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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.
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Solution 1
Let the cost of 1st prize be P.
Cost of 2nd prize = P − 20
And cost of 3rd 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.
Solution 2
In the given problem,
Total amount of money (Sn) = 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.
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