Page 232 - Start Up Mathematics_6

12. In class VI of a school having 50 students, 20 play cricket, 10 play table tennis and 15
play badminton. The remaining students do not play any game. No student is allowed to
play more than one game. Find the ratio of the number of students:

(a) playing cricket to the number who play table tennis.
(b) playing table tennis to the number who play badminton.
(c) playing some game to the number who do not play any game.
(d) playing some game to the total number of students.

Proportion
If two ratios are equal, we say they are in proportion.

If four numbers a, b, c and d are such that the ratio of the first two is equal to the ratio of the last
two, i.e., a : b = c : d, then we say a, b, c and d are in proportion.
The symbol ‘::’ can also be used to denote equality of two ratios, i.e., a : b : : c : d, read as
‘a is to b as c is to d’ also means a, b, c and d are in proportion. Here a and d are called extreme
terms and b and c are called mean terms. The number d is also known as the fourth proportional
to a, b and c.

For example, 3, 4, 18 and 24 are in proportion because 3 : 4 = 18 : 24.

If a, b, c, d are in proportion, which implies, a = c , then
b d
a × d = b × c, i.e., Product of the extremes = Product of the means

Continued Proportion

Three numbers a, b, c are said to be in continued proportion if a, b, b and c are in proportion.
a b
i.e., = ⇒ b = ac
2
b c
If a, b, c are in continued proportion then b is known as the mean proportion of a and c and c is known
as the third proportion.

Example 13: Determine if the following are in proportion:
(a) 3, 12, 5, 20 (b) 10, 20, 30, 40

Solution: (a) 3, 12, 5, 20
3 5
Ratio of 3 to 12 = = 1 : 4 Ratio of 5 to 20 = = 1 : 4
12 20
Since, 3 : 12 = 5 : 20
Therefore, 3, 12, 5, 20 are in proportion.

(b) 10, 20, 30, 40
10 30
Ratio of 10 to 20 = = 1 : 2 Ratio of 30 to 40 = = 3 : 4
20 40
Since, 10 : 20 ≠ 30 : 40
Therefore, 10, 20, 30, 40 are not in proportion.

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