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, S₇ = 700

`7/2[2a+(7-1)d] = 700`

`([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.

Answer 2

In the given problem,

Total amount of money (S_n) = Rs 700

There are a total of 7 prizes and each prize is Rs 20 less than the previous prize. So let us take the first prize as Rs a.

So, the second prize will be Rs a - 20  , third prize will be Rs a - 20 - 20 .

Therefore, the prize money will form an A.P. with first term a and common difference −20.

So, using the formula for the sum of n terms,

`S_n = n/2 [ 2a + (n-1) d]`

We get,

`700 = 7/2 [ 2(a) + (7 - 1) (-20)]`

`700 = 7/2 [ 2a +(6) (-20)]`

`700 = 7/2 (2a - 120)`

700 = 7 (a -60)

On further simplification, we get,

`700/7 =  a - 60`

100 + 60  = a

a = 160

Therefore, the value of first prize is Rs 160.

Second prize = Rs 140

Third prize = Rs 120

Fourth prize = Rs 100

Fifth prize = Rs 80

Sixth prize = Rs 60

Seventh prize= Rs 40

So the values of prizes are 

Rs 160, RS 140, Rs 120, Rs 100, Rs 80, Rs 60, and Rs 40.