Question

a certain sum of money amounts to Rs 57600 in 2 years and Rs 65536 in 4 years at compound interest compounded annually. Find the sum and the rate of interest

Answer 1

Given:

A certain sum of money amounts to Rs 57600 in 2 years and Rs 65536 in 4 years at compound interest compounded annually

To find:

The sum and the rate of interest

Solution:

Let "P" represent the sum of money and "R" represent the rate of interest.

The formula to find the amount in compound interest is given as,

$\boxed{\bold{A = P\: [1+ \frac{R}{100} ]^n}}$

where

A = amount

P = principal

R = rate of interest

n = time period

According to the question and using the formula, we can form two equations such as:

$57600 = P\: [1+ \frac{R}{100} ]^2}}$ ..... (i)

and

$65536 = P\: [1+ \frac{R}{100} ]^4}}$ ...... (ii)

On dividing eq. (ii) by (i), we get

$\frac{65536}{57600} = \frac{P[1+\frac{R}{100}^4 ]}{P[1+\frac{R}{100} ]^2}$

⇒ $\frac{65536}{57600} = [1+\frac{R}{100}]^2$

⇒ $1.1377 = [1+\frac{R}{100}]^2$

taking square root on both sides

⇒ $\sqrt{1.1377} = \sqrt{[1+\frac{R}{100} ]^2}$

⇒ $1.06 = 1+\frac{R}{100}$

⇒ $\frac{R}{100} = 1.06 - 1$

⇒ $R = 0.06 \times 100$

⇒ $\bold{R = 6\%}$

Now, substituting the value of R in eq. (i), we get

$57600 = P\: [1+ \frac{6}{100} ]^2}}$

⇒ $57600 = P\: [ \frac{106}{100} ]^2}}$

⇒ $P\:=\: \frac{57600 \times 100\times 100}{106\times 106}}}$

⇒ $\bold{P = Rs. \: 51263.79}$

Thus, the sum of money is Rs. 51263.79 and the rate of interest is 6%.

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