A sum amounts to Rs. 9,680 in 2 years and Rs. 10,648 in 3 years at a certain rate per annum, interest compounded yearly. The same sum will amount to how much after \(3\frac{3}{4}\) years at double the rate of interest (nearest to a whole number)? 

Given:

Amount after 2 years (A₂) = ₹9,680

Amount after 3 years (A₃) = ₹10,648

Time (t₂) = 2 years

Time (t₃) = 3 years

Time for calculation = 3(3/4) years = 3.75 years

Formula used:

A = P(1 + r/100)^t

Calculations:

From the given data:

A₃/A₂ = (1 + r/100)³ / (1 + r/100)²

⇒ ₹10,648/₹9,680 = (1 + r/100)

⇒ 1.1 = (1 + r/100)

⇒ r = 10%

A₂ = P\((1+\frac{r}{100})^2\)

9680 = P\((1+\frac{10}{100})^2\)

9680 = P\((1.1)^2\)

9680 = P × 1.21

⇒ P = \(\frac{9680}{1.21}\)

⇒ P = ₹8,000

New rate (r') = 2 × original rate = 2 × 10% = 20%

New time (t') = \(3\frac{3}{4}\) years = 3 years + \(\frac{3}{4}\) year

First, calculate the amount after 3 full years with the new rate:

Amount after 3 years = P\((1+\frac{r'}{100})^3\)

⇒ Amount after 3 years = 8000\((1+\frac{20}{100})^3\)

⇒ Amount after 3 years = 8000\((1.2)^3\)

⇒ Amount after 3 years = 8000 × 1.728

⇒ Amount after 3 years = ₹13,824

Now, The simple interest for the remaining \(\frac{3}{4}\) year on this amount:

Interest for fractional year = \(13824 \times \frac{20}{100} \times \frac{3}{4}\)

⇒ Interest for fractional year = \(13824 \times 0.20 \times 0.75\)

⇒ Interest for fractional year = 13824 × 0.15

⇒ Interest for fractional year = ₹2073.6

Final Amount = Amount after 3 years + Interest for fractional year

⇒ Final Amount = ₹13,824 + ₹2073.6

⇒ Final Amount = ₹15,897.6

⇒ Final Amount ≈ ₹15,898

∴ The same sum will amount to ₹15,898 (nearest to a whole number) after \(3\frac{3}{4}\) years at double the rate of interest.