Question

The difference between simple and compound interest on Rs. 6000 for 1 year at \(20 \%\) per annum reckoned half yearly is : (a) 120 (b) 60 (c) 180 (d) 72

Short answer

Answer: The difference between simple interest and compound interest on Rs. 6000 for 1 year at 20% per annum reckoned half-yearly is Rs. 60.

Step-by-step solution

Calculate the Simple Interest

To calculate the simple interest, we'll use the formula: Simple Interest = P × R × T, where P is the principal amount, R is the interest rate (per annum), and T is the time period (in years). So, Simple Interest = Rs. 6000 × 20% × 1 = Rs. 1200.

Calculate the Compound Interest with Half-Yearly Compounding

First, we need to find the interest rate for half-yearly compounding. Since the interest is compounded every 6 months and there are 2 compounding periods in a year, we'll divide the annual interest rate by 2. Half-Yearly Interest Rate = 20% ÷ 2 = 10%. Now, we'll calculate the compound interest using the formula: Compound Amount = P(1 + R/100)^n, where P is the principal amount, R is the interest rate per compounding period, and n is the number of compounding periods. Compound Amount = Rs. 6000 × (1 + 10/100)^2 = Rs. 6000 × (1.1)^2. CompoundAmount = Rs. 6000 × 1.21 = Rs.7260. Now, subtract the principal amount to find the Compound Interest: Compound Interest = Compound Amount - Principal Amount = Rs. 7260 - Rs. 6000 = Rs. 1260.

Find the Difference between the Simple Interest and Compound Interest

We now have the Simple Interest and Compound Interest amounts, so we can find the difference between them: Difference = Compound Interest - Simple Interest = Rs. 1260 - Rs. 1200 = Rs. 60. Thus, the difference between Simple Interest and Compound Interest on Rs. 6000 for 1 year at 20% per annum reckoned half-yearly is Rs. 60. Therefore, the correct answer is (b) 60.

Key concepts

Interest Calculation

Interest calculation is an essential part of understanding how money grows when saved or invested. There are two main types of interest: simple and compound. Both types use different methods to calculate the amount of money you will earn or owe over time.

Simple Interest is calculated using a straightforward formula:

• Simple Interest = Principal (P) × Rate (R) × Time (T).

Here, the Principal is the initial amount of money, the Rate is the annual interest rate, and Time is the period for which the interest is calculated.
For example, if you have Rs. 6000 with a simple interest rate of 20% per year for 1 year, you would calculate the interest as Rs. 6000 × 0.20 × 1 = Rs. 1200. Compound Interest, however, involves interest on both the initial principal and the accumulated interest from previous periods. The formula for compound interest is a bit more complex:

• Compound Amount = P(1 + R/100)^n.

Where n is the number of compounding periods. This method often results in more earnings than simple interest over the same period.

Half-Yearly Compounding

When dealing with compound interest, the frequency of compounding matters greatly. Half-yearly compounding means that interest is calculated and added to the principal every six months.
This is different from annual compounding where the interest is added once per year. With half-yearly compounding, the annual interest rate is divided into two periods.

• For instance, a 20% annual rate becomes 10% per half-year period.

Let's see how this works in practice:

• If you invest Rs. 6000 at 20% per annum with half-yearly compounding, the interest for the first six months will be calculated with a 10% rate. The principal for the next six months includes the first half-year's interest.

This compounding process results in more interest earned compared to calculating annually, as each period's interest is calculated on an increased principal.

Difference Between Interests

Finding the difference between simple and compound interest helps in understanding the benefits of compounding. Simple interest only considers the original amount (principal) over the entire period, while compound interest grows each time interest is added to the principal.

In our given exercise, you can see the effects:

• Simple Interest on Rs. 6000 at 20% for 1 year = Rs. 1200.
• Compound Interest with half-yearly compounding on the same amount and rate leads to Rs. 1260.
• The difference = Rs. 1260 - Rs. 1200 = Rs. 60.

This Rs. 60 represents the additional amount earned due to the compounding effect over simply tallying the interest over one year. It illustrates how choosing compound interest, especially with more frequent compounding, can lead to slightly higher returns.

Principal Amount

The Principal Amount is the foundation of any interest calculation. It's the initial sum of money invested or borrowed before adding interest. Understanding the role of the principal is vital for managing finances effectively.
Before any interest is calculated, whether simple or compound, the principal remains unaltered and is your initial starting point.

• For instance, in our example exercise, Rs. 6000 is the principal amount.

When interest is calculated, it's always in relation to this principal amount. For simple interest, the principal remains the same throughout the time period. For compound interest, however, the principal "grows" as interest is added to it, especially with frequent compounding, like half-yearly intervals.