Simple Interest Calculation Steps

This problem asks us to find the simple interest earned over a specific period, given the final amount, the annual interest rate, and the time duration. We need to work backward from the amount to find the original principal and then calculate the interest.

Given Investment Details

Here's what we know from the question:

• Interest Rate (R) = 12.5% per annum
• Amount after 2 years (A) = ₹7,500
• Time Period (T) = 2 years

Interest Formulas Explained

To solve this, we use the basic formulas for simple interest:

• The formula for the final amount (A) is the sum of the principal (P) and the simple interest (SI):

  $A = P + SI$

• The formula for simple interest (SI) is:

  $SI = \frac{P \times R \times T}{100}$

We can combine these formulas. Substituting the SI formula into the Amount formula:

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

Factoring out P, we get:

$A = P \left( 1 + \frac{R \times T}{100} \right)$

Calculating the Principal Amount

First, we need to find the original principal amount (P). We can rearrange the combined formula to solve for P:

$P = \frac{A}{ \left( 1 + \frac{R \times T}{100} \right) } $

Now, let's plug in the given values:

• $A = 7500$
• $R = 12.5$
• $T = 2$

Calculate the term in the denominator:

$ \frac{R \times T}{100} = \frac{12.5 \times 2}{100} = \frac{25}{100} = 0.25 $

Now substitute this back into the formula for P:

$P = \frac{7500}{1 + 0.25} = \frac{7500}{1.25} $

To make the division easier, we can write 1.25 as a fraction $\frac{5}{4}$:

$P = \frac{7500}{5/4} = 7500 \times \frac{4}{5} $

$P = \frac{7500 \times 4}{5} = 1500 \times 4 = 6000 $

So, the principal amount invested was ₹6,000.

Determining the Simple Interest

Now that we have the principal amount (P = ₹6,000), we can calculate the simple interest (SI) using the SI formula:

$SI = \frac{P \times R \times T}{100}$

Substitute the values:

$SI = \frac{6000 \times 12.5 \times 2}{100}$

$SI = \frac{6000 \times 25}{100}$

$SI = 60 \times 25$

$SI = 1500$

The simple interest earned for the 2-year period is ₹1,500.

Final Answer Verification

The calculated simple interest is ₹1,500. This matches one of the options provided.

Principal (P) = ₹6,000

Interest (SI) = ₹1,500

Amount (A) = P + SI = ₹6,000 + ₹1,500 = ₹7,500. This confirms our calculation is correct.