A certain sum amounts to ₹ 3,825 in 4 years and to ₹ 4,050 in 6 years. Find the rate percent and the sum.

Given:

Amount in 4 years = ₹ 3,825

⇒ P + S.I. of 4 years = = ₹ 3,825 ……….(1)

Amount in 6 years = ₹ 4,050

⇒ P + S.I. of 6 years = ₹ 4,050 ……….(2)

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

S.I. of 2 years = $₹ 4,050 - ₹ 3,825$

= $₹ 225$

S.I. of 1 year = ₹ $\dfrac{225}{2}$

= $₹ 112.5$

S.I. of 4 years = $₹ 112.5 \times 4$

= $₹ 450$

From equation (1), we get:

P + ₹ 450 = ₹ 3,825

P = ₹ 3,825 - ₹ 450

P = ₹ 3,375

Now when P = ₹ 3,375

S.I. = ₹ 112.5

T = 1 year

Let the rate be $r$.

As we know

$\text{S.I.} = \Big(\dfrac{P \times R \times T}{100}\Big)\\[1em] \Rightarrow 112.5 = \Big(\dfrac{3,375 \times r \times 1}{100}\Big)\\[1em] \Rightarrow 112.5 = \dfrac{3,375r}{100}\\[1em] \Rightarrow r = \dfrac{112.5 \times 100}{3375}\\[1em] \Rightarrow r = \dfrac{11250}{3375}\\[1em] \Rightarrow r = \dfrac{10}{3}\\[1em] \Rightarrow r = 3\dfrac{1}{3}$

Hence, principal = $₹ 3,375$ and rate = $3\dfrac{1}{3}\%$