At 12% simple interest per annum a sum of money becomes Rs. 295 in \(1\frac{1}{2}\) years. What was the sum invested?
Rs. 250
Calculating the Original Sum Invested using Simple Interest
This question asks us to find the initial sum of money, also known as the Principal, that was invested. We are given the final amount, the simple interest rate, and the time period. We will use the formulas for simple interest and amount to solve this problem.
Let's identify the given information:
- Amount (\(A\)) = Rs. 295
- Simple Interest Rate (\(R\)) = 12% per annum
- Time (\(T\)) = \(1\frac{1}{2}\) years
We need to find the Principal (\(P\)).
First, let's convert the time from a mixed fraction to a simple fraction or decimal:
\(T = 1\frac{1}{2} \text{ years} = 1 + \frac{1}{2} \text{ years} = \frac{2}{2} + \frac{1}{2} \text{ years} = \frac{3}{2} \text{ years} = 1.5 \text{ years}\)
The formula for Simple Interest (\(SI\)) is:
\(SI = \frac{P \times R \times T}{100}\)
The Amount (\(A\)) received at the end of the time period is the sum of the Principal (\(P\)) and the Simple Interest (\(SI\)):
\(A = P + SI\)
We can substitute the formula for \(SI\) into the formula for \(A\):
\(A = P + \frac{P \times R \times T}{100}\)
We can factor out \(P\) from the right side of the equation:
\(A = P \left(1 + \frac{R \times T}{100}\right)\)
Now, we can plug in the given values into this formula:
\(295 = P \left(1 + \frac{12 \times 1.5}{100}\right)\)
Let's simplify the term inside the parentheses:
\(1 + \frac{12 \times 1.5}{100} = 1 + \frac{18}{100} = 1 + 0.18 = 1.18\)
So the equation becomes:
\(295 = P \times 1.18\)
To find \(P\), we need to divide the Amount by \(1.18\):
\(P = \frac{295}{1.18}\)
Let's perform the division:
\(P = 250\)
So, the sum invested (Principal) was Rs. 250.
Summary of the Calculation
| Given | Value |
|---|---|
| Amount (A) | Rs. 295 |
| Rate (R) | 12% per annum |
| Time (T) | \(1\frac{1}{2}\) years or 1.5 years |
| Formula Used | Explanation |
|---|---|
| \(A = P \left(1 + \frac{R \times T}{100}\right)\) | Amount equals Principal plus Simple Interest |
Substituting values and solving for P:
\(295 = P \left(1 + \frac{12 \times 1.5}{100}\right)\)
\(295 = P \left(1 + \frac{18}{100}\right)\)
\(295 = P (1 + 0.18)\)
\(295 = P \times 1.18\)
\(P = \frac{295}{1.18}\)
\(P = 250\)
The sum invested was Rs. 250.
Revision Table: Key Simple Interest Concepts
| Term | Definition | Formula |
|---|---|---|
| Principal (P) | The initial sum of money invested or borrowed. | - |
| Rate (R) | The percentage of the principal charged as interest per period (usually per year). | - |
| Time (T) | The duration for which the money is invested or borrowed. | - |
| Simple Interest (SI) | Interest calculated only on the initial principal amount. | \(SI = \frac{P \times R \times T}{100}\) |
| Amount (A) | The total sum of money at the end of the period, including principal and interest. | \(A = P + SI\) or \(A = P \left(1 + \frac{R \times T}{100}\right)\) |
Additional Information on Simple Interest Calculations
Simple interest is a basic concept in finance. It is calculated on the original principal only, regardless of the interest earned in previous periods. This is different from compound interest, where interest is calculated on the principal amount and also on the accumulated interest from previous periods.
Understanding how to calculate the principal amount when the final amount, rate, and time are given is a common type of problem. The key is to use the relationship between Amount, Principal, and Simple Interest. By rearranging the formulas, you can find any missing value if the others are known.
Always make sure the time period and the interest rate match. For example, if the rate is per annum, the time should be in years. If time is given in months, convert it to years by dividing by 12. If time is given in days, convert it to years by dividing by 365 (or 366 for a leap year, although often 365 is used unless specified).
In this problem, the time was given as \(1\frac{1}{2}\) years, which is already in years, making the calculation straightforward after converting the mixed fraction.
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