Page 97 - ICSE Math 7

Proportion

Four quantities p, q, r and s are said to be in proportion if p : q = r : s. Maths Info
It is denoted by p : q :: r : s and is read as p is to q as r is to s. The The fourth term of a proportion
numbers used in the proportion are called its terms, where p is the first is called fourth proportional.
term, q is the second term, r is the third term and s is the fourth term.
Also, the first and fourth terms are known as extreme terms or extremes and the second and third
terms are known as mean terms or means. It is not necessary that all the four quantities forming a
proportion are of the same kind. The first two quantities should be of the same kind and the last two
quantities should be of the same kind. For example, 2 kg, 5 kg, ` 10, ` 25 are in proportion because
2 : 5 = 10 : 25. The product of extremes is equal to the product of means. This is also known as
p
r
cross-product rule according to which if = then ps = qr.
q s
Example 12: Are the numbers 18, 14, 45 and 35 in proportion?
18 9 45 9
Solution: 18 : 14 = = and 45 : 35 = =
14 7 35 7
As, 18 : 14 = 45 : 35, therefore 18, 14, 45 and 35 are in proportion.
1
Example 13: Find the fourth proportional to 1 , 4 and .
15 5
Solution: Let the fourth proportional be x.
1 1
So, : 4 = : x
15 5
1 1
∴ × x = 4 × (Product of extremes = Product of means)
15 5
1
⇒ x = 4 × × 15 = 12
5
∴ Fourth proportional is 12.

Continued proportion
Three quantities p, q and r are said to be in continued proportion if p : q = q : r, i.e., if the ratio between
the first and the second quantity is equal to the ratio between the second and the third quantity. If p, q and
p q
2
r are in continued proportion, then = . Therefore, by cross-product rule, q = pr. If p, q and r are in
q r
continued proportion, then q is known as the mean proportional between p and r and r is known as the
third proportional to p and q. For example, 6, 12, 24 are in continued proportion because 6 : 12 = 12 : 24.

Points to remember

• If ps = qr, then four proportions can be formed by taking:
p, s as extremes and q, r as means, namely, p : q = r : s and p : r = q : s
p, s as means and q, r as extremes, namely, q : p = s : r and r : p = s : q
• Three quantities p, q, r of the same kind are said to be in continued proportion if p : q :: q : r, i.e.,
p
q
2
if = ⇒ q = pr here q is known as the mean proportional, p is known as the first proportional
q r
and r is known as the third proportional.
Example 14: Find the mean proportional between 72 and 8.
Solution: Let x be the mean proportional between 72 and 8.
∴ 72, x and 8 are in continued proportion.

83