The average age of fifteen persons is 32 years. If two more persons are added then the average is increased by 3 years. The new persons have an age difference of 9 years between them. The age (in years) of the younger among the new persons is:

Given the average age of fifteen persons is 32 years.

Let the 15 persons be $$\ a_{1}, a_{2}, a_{3}, a_{4}.........a_{15} $$

=> $$ \frac {a_{1}+ a_{2}+a_{3}+a_{4}+.........+a_{15} } {15}$$  = 32

=> $$\ a_{1}+a_{2}+a_{3}+a_{4}+.........+a_{15} = 15 \times 32 $$ 

=> $$\ a_{1}+a_{2}+a_{3}+a_{4}+.........+a_{15} = 480 $$.......(1)

If two more persons are added then the average is increased by 3 years.

Let those persons be $$\ a_{16}, a_{17} $$ respectively

$$ \frac {a_{1}+ a_{2}+a_{3}+a_{4}+.........+a_{15}+a_{16}+a_{17} } {17}$$  = 35

=> $$ \ {a_{1}+ a_{2}+a_{3}+a_{4}+.........+a_{15}+a_{16}+a_{17} } = 35 \times 17 $$

=> $$ \ {a_{1}+ a_{2}+a_{3}+a_{4}+.........+a_{15}+a_{16}+a_{17} }$$ = 595 ......(2)

Substituting (1) in (2) we will get

480 + $$ \ a_{16}+a_{17} $$ = 595

 => $$ \ a_{16}+a_{17} $$ = 595 - 480

=> $$ \ a_{16}+a_{17} $$ = 115 ......(3)

Also given, the new persons have an age difference of 9 years between them

=> $$ \ a_{16} - a_{17} $$ = 9 .......(4)

Solving (3) and (4) we get,

$$\ 2a_{16} $$ = 115+9

=> $$\ 2a_{16} $$ = 124

=> $$\ a_{16} $$ = 62

Hence $$\ a_{17} $$ = 115 -62 = 53