Proportion

Ask a Question, Get an Answer!

Type your question here

Hundreds of tutors are online and ready to help you right now!

Four quantities a, b, c and d are said to be in proportion, if a:b=c:d.

a and d are called extremes of the proportion.

b and c are called means of the proportion.

a:b=c:d

Product of extremes is equal to the product of means in a proportion. In a proportion a:b=c:d, the fourth term 'd' is called the fourth proportional.

The proportion a:b = c:d is often expressed as a:b::c:d and is read as a is to b is the same as c is to d.

Continued proportion

Three quantities are said to be in proportion if the ratio of the first to the second is equal to the ratio of the second and third.

a, b and c are said to be in continued proportion if a:b = b:c

If a, b and c are in continued proportion then b (second quantity) is called the mean proportion and c (third quantity) is called the third proportional of a and b.

Illustrative examples

1) Find the value of x if 2:5::x:15.

Suggested answer:

5x=2 x 15

x = 6

2) What must be added to each of the four numbers 10, 18, 22, 38 to make them a proportion?

Suggested answer:

Let x be added.

380+48x+x²=396+40x+x²

8x=16

x=2

3) Find the fourth proportional to 6, 10 and 12.

Suggested answer:

Let the fourth proportional to be x.

6:10=12:x

6x=120 (Product of extremes is equal to product of means)

x=20

4) Find the mean proportional to 6 and 54.

Suggested answer:

Let x be the mean proportional.

x²= 6 x 54

=2 x 3 x 3=18

x=18

5) Find the third proportional to 5 and 10.

Suggested answer:

Let x be the third proportional to 5 and 10.

5:10=10:x.

5x=10 x 10 (product of extreme is equal to product of means)

6) Find two numbers such that the mean proportional between them is 24 and the third proportional to them is 192.

Suggested answer:

Let the two numbers be x and y.

24 is the mean proportional of x and y

x.y=(24)² …1

192 is the third proportional to x and y.

y³= (24)².24 x 8

y³= (24)³.2³

y=24 x 2=48.

\ the numbers are 12 and 48.

7) If a:b::b:c, prove that a:c=a²:b².

Suggested answer:

a:b::b:c

b²=ac

Suggested answer:

a, b, c are in continued proportion.

Consider the product,

(a+b) (a-b)=a²-b²

=a²-ac

=a(a-c)

(a+b) (a-b)=a(a-c)