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Mathematics

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

AP

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Answer

Let value of highest price be ₹ x. Given, each prize is ₹ 20 less than preceding prize.

Prizes : x, x - 20, x - 40, …….. upto 7 terms.

The above list is an A.P. with first term (a) = x and common difference = -20.

Total prize = ₹ 700.

By formula,

Sum upto n terms = Sn = n2[2a+(n−1)d]\dfrac{n}{2}[2a + (n - 1)d]

Substituting values we get :

⇒S7=72[2×x+(7−1)×−20]⇒700=72×[2x−120]⇒2x−120=700×27⇒2x−120=200⇒2(x−60)=200⇒x−60=100⇒x=₹ 160.\Rightarrow S_7 = \dfrac{7}{2}[2 \times x + (7 - 1) \times -20] \\[1em] \Rightarrow 700 = \dfrac{7}{2}\times [2x - 120] \\[1em] \Rightarrow 2x - 120 = \dfrac{700 \times 2}{7} \\[1em] \Rightarrow 2x - 120 = 200 \\[1em] \Rightarrow 2(x - 60) = 200 \\[1em] \Rightarrow x - 60 = 100 \\[1em] \Rightarrow x = \text{₹ 160}.

Next terms are :

x - ₹ 20 = ₹ 160 - ₹ 20 = ₹140,

₹ 140 - ₹ 20 = ₹120,

₹ 120 - ₹ 20 = ₹100,

₹ 100 - ₹ 20 = ₹80,

₹ 80 - ₹ 20 = ₹60,

₹ 60 - ₹ 20 = ₹40.

Hence, prizes are : ₹160, ₹140, ₹120, ₹100, ₹80, ₹60, ₹40.

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