Simple Interest Calculation and Finding the Invested Sum

The problem asks us to find the original amount of money invested, also known as the principal sum, based on how much more simple interest it would earn if the interest rate were slightly higher over a fixed period.

Understanding the Problem

We are given the following information:

• Type of interest: Simple Interest
• Time period (T): 3 years
• Increase in interest rate: 2% (This means the new rate is the original rate + 2%)
• Increase in interest amount: ₹360 (This is the difference between the interest earned at the new rate and the interest earned at the original rate)
• We need to find the Principal amount (P).

Simple Interest Formula

The formula for Simple Interest (SI) is:

\( \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \)

Where:

• P = Principal amount
• R = Rate of interest per annum
• T = Time period in years

Setting up the Calculation

Let the original rate of interest be R% per annum.

The original simple interest (SI\(_1\)) for 3 years is:

\( \text{SI}_1 = \frac{\text{P} \times \text{R} \times 3}{100} \)

If the rate was 2% higher, the new rate would be (R + 2)% per annum.

The new simple interest (SI\(_2\)) for 3 years at the higher rate is:

\( \text{SI}_2 = \frac{\text{P} \times (\text{R} + 2) \times 3}{100} \)

We are told that the interest fetched is ₹360 more at the higher rate. This means the difference between the new interest and the original interest is ₹360.

\( \text{SI}_2 - \text{SI}_1 = 360 \)

Solving for the Principal (Invested Sum)

Substitute the formulas for SI\(_1\) and SI\(_2\) into the difference equation:

\( \frac{\text{P} \times (\text{R} + 2) \times 3}{100} - \frac{\text{P} \times \text{R} \times 3}{100} = 360 \)

Let's simplify the equation:

\( \frac{3\text{P}(\text{R} + 2) - 3\text{PR}}{100} = 360 \)

Distribute 3P in the first term:

\( \frac{3\text{PR} + 6\text{P} - 3\text{PR}}{100} = 360 \)

Notice that the terms with 3PR cancel each other out:

\( \frac{6\text{P}}{100} = 360 \)

Now, we can solve for P:

\( 6\text{P} = 360 \times 100 \)

\( 6\text{P} = 36000 \)

\( \text{P} = \frac{36000}{6} \)

\( \text{P} = 6000 \)

So, the value of the invested sum (the principal) is ₹6000.

Verification

Let's check the result. Suppose the original rate R was 5%. Then the original interest would be \( \frac{6000 \times 5 \times 3}{100} = \frac{90000}{100} = 900 \). If the rate was 7% (2% higher), the new interest would be \( \frac{6000 \times 7 \times 3}{100} = \frac{126000}{100} = 1260 \). The difference is \( 1260 - 900 = 360 \), which matches the given information.

The calculation confirms that the invested sum is ₹6000.

Final Answer

The value of the invested sum is ₹6000.

Simple Interest Revision Table

Additional Information on Simple Interest Calculations

Simple interest is a straightforward method for calculating interest on a loan or deposit. It is calculated only on the initial principal amount, making it simpler than compound interest, which calculates interest on both the principal and accumulated interest from previous periods.

Key Aspects of Simple Interest:

• Constant Interest Earned: In simple interest, the amount of interest earned each year remains constant, provided the principal and rate are unchanged.
• Dependency: Simple interest directly depends on the principal amount, the interest rate, and the time period. Any change in these factors directly and proportionally affects the simple interest earned or paid.
• Applications: Simple interest is often used for short-term loans, car loans, and some types of savings accounts, although compound interest is more common in long-term investments and loans like mortgages.
• Difference Calculation: As seen in this problem, when comparing simple interest under different rates for the same principal and time, the difference in interest is solely due to the difference in the rate. The formula \( \frac{\text{P} \times (\text{R}_2 - \text{R}_1) \times \text{T}}{100} \) directly gives the difference in interest, where (\( \text{R}_2 - \text{R}_1 \)) is the difference in rates. In this problem, (\( \text{R}_2 - \text{R}_1 \)) = 2%, so the difference in interest is \( \frac{\text{P} \times 2 \times 3}{100} = \frac{6\text{P}}{100} \), which matches our calculation.