Discussion of Question with ID = 094 under Averages

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Question

The cost per unit of a commodity in three successive years is Rs.14/unit, Rs.8/unit and Rs.16/unit. If the annual spending of a family remains fixed, what is the average cost per unit for all the three combined years together?

A

$11{17/29}$.

B

$12{2/3}$.

C

$9{1/2}$.

D

$7{3/5}$.

Soln.
Ans: a

Let the annual spending be Rs. M. The catch in this question is that the spending remains fixed, so the consumption varies from year to year. We shall calculate the total consumption first. Let r1, r2 and r3 be the rates for the three successive years. Consumption in first year = M/r1. Similarly, we get M/r2 and M/r3. So total consumption is $M/{r1} + M/{r2} + M/{r3}$. Money spent in three years is 3M. So the required average = ${3M}/{M/{r1} + M/{r2} + M/{r3}}$ which simplifies to ${3r1r2r3}/{r1r2 + r2r3 + r3r1}$. Putting r1 = 14, r2 = 8, r3 = 16, we get $11{17/29}$. You might be wondering why I derived the formula first. The reason is that sometimes it is better to postpone calculations till the end.


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