Calculating the Interest Rate from Simple and Compound Interest

This problem involves finding the rate of interest given the simple interest (SI) and compound interest (CI) earned on a certain sum over a two-year period at the same rate.

We are given:

• Simple Interest (SI) for 2 years = Rs. 1,000
• Compound Interest (CI) for 2 years = Rs. 1,040
• Time period = 2 years
• The rate of interest is the same for both SI and CI.

Understanding Simple and Compound Interest for 2 Years

Let the principal amount be \( P \) and the rate of interest be \( R \) percent per annum. The time is \( T = 2 \) years.

Simple Interest Calculation

Simple interest is calculated only on the principal amount. The formula for simple interest is:

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

For 2 years, the SI is Rs. 1,000.

\( 1000 = \frac{P \times R \times 2}{100} \)

This gives us: \( P \times R = \frac{1000 \times 100}{2} = 50000 \)

\( PR = 50000 \) (Equation 1)

Also, the simple interest earned in the first year is half of the total simple interest for two years, since the interest is constant each year in simple interest.

SI for 1st year = \( \frac{1000}{2} = 500 \) Rupees.

Compound Interest Calculation

Compound interest is calculated on the principal amount as well as the accumulated interest from previous periods. For the first year, compound interest is the same as simple interest.

CI for 1st year = SI for 1st year = Rs. 500.

The total compound interest for 2 years is Rs. 1,040.

CI for 2 years = CI for 1st year + Interest earned in 2nd year

\( 1040 = 500 + \text{Interest earned in 2nd year} \)

Interest earned in 2nd year = \( 1040 - 500 = 540 \) Rupees.

Finding the Rate of Interest

The key difference between CI and SI for the second year comes from the interest earned on the first year's interest.

The interest earned in the second year under compound interest (Rs. 540) is the interest calculated on the amount at the end of the first year. The amount at the end of the first year is Principal + CI for 1st year = \( P + 500 \). Alternatively, and more simply, the extra interest earned in the second year compared to simple interest is the interest earned on the first year's simple interest.

SI for 2nd year would also be Rs. 500.

The difference between CI for 2nd year and SI for 2nd year is \( 540 - 500 = 40 \) Rupees.

This difference of Rs. 40 is the interest earned on the first year's interest amount (which is Rs. 500) at the rate \( R \) for one year.

Using the simple interest formula for this interest on interest:

Interest on 1st year's interest = \( \frac{\text{1st Year's Interest} \times R \times 1}{100} \)

\( 40 = \frac{500 \times R \times 1}{100} \)

\( 40 = 5 \times R \)

Now, we solve for \( R \):

\( R = \frac{40}{5} \)

\( R = 8 \)

So, the rate of interest is 8 percent per annum.

We can also verify this using a direct formula for the difference between CI and SI for 2 years:

\( \text{CI} - \text{SI} = \frac{P R^2}{100^2} \)

We know \( PR = 50000 \) from Equation 1. We can write \( P = \frac{50000}{R} \).

\( 1040 - 1000 = \frac{\left(\frac{50000}{R}\right) R^2}{10000} \)

\( 40 = \frac{50000 \times R}{10000} \)

\( 40 = 5R \)

\( R = \frac{40}{5} \)

\( R = 8 \)

Both methods yield the same rate of 8 percent per annum.

Summary of Calculation Steps

1. Calculate the Simple Interest for the first year by dividing the total SI for 2 years by 2.
2. Recognize that CI for the first year is equal to SI for the first year.
3. Calculate the interest earned in the second year under compound interest by subtracting the first year's CI from the total CI for 2 years.
4. Find the difference between the interest earned in the second year under CI and the simple interest for one year. This difference is the interest on the first year's interest.
5. Use the simple interest formula to find the rate, considering the principal as the first year's interest, time as 1 year, and the interest as the calculated difference.

The extra Rs. 40 in the second year's compound interest compared to simple interest is the interest earned on the Rs. 500 earned as interest in the first year.

Revision Table: Simple vs. Compound Interest

Additional Information: Compound Interest Formulas

Here are some standard formulas related to compound interest that are useful for solving problems:

• Amount \( A = P(1 + \frac{R}{100})^T \)
• Compound Interest \( \text{CI} = A - P = P[(1 + \frac{R}{100})^T - 1] \)
• Difference between CI and SI for 2 years \( = P(\frac{R}{100})^2 \)
• Difference between CI and SI for 3 years \( = P[(\frac{R}{100})^3 + 3(\frac{R}{100})^2] \)

In our problem, using the 2-year difference formula:

\( \text{CI} - \text{SI} = P(\frac{R}{100})^2 \)

\( 40 = P(\frac{R}{100})^2 \)

We also know \( SI = \frac{PRT}{100} \Rightarrow 1000 = \frac{P \times R \times 2}{100} \Rightarrow PR = 50000 \)

From the difference formula, \( 40 = P \times \frac{R}{100} \times \frac{R}{100} \). Substitute \( PR = 50000 \):

\( 40 = 50000 \times \frac{R}{10000} \)

\( 40 = 5R \)

\( R = 8 \)

This confirms the result obtained by analyzing the year-wise interest.