Understanding and Calculating Simple Interest from Compound Interest

The question provides information about the compound interest earned on a sum of money over a specific period at a given rate and asks for the simple interest on the same sum for the same period and rate. To solve this, we first need to determine the principal sum on which the interest is calculated.

Given Information:

• Compound Interest (CI) = ₹1530
• Time Period (T) = 2 years
• Rate of Interest (R) = 4% per annum

Goal:

Find the Simple Interest (SI) on the same principal, for the same period, and at the same rate.

Step 1: Calculate the Principal Sum (P)

The formula for the amount (A) with compound interest is:

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

The compound interest (CI) is the difference between the amount and the principal:

\(CI = A - P = P \left(1 + \frac{R}{100}\right)^T - P\)

\(CI = P \left[ \left(1 + \frac{R}{100}\right)^T - 1 \right]\)

Substitute the given values into the formula:

\(1530 = P \left[ \left(1 + \frac{4}{100}\right)^2 - 1 \right]\)

\(1530 = P \left[ \left(1 + 0.04\right)^2 - 1 \right]\)

\(1530 = P \left[ \left(1.04\right)^2 - 1 \right]\)

Calculate \((1.04)^2\):

\(1.04 \times 1.04 = 1.0816\)

Now substitute this back into the equation:

\(1530 = P \left[ 1.0816 - 1 \right]\)

\(1530 = P \times 0.0816\)

To find P, divide 1530 by 0.0816:

\(P = \frac{1530}{0.0816}\)

Let's perform the division:

\(P = 18750\)

So, the principal sum is ₹18750.

Step 2: Calculate the Simple Interest (SI)

Now that we have the principal (P), rate (R), and time (T), we can calculate the simple interest using the formula:

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

Substitute the values: P = ₹18750, R = 4%, T = 2 years.

\(SI = \frac{18750 \times 4 \times 2}{100}\)

\(SI = \frac{18750 \times 8}{100}\)

\(SI = \frac{150000}{100}\)

\(SI = 1500\)

The simple interest on the sum for the same period and at the same rate is ₹1500.

Conclusion

The simple interest on the sum of ₹18750 for 2 years at 4% per annum is ₹1500.

Revision Table: Compound vs Simple Interest Calculation

Additional Information: Relationship Between CI and SI

For a principal amount P, rate R, and time T:

• If T = 1 year, CI = SI = \(\frac{PR}{100}\).
• If T > 1 year, CI > SI. The difference between CI and SI arises because in compound interest, the interest earned in the first period is added to the principal, and interest for the subsequent periods is calculated on this new, larger amount. In simple interest, the principal remains the same for interest calculation throughout the period.
• The difference between CI and SI for 2 years is given by the interest on the first year's simple interest at the given rate. Difference \(= CI - SI = P\left(\frac{R}{100}\right)^2 \times T_{effective}\) or more directly for 2 years, Difference \(= P\left(\frac{R}{100}\right)^2\).

In this problem, the difference between CI and SI is \(1530 - 1500 = 30\). Using the difference formula for 2 years: Difference \(= 18750 \times (\frac{4}{100})^2 = 18750 \times (0.04)^2 = 18750 \times 0.0016 = 30\). This confirms our calculations.