A sum of  ₹8000 becomes ₹9500 at 7.5% p.a. in x years at simple interest. What will be the amount of the sum in same time at 20% p.a, if interest is compounded yearly ?

The correct option is ₹12672.

Step 1: Calculate the time period (x) using Simple Interest

Given details:

Principal (P) = ₹8000
Amount (A₁) = ₹9500
Rate of simple interest (R₁) = 7.5% p.a.
Time = x years

Simple Interest (SI) earned:

$SI=A_1-P=9500-8000=1500$

Using the simple interest formula:

$SI=\frac{P\times R_1\times x}{100}$

Substituting the values:

$1500=\frac{8000\times 7.5\times x}{100}$

$1500=600\times x$

$x=\frac{1500}{600}=2.5\text{years}$

Step 2: Calculate the amount with Compound Interest for 2.5 years

Now, we need to find the compound amount for ₹8000 at 20% p.a. for 2.5 years (2 full years + 1/2 year).

- Rate for 1st year = 20%
- Rate for 2nd year = 20%
- Rate for remaining half year = 20% / 2 = 10%

Formula for amount under compounding:

$A=P\times \left(1+\frac{20}{100}\right)\times \left(1+\frac{20}{100}\right)\times \left(1+\frac{10}{100}\right)$

$A=8000\times \frac{6}{5}\times \frac{6}{5}\times \frac{11}{10}$

$A=8000\times 1.2\times 1.2\times 1.1$

$A=12672$

Thus, the amount after 2.5 years compounded yearly at 20% p.a. is ₹12672.