Question

A certain sum amounts to rs 7200 after 2 years and to 8640 after 3 years, interest being compounded annually, find the principal and the rate of interest which are whole numbers

Answer 1

The principal is Rs. 5000.

The rate of interest is 20% p.a..

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Let's understand a few concepts:

To calculate the principal and rate of interest we will use the following formula for the compound interest:

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

where A = Amount, P = Principal, R = Rate of interest and n = no. of years

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Let's solve the given problem:

Let's say,

"P" → sum of money i.e., Principal

"R" → rate of interest

A certain sum amounts to Rs. 7200 after 2 years, so by using the above formula, we get

$7200= P \bigg[1 + \frac{R}{100} \bigg]^2 \:.\:.\:.\:.\:(1)$

A certain sum amounts to Rs. 8640 after 3 years, so by using the above formula, we get

$8640= P \bigg[1 + \frac{R}{100} \bigg]^3 \:.\:.\:.\:.\:(2)$

On dividing equation (2) by (1), we get

$\frac{8640 = P \bigg[1 + \frac{R}{100} \bigg]^3 }{7200 = P \bigg[1 + \frac{R}{100} \bigg]^2 } \\\\\implies \frac{8640}{7200} = \frac{P \bigg[1 + \frac{R}{100} \bigg]^3}{P \bigg[1 + \frac{R}{100} \bigg]^2} \\\\\implies \frac{864}{720} =1 + \frac{R}{100}\\\\\implies \frac{6}{5} =1 + \frac{R}{100}\\\\\\\implies \frac{R}{100} = \frac{6}{5} - 1\\\\\\\implies \frac{R}{100} = \frac{6 - 5}{5} \\\\\implies \frac{R}{100} = \frac{1}{5}\\\\\implies R = \frac{100}{5} \\\\\implies \bold{R = 20\%}$

On substituting R = 20% in equation (1), we get

$7200= P \bigg[1 + \frac{20}{100} \bigg]^2 \\\\\implies 7200= P \bigg[ \frac{120}{100} \bigg]^2 \\\\\implies 7200= P \bigg[ \frac{6}{5} \bigg]^2\\\\\implies 7200= P \times \frac{36}{25} \\\\\implies P = 7200 \times \frac{25}{36} \\\\\implies P = 200 \times 25\\\\\implies \bold{P = Rs.\:5000}$

Thus, the principal is Rs. 5000 and the rate of interest is 20%.

#SPJ3

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