Ratio of two quantities

Ratio of two quantities

The ratio of two quantities shows how one quantity is compared to another quantity. The ratio of two quantities can be written in three ways which are:

1. In words: For example, 2 cows to 3 goats. The word "to" represents the ratio of cows to goats.
2. Using a colon (:): For example, 2 cows to 3 goats can be written as 2 : 3.
3. In fractions: For example, 2 : 3 can be written as $\frac{2}{3}$.

Equivalent ratios

Pairs of ratios of two quantities can be compared. Sometimes, two ratios may seem different but are equivalent. Their equivalence can be observed when the ratios are simplified to their lowest terms.

For example, 2 : 3 = 8 : 12. The right-side ratio becomes 2 : 3 when simplified to its lowest term. Therefore, it is equivalent to the left-side ratio.

To simplify the ratios, the following steps are useful:

1. Find the Greatest Common Factor (G.C.F) of the given numbers.
2. Divide each number in a given ratio by the G.C.F of the numbers, and then simplify.

Example 1

A teacher's guide has 360 pages and a pupil's book has 480 pages.

(a) Write down the ratio of the number of pages in the teacher's guide to that of pupil's book.

(b) Find the ratio of the number of pages in the pupil's book to the total number of pages in the two books.

Solution

(a) Write the number of pages in the teacher's guide followed by the number of pages in the pupil's book separated by a colon (:). That is, Teacher's guide : Pupil's book = 360 : 480

Find the G.C.F of the given numbers, and then divide each number by the G.C.F. In this case, the G.C.F of 360 and 480 is 120.

Thus, $360 \div 120 : 480 \div 120 = 3 : 4$

Therefore, the ratio of the pages in the teacher's guide to the pupil's book is 3 : 4.

(b) Find the total number of pages in the two books: $360 + 480 = 840$.

Using colon, separate the page numbers of the pupil's book from the total number of pages in the two books.

Start with the number of pages in the pupil's book, followed by the total number of pages in both books. That is,

Pupil's book: Total number of pages in both books = 480 : 840.

Find the G.C.F of the numbers involved, and then divide each number by the G.C.F. In this case, the G.C.F of 480 and 840 is 120.

Thus, $480 \div 120 : 840 \div 120 = 4 : 7$.

Therefore, the ratio of the pages in the pupil's book to the total number of pages in the two books is 4 : 7.

Example 2

A class has 5 girls and 7 boys.

(a) Write the ratio of the number of girls to that of boys.

(b) Find the ratio of girls to the total number of pupils.

Solution

(a) Find the G.C.F of the numbers given, and then divide all numbers by the G.C.F of 5 and 7 which is 1.

Number of girls : Number of boys = 5 : 7

The G.C.F of 5 and 7 is 1

So, $5 \div 1 : 7 \div 1 = 5 : 7$.

Therefore, the ratio of girls to boys is 5 : 7.

(b) The total number of pupils is $5 + 7 = 12$.

Therefore, the ratio of girls to the total number of pupils is 5 : 12.

Example 3

Simplify 16 : 28.

Solution

The GCF of 16 and 28 is 4.

Divide by G.C.F throughout:

$16 \div 4 : 28 \div 4 = 4 : 7$.

Alternative solution 1

$16 : 28 = 4 \times 4 : 4 \times 7 = 4 \times \frac{4}{4} : 4 \times \frac{7}{4} = 4 : 7$.

Alternative solution 2

Write the numbers in fraction. That is, $\frac{16}{28}$, and then simplify by using the common factors:

$\frac{16}{28} = \frac{8}{14} = \frac{4}{7}$.

Therefore, 16 : 28 = 4 : 7.

Example 4

Simplify 35 : 90.

Solution

The G.C.F of 35 and 90 is 5. Thus, divide each number by 5.

$35 \div 5 : 90 \div 5 = 7 : 18$.

Therefore, 35 : 90 = 7 : 18.

Example 5

Simplify 18 : 12.

Solution

The G.C.F of 18 and 12 is 6. Thus, divide each number by 6.

$18 \div 6 : 12 \div 6 = 3 : 2$.

Therefore, 18 : 12 = 3 : 2