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.)
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