Understanding the Problem: Compound vs. Simple Interest

The question asks us to find the simple interest on a certain sum for a specific time and rate, given information about the difference in compound interest calculation methods for one year on the same sum and rate. We are given the annual interest rate, the time period for the interest difference calculation, the difference amount, and the time period for which simple interest needs to be calculated.

Let's break down the information provided:

• Rate of Interest (R): 10% per annum (p.a.)
• Time for CI Difference: 1 year
• Difference in CI (Half-yearly vs. Yearly): Rs. 88.80
• Time for SI Calculation (T'): \(1\frac{2}{3}\) years

Our first step is to find the principal sum (P) using the compound interest information. Once we have P, we can calculate the simple interest for \(1\frac{2}{3}\) years at 10% p.a.

Calculating Compound Interest Under Different Compounding Frequencies

For one year at 10% p.a.:

• Compounded Yearly: The interest is calculated once at the end of the year. The rate applied is the full annual rate, 10%.
• Compounded Half-yearly: The interest is calculated twice a year. The rate applied per period is half the annual rate (10%/2 = 5%), and the number of periods in one year is 2.

Let P be the principal sum.

CI compounded Yearly for 1 year:

The amount after 1 year is given by the formula: \(A = P(1 + \frac{R}{100})^T\)

Here, R = 10%, T = 1 year.

\(A_{\text{yearly}} = P(1 + \frac{10}{100})^1 = P(1 + 0.1)^1 = P(1.1)\)

The compound interest is \(CI_{\text{yearly}} = A_{\text{yearly}} - P = 1.1P - P = 0.1P\)

CI compounded Half-yearly for 1 year:

The rate per half-year is \(R' = \frac{10\%}{2} = 5\%\). The number of periods in 1 year is \(n = 1 \times 2 = 2\).

The amount after 1 year is given by the formula: \(A = P(1 + \frac{R'}{100})^n\)

\(A_{\text{half-yearly}} = P(1 + \frac{5}{100})^2 = P(1 + 0.05)^2 = P(1.05)^2\)

\(A_{\text{half-yearly}} = P(1.1025)\)

The compound interest is \(CI_{\text{half-yearly}} = A_{\text{half-yearly}} - P = 1.1025P - P = 0.1025P\)

Finding the Principal Sum (P)

The difference between the two compound interests is given as Rs. 88.80.

\(CI_{\text{half-yearly}} - CI_{\text{yearly}} = 88.80\)

\(0.1025P - 0.1P = 88.80\)

\(0.0025P = 88.80\)

To find P, we divide 88.80 by 0.0025:

\(P = \frac{88.80}{0.0025} = \frac{88.80}{\frac{25}{10000}}\)

\(P = 88.80 \times \frac{10000}{25} = 88.80 \times 400\)

\(P = 35520\)

So, the principal sum is Rs. 35,520.

Calculating Simple Interest

Now we need to calculate the simple interest on the sum P = Rs. 35,520 for \(1\frac{2}{3}\) years at the same rate of 10% p.a.

The formula for Simple Interest is: \(SI = \frac{P \times R \times T'}{100}\)

Here, P = 35520, R = 10%, and \(T' = 1\frac{2}{3}\) years.

Convert the mixed fraction time into an improper fraction: \(1\frac{2}{3} = \frac{(1 \times 3) + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3}\) years.

Now, substitute the values into the SI formula:

\(SI = \frac{35520 \times 10 \times \frac{5}{3}}{100}\)

\(SI = \frac{35520 \times 10 \times 5}{100 \times 3}\)

\(SI = \frac{35520 \times 50}{300}\)

Cancel out common factors (like 100 from numerator and denominator):

\(SI = \frac{35520 \times 50/100}{300/100} = \frac{35520 \times 0.5}{3}\)

Alternatively, cancel out 100 directly:

\(SI = \frac{35520 \times 10 \times 5}{100 \times 3} = \frac{35520 \times 5}{10 \times 3} = \frac{3552 \times 5}{3}\)

Divide 3552 by 3:

\(3552 \div 3 = 1184\)

\(SI = 1184 \times 5\)

\(SI = 5920\)

The simple interest on the same sum for \(1\frac{2}{3}\) years at the same rate is Rs. 5,920.

Conclusion

By first using the difference in compound interest calculated yearly and half-yearly for one year, we found the principal sum. Then, we used this principal sum to calculate the simple interest for the specified time period and rate.

Revision Table: Compound and Simple Interest Basics

Additional Information: Interest Calculation Methods

Interest is a fee paid by a borrower of funds to the lender in return for the use of the money. It is typically calculated as a percentage of the principal sum.

• Simple Interest: This is the most basic form of interest. It is calculated only on the initial principal amount. It is commonly used for short-term loans and some savings accounts.
• Compound Interest: This is interest calculated on the initial principal and also on the accumulated interest from previous periods. It is often referred to as "interest on interest," and it can lead to significantly larger amounts over time compared to simple interest, especially over longer periods.
• Compounding Frequency: This refers to how many times per year the accumulated interest is added back to the principal to calculate future interest. Common frequencies include annually, half-yearly, quarterly, monthly, and daily. The more frequent the compounding, the higher the compound interest earned or paid for a given annual rate. The difference between compound interest compounded at different frequencies increases with the principal amount, the interest rate, and the time period.

Understanding the difference between simple and compound interest and how compounding frequency affects returns or costs is crucial for financial planning and analysis.