Question

In how many years will ₹ 3375 become ₹ 4096 at ${\displaystyle 13\frac{1}{3}}$ p.a if the interest is compounded half-yearly?

Answer 1

Principal = ₹ 3375

Amount = ₹ 4096

r = ${\displaystyle 13\frac{1}{3}\% \text{p.a}}$

= ${\displaystyle \frac{40}{3}\% \text{p.a}}$

Compounded half-yearly r = ${\displaystyle \frac{\frac{40}{3}}{2}=\frac{20}{3}\%}$ half-yearly

Let no. of years be n

For compounding half-yearly, formula is

A = ${\displaystyle \text{P}{(1+\frac{\text{r}}{100})}^{2\text{n}}}$

∴ 4096 = ${\displaystyle 3375+{(1+\frac{\frac{20}{3}}{100})}^{2\text{n}}}$

∴ ${\displaystyle \frac{4096}{3375}={(1+\frac{20}{3\times 100})}^{2\text{n}}}$

= ${\displaystyle {(1+\frac{1}{15})}^{2\text{n}}}$

${\displaystyle {\left(\frac{15+1}{15}\right)}^{2\text{n}}={\left(\frac{16}{15}\right)}^{2\text{n}}}$

⇒ ${\displaystyle \frac{4096}{3375}={\left(\frac{16}{15}\right)}^{2\text{n}}}$

Taking cubic root on both sides,

${\displaystyle {\left(\frac{16}{15}\right)}^{\frac{2\text{n}}{3}}=\frac{\sqrt[3]{4096}}{\sqrt[3]{3375}}=\frac{16}{15}}$

∴ ${\displaystyle \frac{2\text{n}}{3}}$ = 1

∴ n = ${\displaystyle \frac{3}{2}}$ = 1.5 years