A sum of money was invested at the rate of 7.5% simple interest per annuum for 4 years. If the investments were for 5 years, the interest earned would have been Rs. 375 more. What was the initial sum invested?
Rs. 5,000
Understanding Simple Interest and the Problem
This question deals with simple interest earned on an investment. Simple interest is calculated only on the principal amount. The formula for simple interest is:
\( \text{SI} = \frac{P \times R \times T}{100} \)
Where:
- \(P\) is the Principal amount (the initial sum invested).
- \(R\) is the Rate of interest per annum.
- \(T\) is the Time period in years.
- \(\text{SI}\) is the Simple Interest earned.
The problem provides us with the interest rate, two different time periods for investment, and the difference in the simple interest earned between these two periods. We need to find the initial sum invested, which is the Principal (\(P\)).
Setting up the Simple Interest Calculation
Let the initial sum invested be \(P\) (in Rupees).
The given Rate of Interest is \(R = 7.5\%\) per annum.
Case 1: Investment for 4 Years
Time period, \(T_1 = 4\) years.
Simple Interest earned in 4 years, \(\text{SI}_1\):
\( \text{SI}_1 = \frac{P \times R \times T_1}{100} = \frac{P \times 7.5 \times 4}{100} \)
\( \text{SI}_1 = \frac{30P}{100} \)
Case 2: Investment for 5 Years
Time period, \(T_2 = 5\) years.
Simple Interest earned in 5 years, \(\text{SI}_2\):
\( \text{SI}_2 = \frac{P \times R \times T_2}{100} = \frac{P \times 7.5 \times 5}{100} \)
\( \text{SI}_2 = \frac{37.5P}{100} \)
Calculating the Initial Sum Invested
The problem states that if the investment were for 5 years instead of 4 years, the interest earned would have been Rs. 375 more. This means the difference between the simple interest earned in 5 years (\(\text{SI}_2\)) and the simple interest earned in 4 years (\(\text{SI}_1\)) is Rs. 375.
\( \text{SI}_2 - \text{SI}_1 = 375 \)
Substitute the expressions for \(\text{SI}_1\) and \(\text{SI}_2\):
\( \frac{37.5P}{100} - \frac{30P}{100} = 375 \)
Combine the terms on the left side:
\( \frac{(37.5 - 30)P}{100} = 375 \)
\( \frac{7.5P}{100} = 375 \)
Now, solve for \(P\). Multiply both sides by 100:
\( 7.5P = 375 \times 100 \)
\( 7.5P = 37500 \)
Divide both sides by 7.5:
\( P = \frac{37500}{7.5} \)
To make the division easier, we can multiply the numerator and denominator by 10 to remove the decimal:
\( P = \frac{375000}{75} \)
Now, perform the division:
\( P = 5000 \)
So, the initial sum invested was Rs. 5,000.
Let's verify the difference:
- \( \text{SI}_1 = \frac{5000 \times 7.5 \times 4}{100} = \frac{5000 \times 30}{100} = \frac{150000}{100} = 1500 \)
- \( \text{SI}_2 = \frac{5000 \times 7.5 \times 5}{100} = \frac{5000 \times 37.5}{100} = \frac{187500}{100} = 1875 \)
- Difference = \( \text{SI}_2 - \text{SI}_1 = 1875 - 1500 = 375 \)
The calculated difference matches the value given in the problem (Rs. 375), confirming our calculated principal amount is correct.
Summary of Calculation Steps
| Step | Description | Calculation |
|---|---|---|
| 1 | Identify Variables | R = 7.5%, T1 = 4 years, T2 = 5 years, SI2 - SI1 = 375 |
| 2 | Write SI Formula | \( \text{SI} = \frac{P \times R \times T}{100} \) |
| 3 | Calculate SI1 | \( \text{SI}_1 = \frac{P \times 7.5 \times 4}{100} = \frac{30P}{100} \) |
| 4 | Calculate SI2 | \( \text{SI}_2 = \frac{P \times 7.5 \times 5}{100} = \frac{37.5P}{100} \) |
| 5 | Set up Difference Equation | \( \text{SI}_2 - \text{SI}_1 = 375 \) |
| 6 | Solve for P | \( \frac{37.5P}{100} - \frac{30P}{100} = 375 \implies \frac{7.5P}{100} = 375 \implies P = \frac{37500}{7.5} = 5000 \) |
Revision Table: Simple Interest Concepts
| Term | Definition | Formula |
|---|---|---|
| Principal (P) | The initial amount of money invested or borrowed. | - |
| Rate (R) | The percentage at which interest is calculated per annum. | - |
| Time (T) | The duration for which the money is invested or borrowed, usually in years. | - |
| Simple Interest (SI) | Interest calculated only on the principal amount. | \( \text{SI} = \frac{P \times R \times T}{100} \) |
| Amount (A) | The total sum after adding interest to the principal. | \( A = P + \text{SI} \) |
Additional Information: Simple vs. Compound Interest
It's important to distinguish between simple interest and compound interest, although this problem specifically uses simple interest.
- Simple Interest: As seen in this problem, simple interest is calculated only on the original principal amount. The interest earned in each period is constant if the principal and rate remain the same.
- Compound Interest: In compound interest, the interest earned in each period is added to the principal for the next period's calculation. This means interest is earned on both the original principal and the accumulated interest from previous periods. The formula for compound interest is \( A = P(1 + \frac{R}{100})^T \), where \(A\) is the amount after \(T\) years. Compound interest grows faster than simple interest over time.
Understanding which type of interest is being used is crucial for solving problems correctly.
Similar Questions
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Important Questions from Simple Interest
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If the interest on a sum of Rs.1200 is more than the interest on Rs.1000 by Rs.120 in three years, then what is the rate of interest per annum?.
The difference between the simple interest received from two banks on Rs. 500 for two years is Rs. 2.50. What is the difference between their rates?
P borrowed an amount from Q at a simple interest of 10% p.a. At the end of the year, P paid back Rs 99000 which was 90% of what she owed. How much money did P borrow?
