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 prize.

Given:

A sum of Rs. 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. Each prize is Rs. 20 less than its preceding prize.

To do:

We have to find the value of each prize.

Solution:

Total amount $=Rs.\ 700$

Number of prizes \( n=7 \)

Let the first prize be \( \left(a_{1}\right)=a \)

This implies,

Second prize \( \left(a_{2}\right)=a-20 \)

Third prize \( \left(a_{3}\right)=a-40 \)

Common difference \( (d)=a-20-a=-20 \)

We know that,

\( \mathrm{S}_{n}=\frac{n}{2}[2 a+(n-1) d] \)

\( \Rightarrow 700=\frac{7}{2}[2 a+(7-1)(-20) \)

\( \Rightarrow 700=\frac{7}{2}[2 a-120] \)

\( \Rightarrow 2 a-120=\frac{700 \times 2}{7}=200 \)

\( \Rightarrow 2 a=200+120=320 \)

\( \therefore a=\frac{320}{2}=160 \)

This implies,

First prize \( = Rs.\ 160 \)

Second prize \( = Rs.\ (160-20) = Rs.\ 140 \), Third prize \( = Rs.\ 120 \)

Fourth prize \( = Rs.\ 100 \), Fifth prize \( = Rs.\ 80 \)

Sixth prize \( = Rs.\ 60 \), Seventh prize \( = Rs.\ 40 \)

Therefore, the value of each prize (from 1st to 7th) is Rs. 160, Rs. 120, Rs. 100, Rs. 80, Rs. 60 and Rs. 40.

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Updated on: 10-Oct-2022

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