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 Concept:
The problem explores the fundamental differences between simple interest and compound interest over a multi-year period.
Interest is the cost of borrowing money or the reward for saving it.
Simple Interest (SI) is calculated only on the principal amount, making it a linear growth model.
Compound Interest (CI) is calculated on the principal plus any accumulated interest from previous periods, leading to exponential growth.
In competitive exams, comparing these two is common because CI is always greater than or equal to SI for the same rate and time.
Here, the total CI for 3 years is known, and we need to bridge back to the principal before calculating the corresponding SI.
Step 2: Key Formula or Approach:
We utilize the standard formulas for both types of interest calculation:
The formula for Compound Interest is:
\[ CI = P \left[ \left(1 + \frac{R}{100}\right)^T - 1 \right] \]
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where \( P \) is the Principal, \( R \) is the rate of interest, and \( T \) is the time in years.
Step 3: Detailed Explanation:
We start by extracting the given information: \( CI = 33,100 \), \( R = 10% \), and \( T = 3 \) years.
Our first goal is to determine the principal amount \( P \).
Substitute the values into the CI formula:
\[ 33,100 = P \left[ \left(1 + \frac{10}{100}\right)^3 - 1 \right] \]
Simplify the fraction inside the parentheses: \( 1 + 0.1 = 1.1 \).
Now, calculate the cube of 1.1: \( 1.1 \times 1.1 \times 1.1 = 1.331 \).
The equation becomes:
\[ 33,100 = P [1.331 - 1] \]
\[ 33,100 = P [0.331] \]
To find \( P \), divide the total interest by the decimal factor:
\[ P = \frac{33,100}{0.331} \]
Multiplying both the numerator and denominator by 1000 to remove the decimal:
\[ P = \frac{33,100,000}{331} \]
\[ P = 1,00,000 \]
The principal sum is Rs 1,00,000.
Now that we have the principal, we can compute the Simple Interest for the same duration and rate.
\[ SI = \frac{1,00,000 \times 10 \times 3}{100} \]
Cancel the two zeros in the denominator with the numerator:
\[ SI = 1,000 \times 10 \times 3 \]
\[ SI = 30,000 \]
The simple interest earned on the same sum 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.