# CLASS-10RATIO & PROPORTION - MEAN PROPORTION

Mean Proportion (or Geometric Mean)

Let a, b, c be in continued proportion

a           b

Then, ------- = -------

b           c

=> b² = ac

=> b = √ac

Here, b is called the mean proportion or geometric mean between a & c

Example.1) Find the mean proportional to 5, and 80

Ans.) Hence the mean proportion between 5 & 80 = √5 X 80

= √400 = √(20)²

= 20         (Ans.)

Example.2) Find two numbers whose mean proportion is 28 and the third proportional is 224.

Ans.) Let, the required numbers be a & b. Then

Mean proportion between them is √ab

As per the given condition, √ab = 28

=> (√ab)² = 28²

=> ab = 28² ……….(i)

And, also it is given that the third proportional to a & b is 224

So, a : b : : b : 224

a            b

=> ------- = -------    [so, Product of extremes = Product of means]

b           224

=>  224a = b²

b²

=> ------- = 224 …………….(ii)

a

On multiplying the corresponding sides of (i) and (ii), we get –

b²

(ab X -------) = 28 X 28 X 224

a

=> b³ = 28 X 28 X 28 X 8

=> (b)³ = (28 X 2)²

=> b = 56

Substituting b = 56 in (i), we get –

ab = 28²

=> 56a = 28 X 28

28 X 28

=> a = ----------- = 14

56

Thus a = 14, and b = 56

Hence the required number is 14 & 56.      (Ans.)