A certain sum invested at compound interest amounts to ₹19965 at 10% p.a. in 3 years. The same sum will amount to ₹x at the same rate in $$2\frac{2}{5}$$ years. If the interest is compounded yearly in both cases, what is the value of x?

A certain sum invested at compound interest amounts to ₹19965 at 10% p.a. in 3 years.

Let's assume the principal amount is 'P'.

R = rate of interest

T = time

$$19965=P\left(1+\frac{R}{100}\right)^T$$

$$19965=P\left(1+\frac{10}{100}\right)^3$$

$$19965=P\left(1+\frac{1}{10}\right)^3$$

$$19965=P\left(\frac{11}{10}\right)^3$$

$$19965=\frac{1331P}{1000}$$

$$15=\frac{P}{1000}$$

P = ₹15000

The same sum will amount to ₹x at the same rate in $$2\frac{2}{5}$$ years.

For 2 years, the rate of interest is 10% for each year.

Now for $$\frac{2}{5}$$ years rate of interest = $$\left(\frac{2}{5}\times\ 10\right)\%$$ = 4%

x = 15000 of (100+10)% of (100+10)% of (100+4)%

= 15000 of 110% of 110% of 104%

= $$15000\times1.1\times1.1\times1.04$$

= ₹18876