At 8% simple interest per annum a sum of money becomes Rs. 300 in \(2\frac{1}{2}\) years. What was the sum invested?
Rs. 250
Calculating Principal Amount with Simple Interest
The problem asks us to find the original sum of money invested, also known as the principal amount, given the final amount received after a certain period at a specific simple interest rate. This is a classic simple interest calculation problem.
Understanding Simple Interest Concepts
Simple interest is calculated only on the initial principal amount. The formula for simple interest (SI) is:
\( \text{SI} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} \)
The total amount (A) received at the end of the period is the sum of the principal (P) and the simple interest (SI):
\( \text{A} = \text{P} + \text{SI} \)
Given Information
We are provided with the following details:
- Amount (A) = Rs. 300
- Rate of Interest (R) = 8% per annum
- Time (T) = \(2\frac{1}{2}\) years
Let's convert the time into a decimal for easier calculation:
\( \text{T} = 2\frac{1}{2} \text{ years} = 2.5 \text{ years} \)
We need to find the Principal (P).
Step-by-Step Calculation of Principal
We know that \( \text{A} = \text{P} + \text{SI} \). We can substitute the formula for SI into this equation:
\( \text{A} = \text{P} + \frac{\text{P} \times \text{R} \times \text{T}}{100} \)
Now, let's plug in the given values:
\( 300 = \text{P} + \frac{\text{P} \times 8 \times 2.5}{100} \)
Simplify the term in the numerator:
\( 8 \times 2.5 = 20 \)
Substitute this back into the equation:
\( 300 = \text{P} + \frac{20\text{P}}{100} \)
Simplify the fraction \( \frac{20}{100} \):
\( \frac{20}{100} = \frac{1}{5} \)
So the equation becomes:
\( 300 = \text{P} + \frac{\text{P}}{5} \)
Now, we can take P common from the right side of the equation:
\( 300 = \text{P} \left( 1 + \frac{1}{5} \right) \)
Combine the terms inside the parenthesis:
\( 1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5} \)
The equation is now:
\( 300 = \text{P} \left( \frac{6}{5} \right) \)
To find P, we need to isolate it. Multiply both sides by \( \frac{5}{6} \):
\( \text{P} = 300 \times \frac{5}{6} \)
Perform the multiplication:
\( \text{P} = \frac{300 \times 5}{6} \)
\( \text{P} = \frac{1500}{6} \)
\( \text{P} = 250 \)
So, the principal amount invested was Rs. 250.
Summary of Results
| Parameter | Value |
|---|---|
| Amount (A) | Rs. 300 |
| Rate (R) | 8% p.a. |
| Time (T) | 2.5 years |
| Principal (P) | Rs. 250 |
The sum invested that becomes Rs. 300 at 8% simple interest in \(2\frac{1}{2}\) years is Rs. 250.
Revision Table: Simple Interest Formulas
| Concept | Formula | Description |
|---|---|---|
| Simple Interest (SI) | \( \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \) | Interest earned only on the principal. |
| Amount (A) | \( \text{A} = \text{P} + \text{SI} \) | Total money after adding interest to the principal. |
| Principal (P) | \( \text{P} = \frac{\text{SI} \times 100}{\text{R} \times \text{T}} \) | Original amount invested or borrowed (derived from SI formula). |
| Principal (P) | \( \text{P} = \frac{\text{A} \times 100}{100 + \text{R} \times \text{T}} \) | Original amount when Amount, Rate, and Time are known. |
Additional Information: Simple vs. Compound Interest
It's important to distinguish simple interest from compound interest. While simple interest is calculated only on the initial principal, compound interest is calculated on the principal amount and also on the accumulated interest of previous periods. This means compound interest grows much faster over time compared to simple interest for the same rate and principal.
- Simple Interest: Interest calculation is straightforward and constant each period based on the original principal.
- Compound Interest: Interest is added to the principal, and the next period's interest is calculated on this new, larger amount. This effect is often called "interest on interest."
Problems involving simple interest are typically easier as they follow a linear growth pattern for the interest earned.
Similar Questions
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Important Questions from Simple Interest
If ₹12,800 is invested in a bank for 5 years at the rate of 9% per annum simple interest. what amount is returned by the bank?
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If the simple interest for five years is equal is 35% of the principal, that rate of interest is:
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