Question

9. A certain sum of money amounts to Rs 2,646 in 2 years and to Rs 2,778.30 in 3 years at the same
annual compound interest rate. Find the rate of interest and the sum.​

Answer 1

Given

(i) A certain sum of money amounts to Rs 2646 in 2 years

(ii) It then amounts to Rs 2778.30 in 3 years

To find:

(I) Rate of interest and sum.

Solution:

The amount (A) in compound interest is given as:

A = P(1 + r/n)^(nt)

Since interest is compounded annually, n=1.

For first case, A = Rs 2646, t = 2.

2646 = P(1 + r)^2 ...(1)

For second case,

A = Rs 2778.30, t = 3.

2778.30 = P(1 + r)^3 ...(2)

Dividing (2) by (1), we get,

(2778.30/2646) = (1+r)

=> 1 + r = 21/20

=> r = 1/20

=> r = (100/20)% = 5%

So, rate = 5% p.a.

P(1 + 1/20)^2 = 2646

=> 441P/400 = 2646

=> P = Rs 2400

So, sum is Rs 2400 and rate of interest is 5% p.a.

Answer 2

Given:

A certain sum of money amounts to Rs 2,646 in 2 years and to Rs 2,778.30 in 3 years at the same  annual compound interest rate

To find:

The rate of interest and the sum

Formula to be used:

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

where

A = Amount

P = Principal or sum of money

R = Rate of interest

n = no. of years

Solution:

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

According to the question and the above formula, we can form equations as,

2646 = P $[1 + \frac{R}{100} ]^2$ ....... (i)

and

2778.30 = P $[1 + \frac{R}{100} ]^3$ ....... (ii)

Now, dividing equation (ii) by (i), we get

$\frac{2778.30}{2646}$ = $\frac{[1 + \frac{R}{100}]^3}{[1+ \frac{R}{100}]^2 }$

⇒ 1.05 = $[1 + \frac{R}{100} ]$

⇒ 1.05 - 1 = $\frac{R}{100}$

⇒ 0.05 = $\frac{R}{100}$

⇒ R = 0.05 * 100

⇒ R = 5%

We will substitute the value of R = 5% in eq. (i), we get

2646 = P $[1 + \frac{5}{100} ]^2$

⇒ 2646 = P $[1 + \frac{1}{20} ]^2$

⇒ 2646 =  P $[ \frac{21}{20} ]^2$

⇒ P = $2646 * \frac{20}{21} * \frac{20}{21}$

⇒ P = Rs. 2400

Thus, the rate of interest is 5% and the sum of money is Rs. 2400.

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