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Solution:

15 is the mean proportional between the number.

(10 + x) : (16 + x) :: (22 + x) : (32 + x)

(10 + x)/(16 + x) = (22 + x)/(32 + x)

We can solve this using cross multiplication or Using Componendo or Dividendo but hare we can solve this using equal difference between 16 – 10 = 6 and 32 – 22 = 10. For makes calculation easy and saves time.

6 × 5 = 30 and 10 × 3 = 30

Now,

5 (10 + x) = 3 (22 + x)

50 + 5x = 66 + 3x

2x = 16

x = 8

Now,

Let the mean proportional between the numbers (x + 1) and (3x + 1) be A, then

(x + 1) : A :: A : (3x + 1)

(8 + 1)/A = A/(3 × 8 + 1)

A2 = 9 × (24 + 1)

A = √(9 × 25)

A = 3 × 5 = 15