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 the 1st prize be \(P\). The cost of the 2nd prize is \( P - 20 \), and the cost of the 3rd prize is \(P - 40\). These prize costs form an arithmetic progression (A.P.) with a first term \(a = P\) and a common difference \(d = -20\). Given that the sum of the first 7 terms is \(S_7 = 700\), we have the equation \(\frac 72 [2a + (7-1)d] = 700\).

Simplifying the equation, we get \(\frac {[2a + (6)(-20)]}{2} = 100\), which further reduces to \(a + 3(-20) = 100\). This leads to \(a - 60 = 100\), and thus \(a = 160\). Therefore, the costs of the prizes are Rs. 160, Rs. 140, Rs. 120, Rs. 100, Rs. 80, Rs. 60, and Rs. 40.