Rupees 4,000 amounts to ₹ 5,000 in 8 years; in what time will ₹ 2,100 amount to ₹ 2,800 at the same rate?

Given:

P = ₹ 4,000

A = ₹ 5,000

T = 8 years

Let the rate be $r$.

As we know, $\text{A = P + S.I.}\\[1em] \Rightarrow ₹ 5,000 = ₹ 4,000 + S.I.\\[1em] \Rightarrow S.I. = ₹ 5,000 - ₹ 4,000 \\[1em] \Rightarrow S.I. = ₹ 1,000$

And

$\text{S.I.} = \Big(\dfrac{P \times R \times T}{100}\Big)\\[1em] \Rightarrow ₹ 1,000 = \Big(\dfrac{₹ 4,000 \times r \times 8}{100}\Big)\\[1em] \Rightarrow ₹ 1,000 = 40 \times r \times 8\\[1em] \Rightarrow ₹ 1,000 = 320 \times r\\[1em] \Rightarrow r = \dfrac{1,000}{320}\%\\[1em] \Rightarrow r = \dfrac{25}{8}\%$

When P = $₹ 2,100$

A = $₹ 2,800$

R = $\dfrac{25}{8}\%$

Let the time be $t$.

As we know,

$\text{A = P + S.I.}\\[1em] \Rightarrow ₹ 2,800 = ₹ 2,100 + S.I.\\[1em] \Rightarrow S.I. = ₹ 2,800 - ₹ 2,100 \\[1em] \Rightarrow S.I. = ₹ 700$

And

$\text{S.I.} = \Big(\dfrac{P \times R \times T}{100}\Big)\\[1em] \Rightarrow 700 = \Big(\dfrac{2,100 \times 25 \times t}{8 \times 100}\Big)\\[1em] \Rightarrow 700 = \dfrac{21 \times 25 \times t}{8}\\[1em] \Rightarrow 700 = \dfrac{525 \times t}{8}\\[1em] \Rightarrow t = \dfrac{700 \times 8}{525}\\[1em] \Rightarrow t = \dfrac{32}{3}\\[1em] \Rightarrow t = 10\dfrac{2}{3}\\[1em] \Rightarrow t = 10\dfrac{8}{12}$

Hence, the time = 10 years and 8 months.