3.2 Ratios
Looking again at the chocolate bars in Figure 3.1, you can see that 3 out of 8 pieces are darkened in the first, 6 out of 16 in the second, 9 out of 24 in the third and 15 out of 40 in the fourth. These pairs of numbers, which form equivalent fractions, are said to be in the same ratio. So any pairs of numbers that form equivalent fractions are in the same ratio.
Which of the following pairs of numbers are in the same ratio as (6, 20)?
6/20 = 3/10 = 12/40 = 30/100, so these are all equivalent fractions, and therefore the pairs of numbers from which they are formed are in the same ratio. Using the lowest whole numbers, we say that these pairs of numbers are all in the ratio 3 to 10. The fractions 12/30 (= 6/15) and 24/100 (= 6/25) are not equivalent to 6/20, so (12, 30) and (24, 100) are not in the same ratio as (6, 20).
Ratios are often written as two numbers separated by a colon (:). So, a fraction such as 3/10 is equivalent to a ratio of 3 to 10 and this is often written as 3 : 10.
You need to take care when asked to give the ratio of two numbers, as the following example shows.
Suppose 2 out of 10 people in the UK drink bottled water. What is the ratio of people who drink bottled water to those who don’t?
The ratio is 2 : 8.
Did you fall into the trap and answer 2 : 10? This is the ratio of people who drink bottled water to the total number of people. Of course if we’d asked what is the ratio of people who don’t drink bottled water to those who do, the answer would have been 8 : 2. So, always read the question carefully!
Ratios are particularly useful where the relative proportions of two or more parts of a whole are being considered. For example, the ratio of males to females in the general population of the UK is about 1 : 1.
Question 3.2
In a random group of students, 15 were men and 8 were women.
a.What was the ratio of women to men in the group?
The ratio of women to men in the group is 8 : 15.
b.What fraction of the total group were women?
The total number of students is , so of the group are women.
It is sometimes helpful to express a ratio in the simplest possible form, so the ratio 2 : 8 would be expressed as 1 : 4 (dividing each number by 2) and the ratio 60 : 40 would be expressed as 3 : 2 (dividing each number by 20).
Express the ratio 25 : 20 in the simplest possible form involving whole numbers.
25 : 20 is the same as 5 : 4 (dividing both numbers by 5).
In addition, you may sometimes be required to express a ratio as ‘something : 1’, where the ‘something’ is no longer necessarily a whole number. To convert a ratio to the ‘something : 1’ form, divide both numbers by the number on the right hand side of the ratio, so 3 : 2 becomes 1.5 : 1 and 1 : 4 becomes 0.25 : 1.
Express the ratio 5 : 4 in the form ‘something : 1’.
5 : 4 is the same as 1.25 : 1 (dividing both numbers by 4).
Express the ratio 365 : 20 in the form X : 1 where X is rounded to the nearest whole number.
365 : 20 is the same as 18.25 : 1 (dividing both numbers by 20). When rounded to the nearest whole number this gives a ratio of 18 : 1.