Question

If the compound interest on a certain sum at $10%$ per annum for 3 years is Rs 33,100, find the simple interest on the same sum at the same rate and for the same period?

• A. Rs 25,000
• B. Rs 28,000
• C. Rs 30,000
• D. Rs 32,000

Hint:

For a rate of $10%$ per annum over 3 years, the effective compound interest rate is always $33.1%$, while the effective simple interest rate is $30%$.
Using this ratio directly:
\[ SI = \frac{30%}{33.1%} \times CI \]
\[ SI = \frac{30}{33.1} \times 33,100 = 30 \times 1,000 = \text{Rs }30,000 \]
This saves calculation time in exams!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem provides the compound interest earned on a certain principal sum over a duration of 3 years at a rate of $10%$ per annum.
To solve this, we must first determine the original principal sum using the compound interest formula.
Once the principal is obtained, we will calculate the simple interest on this same principal for the same rate and period.

Step 2: Key Formula or Approach:
The formula for Compound Interest (\(CI\)) is given by:
\[ CI = P \left[ \left(1 + \frac{R}{100}\right)^T - 1 \right] \]
The formula for Simple Interest (\(SI\)) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
\(P\) is the Principal sum.
\(R\) is the Rate of interest per annum.
\(T\) is the Time period in years.

Step 3: Detailed Explanation:
$\bullet$ Calculating the Principal (P):
We are given:
\(CI = \text{Rs }33,100\)
\(R = 10%\) per annum
\(T = 3\) years
Substituting these values into the compound interest formula:
\[ 33,100 = P \left[ \left(1 + \frac{10}{100}\right)^3 - 1 \right] \]
\[ 33,100 = P \left[ \left(\frac{11}{10}\right)^3 - 1 \right] \]
\[ 33,100 = P \left[ \frac{1331}{1000} - 1 \right] \]
\[ 33,100 = P \left[ \frac{331}{1000} \right] \]
Solving for \(P\):
\[ P = \frac{33,100 \times 1000}{331} \]
\[ P = 100 \times 1000 = \text{Rs }1,00,000 \]
The principal sum is Rs 1,00,000.
$\bullet$ Calculating the Simple Interest (SI):
Now, we calculate the Simple Interest on the same principal (\(P = 1,00,000\)), at the same rate (\(R = 10%\)), for the same period (\(T = 3\) years):
\[ SI = \frac{1,00,000 \times 10 \times 3}{100} \]
\[ SI = 1,000 \times 30 = \text{Rs }30,000 \]
The simple interest earned is Rs 30,000.

Step 4: Final Answer:
The simple interest on the same sum at the same rate and for the same period is Rs 30,000.