A sum of money lent at simple interest amounts to ₹ 3,224 in 2 years and ₹ 4,160 in 5 years. Find the sum and the rate of interest.

Given:

Amount in 2 years = ₹ 3,224

⇒ P + S.I. of 2 years = = ₹ 3,224 ……….(1)

Amount in 5 years = ₹ 4,160

⇒ P + S.I. of 5 years = ₹ 4,160 ……….(2)

Subtracting equation (1) from (2), we get

S.I. of 3 years = ₹ 4,160 - ₹ 3,224 = ₹ 936

S.I. of 1 year = ₹ $\dfrac{936}{3}$ = $₹ 312$

S.I. of 2 years = $₹ 312 \times 2$ = $₹ 624$

From equation (1), we get:

P + ₹ 624 = ₹ 3,224

P = ₹ 3,224 - ₹ 624

P = ₹ 2,600

Now when P = ₹ 2,600

S.I. = ₹ 312

T = 1 year

Let the rate be $r$.

As we know,

$\text{S.I.} = \Big(\dfrac{P \times R \times T}{100}\Big)\\[1em] \Rightarrow 312 = \Big(\dfrac{2,600 \times r \times 1}{100}\Big)\\[1em] \Rightarrow 312 = \dfrac{2,600r}{100}\\[1em] \Rightarrow 312 = 26r\\[1em] \Rightarrow r = \dfrac{312}{26}\\[1em] \Rightarrow r = 12$

Hence, principal = $₹ 2,600$ and rate = $12\%$