For the same principal amount, the compound interest for two years at 5% per annum exceeds the simple interest for three years at 3% per annum by Rs 1125. Then the principal amount in rupees is

The correct answer is 90000.

Let us denote the principal amount as $P$ rupees.

Step 1: Calculate the Compound Interest (CI)
The rate of interest is 5% per annum, and the time period is 2 years.
The formula for Compound Interest is given by:

$$\mathrm{CI}=P\left[{\left(1+\frac{r}{100}\right)}^n-1\right]$$
Substituting the values of $r=5$ and $n=2$:

$$\mathrm{CI}=P\left[{\left(1+\frac{5}{100}\right)}^2-1\right]$$
$$\mathrm{CI}=P\left[{\left(1.05\right)}^2-1\right]$$
$$\mathrm{CI}=P\left[1.1025-1\right]=0.1025P$$

Step 2: Calculate the Simple Interest (SI)
The rate of interest is 3% per annum, and the time period is 3 years.
The formula for Simple Interest is given by:

$$\mathrm{SI}=\frac{P\times R\times T}{100}$$
Substituting the values of $R=3$ and $T=3$:

$$\mathrm{SI}=\frac{P\times 3\times 3}{100}=\frac{9P}{100}=0.09P$$

Step 3: Solve for the Principal Amount (P)
We are given that the compound interest exceeds the simple interest by Rs 1125:

$$\mathrm{CI}-\mathrm{SI}=1125$$
Substituting the expressions derived in Step 1 and Step 2:

$$0.1025P-0.09P=1125$$
$$0.0125P=1125$$
$$P=\frac{1125}{0.0125}$$
$$P=\frac{1125\times 10000}{125}$$
Since $1125÷125=9$, we have:

$$P=9\times 10000=90000$$

Thus, the principal amount is Rs 90000.