Concept of Proportion

In concept of proportion we will learn how a proportion is an expression which states that the two ratios are in equal.

A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. When two ratios are equal, then that type of equality of ratios is called proportion and their terms are said to be in proportion.

For example: If the cost of 5 pens is $ 40 and that of 7 pens is $ 56, then ratio of pens is 5 : 7 and the ratio of their costs is $ 40 : $ 56.

Thus, 5 : 7 = 40 : 56.

Therefore, the terms 5, 7, 40 and 56 are in proportion.

Definition of Proportion:

When four quantities are so related that the ratio between the first and the second quantities is equal to the ratio between the third and the fourth quantities; the quantities are said to be in proportion.

Thus, proportion is equality of two ratios.

In order to represent a proportion; either put the sign of equality (=) between the two ratios or put a double colon (::).

Thus, 3, 4, 9 and 12 are in proportion and is expressed as:

3 : 4 = 9 : 12 or 3 : 4 :: 9 : 12

Consider the following examples to understand the concept of proportion:

(i) What is the ratio of the number of boys to the number of girls in a group of 8 boys and 12 girls?

The required ratio = number of boys/number of girls

     = $\frac{8}{12}$

     = $\frac{2 × 4}{3 × 4}$

      = $\frac{2}{3}$

(ii) What is the ratio of the number of boys to the number of girls in another group of 18 boys and 27 girls?

The required ratio = number of boys/number of girls

                                  = $\frac{18}{27}$

                                  = $\frac{2 × 9}{3 × 9}$ 

                                  = $\frac{2}{3}$

It is observed, in the above explanations (i) and (ii), that the ratios $\frac{8}{12}$ and $\frac{18}{27}$ are equal.

i.e., $\frac{8}{12}$ = $\frac{18}{27}$

or, 8 : 12 = 18 : 27

Such an equality of two ratios is called a proportion and is read as “8 is to 12 as 18 is to 27”.

The numbers 8, 12, 18 and 27 that are used in the proportion, are called its terms, i.e., 8 is the first terms, 12 is the second term, 18 is the third term and 27 is the fourth term of the proportion 8 : 12 = 18 : 27.

Similarly, suppose 5, 12, 25, and 60 are in proportion which is written as 5 : 12 : : 25 : 60 and is read as 5 is to 12 as 25 is to 60.

Also, 5 : 12 = 25 : 60

⇒ $\frac{5}{12}$ = $\frac{25}{60}$

⇒ $\frac{5}{12}$ = $\frac{5}{12}$

Are 3, 5, 18 and 30 in proportion?

The numbers, 3, 5, 18, 30 are in proportion because,

$\frac{3}{5}$ = $\frac{18}{30}$

⇒ $\frac{3}{5}$ = $\frac{3}{5}$

Solved Examples on Concept of Proportion:

1. Consider the ratios 21 : 49 and 33 : 77, and find if they are in proportion.

Solution:

        21 : 49 = $\frac{21}{49}$ = $\frac{21 ÷ 7}{49 ÷ 7}$ = $\frac{3}{7}$ = 3 : 7

and  33 : 77 = $\frac{33}{77}$ = $\frac{33 ÷ 11}{77 ÷ 11}$ = $\frac{3}{7}$ = 3 : 7

21 : 49 = 33 : 49

21 : 49 :: 33 : 77

Hence, 21, 49, 33 and 77 are in proportion.

Worksheet on Basic Concept of Proportion:

1. Determine if the following numbers are in proportion.

(i) 5, 25, 30, 150

(ii) 18, 36, 72, 144

(iii) 16, 30, 24, 45

(iv) 75, 150, 3, 18

(ν) 200, 300, 400, 600

(vi) 224, 34, 68, 112

2. Write True (T) or False (F) against each of the following statements:

(i) $ 200 : $ 300 :: 26 kg : 39 kg

(ii) 50 g : 625 g :: 4 kg: 100 kg

(iii) 39 persons : 195 persons :: $ 7 : $ 35

(iv) 25 ℓ : 1 ℓ :: 40 cm : 16 cm

(v) 16 m : 64 m :: 18 hours : 72 hours

(vi) 80 ml : 5ml :: 192 kg : 12 kg

3. Determine if the following ratios form a proportion.

(i) 50 cm : 25 m and 20 mℓ : 1 ℓ

(ii) 12 : 60 and 10.8 km : 54 km

(iii) 60 persons : 50 persons and 180 ℓ : 150 ℓ

(iv) 300 g : 560 g and 4 m : 7 m

4. Find the value of x in each of the following proportions.

(i) 6 : x :: 78 : 65

(ii) 18 : x :: 27 : 3

(iii) x : 45 :: 52 : 39

(iv) 7 : 24 :: x : 360

(v) 3 : 17 :: 45 : x

(vi) 800 : 300 :: x : 210

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