A sum of money becomes $\dfrac{5}{4}$ of itself in 5 years. The rate of interest is:

1. 10%

2. 5%

3. 8%

4. 15%

Given:

T = 5 years

A = ₹ $\dfrac{5}{4}$ of P

Let the principal be $P$ and the rate be $r$.

As we know,

$\text{A = P + S.I.}\\[1em] \Rightarrow\dfrac{5}{4}P = P + S.I.\\[1em] \Rightarrow\text{S.I.} = \dfrac{5}{4}P - P\\[1em] \Rightarrow\text{S.I.} = \dfrac{5}{4}P - \dfrac{4}{4}P\\[1em] \Rightarrow\text{S.I.} = \dfrac{(5 - 4)}{4}P\\[1em] \Rightarrow\text{S.I.} = \dfrac{1}{4}P$

And

$\text{S.I.} = \Big(\dfrac{P \times R \times T}{100}\Big)\\[1em] \Rightarrow \dfrac{1}{4} \times P = \Big(\dfrac{P \times r \times 5}{100}\Big)\\[1em] \Rightarrow \dfrac{1}{4}P = \Big(\dfrac{5Pr}{100}\Big)\\[1em] \Rightarrow \dfrac{1}{4}\cancel{P} = \Big(\dfrac{5r\cancel{P}}{100}\Big)\\[1em] \Rightarrow \dfrac{1}{4} = \Big(\dfrac{5r}{100}\Big)\\[1em] \Rightarrow \dfrac{1}{4} = \Big(\dfrac{r}{20}\Big)\\[1em] \Rightarrow r = \Big(\dfrac{1 \times 20}{4}\Big)\\[1em] \Rightarrow r = \Big(\dfrac{20}{4}\Big)\\[1em] \Rightarrow r = 5$

Hence, option 2 is the correct option.