geometry and Measurements
Activity 1.1
Ratio
A ratio shows the relative sizes of two or more values.
Ratios can be shown in different ways: using the ":" to separate example values, using
the "/" to separate one value from the total, as a decimal, after dividing one value by
the total, as a percentage, after dividing one value by the total.
We will start our discussion of ratio by considering the following sets:
A= { Omar, Sammy, Laarni } ; n(A) = 3 children
B= { OOOOOO } ; n(B) = 6 guavas
Can we compare the size of the two sets by one-to-one correspondence? No,
because the number of Set B is greater than of Set A.
Observe that if we regroup the elements of Set B so that we can pair each group
with the elements of A, we will have
A= { Omar, Sammy, Laarni }
B= { OOOOOO }
Is there a one-to-one correspondence now between the two sets? No. The
correspondence is not one-to-one rather it is one-to-two. This means that for every
child in Set A there are two O’s in set B. The set of children as 2 is related to 1.
Such relation between the elements of the two sets is called ratio. We therefore
say that the ratio of A to B is “1 to 2” and can be written as
1
2
or 1:2. The notation “1:2”
is read as “one is to two.” A ratio may be written as a common fraction or by using the
symbol :. Thus, we can say that:
The ratio of the number a to number b, where b is not equal to zero is
𝑎
𝑏
or a:b.
Example 1: There are 20 males and 23 females in a class. What is the ratio of the
males to the females in the class? Females to males?
The ratio of the males to the females is 20 is to 23 or 20:23. The ratio of the
females to the males is 23 is to 20 or 23:20.
Example 2: If 4 girls received a box with 24 pieces of candies, what is the ratio of the
girls to the candies?
The ratio of the girls to the candies to 4:24. This ratio may be written as the
fraction
4
24
which can simplified as
simplest form is
1
6
4
24
1. 4
=6
. 4
1
= 6. Thus the ratio of girls to the candies in
or 1:6.
Check-up Questions.
1. Express the following fractions as a ratio of the form a:b where a is the
numerator and b the denominator of the fractions:
a.
2
3
b.
4
5
c.
7
6
d.
20
10
2. Write the following ratios in simplest form: 1) expressing it as a common
fraction; and 2) using the ratio symbol
a. 20 to 40
b. 24 to 18
c. 32 to 64
d. 10 children to 5 books
Proportion
Our example in the previous section shows that the ratio of the girls to the
candies is 4:24 or in simplest form 1:6 that is
4
24
=
1
6
or 4:24= 1:6. This statement is an
example of a 6proportion. We defines proportion as statement that indicates the
equality of two ratios.
In our study of fraction if
𝑎
𝑏
=
𝑐
𝑑
, where b and d are not equal to zero, then a . d=
b . c. the proportion can be written as a:b=c:d or ad = bc, read as “a is to b as c is to
d.” In the proportion a : b = c : d, the outer numbers a and d are called the extremes
and the inner numbers b and c are called the means. That is:
We can say therefore that in a proportion the product of the means is equal to
the product of the extremes. This principle is very useful in problem solving as shown
in the succeeding examples.
Example 1: A housewife found out that with the rice she has just bought, she needs 4
cups of water for every 3 cups of rice. If she will cook 12 cups, how many cups of water
will she need?
This proportion is expressed:
4 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟
3 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑟𝑖𝑐𝑒
=
𝑥 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟
12 𝑐𝑢𝑝𝑠 𝑜 𝑟𝑖𝑐𝑒
or 4 : 3 : : x : 12
3x = 4 . 12
3x= 48
𝑥=
48
3
X= 16 cups of water
Example 2: If a man was able to travel 250 kilometers in 5 hours driving, how many
kilometers would he cover in 9 hours at the same rate of speed?
This proportion is expressed:
250 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟𝑠
𝑥 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟𝑠
=
5 ℎ𝑜𝑢𝑟𝑠
9 ℎ𝑜𝑢𝑟𝑠
Or
250 : 5 = x : 9
5x = 250 (9)
𝑥=
2250
5
X = 450 kilometers
Check-up Questions
1. Identify which of the following are proportions:
a. 4 : 2 = 5 : 3
b. 5 : 6 = 10 : 12
c. 14 : 7 : : 10 : 5
d.
3
5
=
12
20
2. Solve for x in the following proportions:
a.
𝑥
5
=
10
6
b. 4 : 8 = x : 120
c. X : 3 = 12 : 18
d. 5 : x = 30 : 42