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A machine being used by a company is estimated to have a life of 15 years. At that time, the new machine would cost Rs. 74,000 and the scrap of the old machine would yield Rs. 4,600 only. A sinking fund is created for replacing the machine at the end of its life. What approx. sum should be invested by the company at the end of each year to accumulate at 6% per annum?

A machine being used by a company is estimated to have a life of 15 years. At that time, the new machine would cost Rs. 74,000 and the scrap of the old machine would yield Rs. 4,600 only. A sinking fund is created for replacing the machine at the end of its life. What approx. sum should be invested by the company at the end of each year to accumulate at 6% per annum? Correct Answer Rs. 2984

GIVEN :

Now as the machine costs Rs. 74000 and the scrap value at the end of 15 years is 4600

∴ The amount which is needed at the end of 15 years to purchase the machine will be 74000 - 4600 = 69400 Rs.

 

ASSUMPTION :

Now assume the amount P is deposited at the end of every year and the interest is also accumulated for the amount deposited.

 

CALCULATION :

69400 = P + P (1 + 6/100) + P (1 + 6/100)2 + P (1 + 6/100)3 + ………….. + P (1 + 6/100)14

∴ 69400 = P + P (1.06) + ………… P (1.06)13 + P (1.06)14

∴ 69400 = P (1 + (1.06) + ………… (1.06)13 + (1.06)14)       ------ eq. (i)

The right hand side of the equation is a Geometric progression with a = 1 and r = 1.06

The sum of GP can be given as

S = a (rn - 1)/(r - 1)

Where a = first term

R = common ratio

n = Number of terms in a series

S = 1(1.0615 - 1)/(1.06 - 1)

∴ S = 1.396/0.06

∴ S = 23.26

∴ The eq. (i) will become as

69400 = P × 23.26

∴ P = 2983.66

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