Question

A sum of Rs. $700$ is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs. $20$ less than its preceding prize, find the value of each of the prizes.

Answer 1

Let the cost of $1^{st}$ prize be $P$.
Cost of $2^{nd}$ prize $=P-20$
And cost of $3^{rd}$ prize $P-40$
It can be observed that the cost of these prizes are in an A.P. having common difference as $-20$ and first term as $P$.

$a=P,d=-20$

Given that,

$S_7=700$

$\frac{7}{2}[2a+(7-1\left)\right(-20\left)\right]$

$\frac{2a+6(-20)}{2}=100$

$a+3(-20)=100$

$a-60=100$

$a=160$

Therefore, the value of each of the prizes was Rs $160,$ Rs $140,$ Rs $120,$ Rs $100,$ Rs $80,$ Rs $60$ and Rs $40$.