When x is added to each of 9, 15, 21 and 31 the numbers so obtained are in proportion. What is the mean proportional between the numbers (3x - 2) and (5x + 4)?

Solution (By Examveda Team)

$$\eqalign{ & 45 + 5x = 63 + 3x \cr & 2x + 18 \cr & x = 9 \cr & {\text{Mean proportional}} = \sqrt {\left( {3x - 2} \right)\left( {5x + 4} \right)} \cr & = \sqrt {\left( {9 \times 3 - 2} \right)\left( {5 \times 9 + 4} \right)} \cr & = \sqrt {25 \times 49} \cr & = 5 \times 7 \cr & = 35 \cr} $$