Question

Ganesh makes a fixed deposit of Rs 40000 in a bank for a year at \(20 \%\) p. a simple interest. Harish makes a fixed deposit of an equal sum for the same period and at the same rate of interest, interest being compounded half yearly. Find the difference between maturity values of investment of Harish and Ganesh. (in \(\mathrm{Rs}\) s). (1) 400 (2) 360 (3) 440 (4) 300

Short answer

Answer: 400 Rs.

Step-by-step solution

Calculate the simple interest for Ganesh's investment

To calculate the simple interest on Ganesh's investment, we will use the formula: Simple interest \(= P \cdot R \cdot T\), where \(P\) is the principal amount, \(R\) is the rate, and \(T\) is the time (in years). Here, \(P = 40000 \ \mathrm{Rs}\), \(R = 20 \%\), and \(T = 1\) year. So, the simple interest will be: Simple interest \(= 40000 \cdot \frac{20}{100} \cdot 1 = 8000 \ \mathrm{Rs}\).

Calculate Ganesh's maturity value

Maturity value is the total amount gained after adding the simple interest to the principal amount. For Ganesh, the maturity value will be: Maturity value \(= \mathrm{Principal \ amount + Simple \ interest} = 40000 + 8000 = 48000 \ \mathrm{Rs}\).

Calculate the compound interest for Harish's investment

To calculate the compound interest on Harish's investment, compounded half yearly, we will use the formula: Compound amount \(= P \cdot (1+ \frac{R}{n \cdot 100})^{n \cdot T}\), where \(P\) is the principal amount, \(R\) is the rate, \(T\) is the time (in years), and \(n\) is the number of times the interest is compounded per year. Here, \(P = 40000 \ \mathrm{Rs}\), \(R = 20 \%\), \(T = 1\) year, and interest is compounded half yearly, so \(n = 2\). So the compound amount will be: Compound amount \(= 40000 \cdot (1+ \frac{20}{2 \cdot 100})^{2 \cdot 1} = 40000 \cdot (1+\frac{1}{10})^2 = 40000 \cdot \frac{121}{100} = 48400 \ \mathrm{Rs}\).

Find the difference between maturity values

Now, we can find the difference between the maturity values of investment of Harish and Ganesh, which is: Difference \(= \mathrm{Maturity \ value \ of \ Harish - Maturity \ value \ of \ Ganesh} = 48400 - 48000 = 400 \ \mathrm{Rs}\). So, the difference between maturity values of investment of Harish and Ganesh is \(400 \ \mathrm{Rs}\). The correct answer is (1).

Key concepts

Simple Interest Calculation

Simple interest is a financial concept used to calculate the additional money earned or paid on a principal amount over a specific period. The calculation is straightforward, applying a fixed interest rate to the principal amount across the entire period. In Ganesh's case, the formula used is :
\[\text{Simple Interest} = P \times \frac{R}{100} \times T\]
where \(P\) is the principal amount, \(R\) is the annual interest rate, and \(T\) is the time in years. For students, it's important to remember that these factors must be in their annual forms to calculate the yearly simple interest earned correctly.

Compound Interest Calculation

Compound interest is what makes investments grow faster over time. Unlike simple interest, compound interest earns interest not only on the initial principal but also on the accrued interest from previous periods. In Harish's scenario, the bank compounds interest half-yearly. The compound interest calculation is a bit more complex than simple interest:

\[\text{Compound Amount} = P \times \left(1 + \frac{R}{n\times 100}\right)^{n\times T}\]

where \(P\) is the principal, \(R\) is the annual interest rate, \(n\) is the number of compounding periods per year, and \(T\) is the time in years. The power of compounding is evident as each interest addition creates a new, larger base for the next interest calculation.

Maturity Value of Investments

The maturity value is the total amount you receive when an investment reaches its maturity date, including the initial principal plus all the interest it has accrued. For Ganesh, investing with simple interest, the maturity value was calculated by adding the simple interest to the principal.
\[\text{Maturity Value} = \text{Principal Amount} + \text{Simple Interest}\]
For Harish, who chose compound interest, the maturity value was already computed in the compound amount as the formula accounts for the principal and the compounded interest over the period.

Interest Compounding Periods

The interest compounding period is crucial when calculating compound interest. It determines how often interest is added to the principal. The more frequent the compounding, the greater the amount of compound interest. It's common to have yearly, semiannually, quarterly, or monthly compounding. In Harish's case, the bank opted for semiannual compounding, meaning the interest was compounded every six months (twice a year), as reflected by \(n = 2\) in the compound amount formula above. It's essential to know the compounding period to accurately predict the growth of your investment.