Calculating Simple Interest on a Principal Amount

This problem requires us to first find the initial principal amount using the information given about the first investment scenario, and then calculate the simple interest for a second scenario using that same principal.

Step 1: Finding the Principal Amount

We are given that a sum grows to Rs. 1,13,880 in 5 years at a simple interest rate of 6% per annum. The amount (A) received at the end of the investment period is the sum of the principal (P) and the simple interest (SI) earned.

The formula for simple interest is:

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

Where:

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

The formula for the amount is:

\( A = P + SI \)

Substituting the SI formula into the amount formula:

\( A = P + \frac{P \times R \times T}{100} \)

Given values for the first scenario:

• \(A = 1,13,880\)
• \(R = 6\%\)
• \(T = 5\) years

Substituting these values into the amount formula:

\( 1,13,880 = P + \frac{P \times 6 \times 5}{100} \)

\( 1,13,880 = P + \frac{30P}{100} \)

\( 1,13,880 = P + 0.3P \)

\( 1,13,880 = 1.3P \)

Now, solve for P:

\( P = \frac{1,13,880}{1.3} \)

To make division easier, we can write 1.3 as \( \frac{13}{10} \):

\( P = \frac{1,13,880}{\frac{13}{10}} = 1,13,880 \times \frac{10}{13} \)

\( P = \frac{11,38,800}{13} \)

Performing the division:

\( 1138800 \div 13 \)

\( 113 \div 13 \approx 8 \) ( \( 13 \times 8 = 104 \) )

\( 113 - 104 = 9 \)

\( 98 \div 13 \approx 7 \) ( \( 13 \times 7 = 91 \) )

\( 98 - 91 = 7 \)

\( 78 \div 13 = 6 \) ( \( 13 \times 6 = 78 \) )

So, \( \frac{11,38,800}{13} = 87600 \)

The principal amount \(P = 87,600\).

Step 2: Calculating Simple Interest for the Second Scenario

Now we need to calculate the simple interest for the same principal amount \(P = 87,600\) invested for 8 years at a rate of 9% per annum.

Given values for the second scenario:

• \(P = 87,600\)
• \(R = 9\%\)
• \(T = 8\) years

Using the simple interest formula:

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

Substituting the values:

\( SI = \frac{87,600 \times 9 \times 8}{100} \)

\( SI = \frac{87,600 \times 72}{100} \)

Cancel out the zeros from 87,600 and 100:

\( SI = 876 \times 72 \)

Calculating the product:

\( 876 \times 72 \)

\( 876 \times (70 + 2) = 876 \times 70 + 876 \times 2 \)

\( 876 \times 70 = 61320 \)

\( 876 \times 2 = 1752 \)

\( SI = 61320 + 1752 \)

\( SI = 63072 \)

The simple interest for the second scenario is Rs. 63,072.

Summary of Calculation Steps

• Used the given amount, rate, and time from the first scenario to find the principal amount (P).
• Calculated the simple interest for the second scenario using the principal (P) found, the new rate (9%), and the new time (8 years).

Revision Table: Key Formulas

Additional Information on Simple Interest

Simple interest is a method of calculating interest where the interest amount is fixed over the investment period. It is calculated only on the initial principal amount. This differs from compound interest, where interest is calculated on the principal amount plus any accumulated interest from previous periods.

Factors that influence the simple interest earned are:

• Principal Amount (P): The initial sum of money invested or borrowed. A higher principal earns more interest, all else being equal.
• Rate of Interest (R): The percentage at which interest is calculated annually. A higher rate earns more interest.
• Time Period (T): The duration for which the money is invested or borrowed. Interest is directly proportional to the time period in simple interest calculations.

Simple interest is commonly used for short-term loans and is easier to calculate compared to compound interest.