Question

The ages of boys in a group are in A. P. with the common differences of 3 months. the ages of the youngest boy in a group is 12 years. The sum of the ages of all the boys in the group is 375 years. find the number of boys in group.

Answer 1

Here is the required answer

Answer 2

$\textbf{\underline{\underline{According\:to\:the\:Question}}}$

Assumption

$\textbf{\underline{No\;of\;boys\;be\;p}}$

a = 12

$\textbf{\underline{Common\; difference}}$

= 3 months

$\tt{\rightarrow\dfrac{3}{12}years}$

$\tt{\rightarrow\dfrac{1}{4}years}$

Sn = 375 years

${\boxed{\sf\:{S_{n}=\dfrac{n}{2}[2a+(n-1)d]}}}$

$\tt{\rightarrow 375=\dfrac{1n}{2}[2\times 12+(n-1)1/4]}$

$\tt{\rightarrow 375\times 2=n[24+2(n-1)/4]}$

$\tt{\rightarrow 750=[\dfrac{24n}{1}+\dfrac{n^2-n}{4}]}$

$\tt{\rightarrow 750=\dfrac{96n+n^2-n}{4}}$

Put p in the place of n as per assumption

750 × 4 = p² + 95n

p² + 95p - 3000 = 0

p² + 120 - 25p = 3000 = 0

p(p + 120) - 25(p + 120) = 0

(p + 120) = 0

(p - 25) = 0

p = - 120

p = 25

$\textbf{\underline{Negative\;value\;is\;rejected}}$

Therefore

${\boxed{\sf\:{Total\;no\;of\;boys\;=\;25}}}$