Question

If the simple interest on a certain sum of money for $3\frac{1}{2}$ years at 12% per annum is Rs 60 less than the simple interest on the same sum for $4\frac{1}{2}$ years at 10% per annum, then the sum (in Rs) is:

• A. 2000
• B. 200
• C. 6000
• D. 2060

Hint:

Using effective interest rate percentage ($R \times T$) simplifies the equation.
Case 1: $12 \times 3.5 = 42\%$.
Case 2: $10 \times 4.5 = 45\%$.
The difference is $45\% - 42\% = 3\%$.
Since $3\% = 60$, then $100\%$ (the full sum) is: \[ \frac{60}{3} \times 100 = 2000 \] This calculation can easily be done mentally during the exam.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This problem involves comparing two simple interest scenarios on the same principal sum ($P$).
In the first case, the money is lent for $3.5$ years at 12% per annum.
In the second case, the same money is lent for $4.5$ years at 10% per annum.
We are given that the interest from the first case is Rs 60 less than the interest from the second case. We need to find the principal sum ($P$).

Step 2: Key Formula or Approach: The simple interest ($SI$) formula is: \[ SI = \frac{P \times R \times T}{100} \] We set up the equations for both cases: \[ SI_1 = \frac{P \times R_1 \times T_1}{100} \] \[ SI_2 = \frac{P \times R_2 \times T_2}{100} \] We are given that: \[ SI_2 - SI_1 = 60 \]

Step 3: Detailed Explanation:

• Calculate the Effective Percentage Interest for Case 1: Rate $R_1 = 12\%$ per annum
Time $T_1 = 3\frac{1}{2}\text{ years} = 3.5\text{ years}$
The interest earned in Case 1 as a percentage of the principal is: \[ R_1 \times T_1 = 12 \times 3.5 = 42\% \text{ of } P \] So: \[ SI_1 = 0.42P \]

• Calculate the Effective Percentage Interest for Case 2: Rate $R_2 = 10\%$ per annum
Time $T_2 = 4\frac{1}{2}\text{ years} = 4.5\text{ years}$
The interest earned in Case 2 as a percentage of the principal is: \[ R_2 \times T_2 = 10 \times 4.5 = 45\% \text{ of } P \] So: \[ SI_2 = 0.45P \]

• Set up and Solve the Equation: We are given: \[ SI_2 - SI_1 = 60 \] Substitute the percentage values: \[ 45\% \text{ of } P - 42\% \text{ of } P = 60 \] \[ 3\% \text{ of } P = 60 \] \[ \frac{3}{100} \times P = 60 \] Solve for $P$: \[ P = \frac{60 \times 100}{3} = 20 \times 100 = 2000 \] So, the principal sum is Rs 2000.

Step 4: Final Answer:
The sum of money is Rs 2000.
Therefore, the correct option is (A).