The compound interest on sum of money in 2 years and 4 years are Rs. 5560 and Rs. 12066.60 respectively. Find the rate of interest compounded yearly and the sum. 

Solution:
Given,

We have,
Principal or the sum = Rs P
Rate of Compound Interest = R% per annum

Now, 

In Case I,


Duration of time (T) = 2 years
Compound Interest (CI) = Rs 5560 

So, CI = $P [ \left ( 1+\frac{R}{100} \right )^T -1]$

or, 5560 = $P [ \left ( 1+\frac{R}{100} \right )^2 -1]$

or, 5560 = $P [ \left ( \frac{100+R}{100} \right )^2 -1]$

or, 5560 = $P \left ( \frac{(100+R)^2 -100^2}{100^2} \right)$

or, P = $\frac{5560 * 100^2}{(100+R)^2-100^2}$ ----- (i)



And,

In Case II,


Duration of time (T) = 4 years
Compound Interest (CI) = Rs 12066.60

So, CI = $P [ \left ( 1+\frac{R}{100} \right )^T -1]$

or, 12066.60 = $P [ \left ( 1+\frac{R}{100} \right )^4 - 1]$

or, 12066.60 = $P [ \left ( \frac{100+R}{100} \right )^4 -1]$

or, 12066.60 = $P \left ( \frac{(100+R)^4 -100^4}{100^4} \right)$

or, P = $\frac{12066.60 * 100^4}{(100+R)^4-100^4}$ ------ (ii)


Since, values of P are equal, the equation (i) and (ii) can be written as:

$\frac{5560 * 100^2}{(100+R)^2-100^2}$  = $\frac{12066.60 * 100^4}{(100+R)^4-100^4}$

or, $\frac{5560 * 100^2}{(100+R)^2-100^2}$ = $\frac{12066.60 * 100^4}{\{(100+R)^2-100^2\}\{(100+R)^2+100^2\}}$

or, $\frac{\{(100+R)^2-100^2\}\{(100+R)^2+100^2\}}{\{(100+R)^2-100^2\}} = \frac{12066.60 * 100^4}{5560 * 100^2}$

or, $(100+R)^2 +100^2 = 21702.51$

or, $100^2 +200R +R^2 +100^2 = 21702.51$

or, $R^2 +200R +20000 = 21702.51$

or, $R^2 +200R - 1702.51 = 0$ ----- (iii)


Comparing equation (iii) with ax² +bx +c = 0


We get,

a = 1, b = 200, c = -1702.51, x = R


Using formula of quadratic equation;

$x = \frac{-b +_{-} \sqrt{b^2-4ac}}{2a}$

Since, our rate will we positive, we remove the '-' sign,

or, $R = \frac{-200 + \sqrt{200^2-4(1)(-1702.51)}}{2(1)}$

= $\frac{ -200 + \sqrt{46810.04}}{2}$

= $\frac{ -200 + 216.35}{2}$

So, R = 8.17

$\therefore R = 8.17%$


Again,

Put value of R in equation (i)

P = $\frac{5560 * 100^2}{(100+8.17)^2-100^2}$ 

= Rs 32691.5

Therefore, the required rate of interest for the given cases is 8.17% per annum and the sum of money or principal is equal to Rs 32691.5.


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