Understanding Simple Interest Calculation

This problem asks us to find the annual rate of simple interest at which a sum of money grows to \( \frac{8}{7} \) times its original value in 2 years. Simple interest is calculated only on the principal amount.

Simple Interest Formulas

The key formulas for simple interest are:

• Simple Interest (SI) = \( \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} \)
• Amount (A) = Principal (P) + Simple Interest (SI)

From the second formula, we can also write SI = Amount (A) - Principal (P).

Setting up the Problem

Let the original sum of money be the Principal, denoted by \(P\). The problem states that the sum of money becomes \( \frac{8}{7} \) of itself in 2 years. This means the Amount (A) after 2 years is \( \frac{8}{7} \) times the Principal.

• Principal = \(P\)
• Amount (A) = \( \frac{8}{7} P \)
• Time (T) = 2 years
• We need to find the Rate (R) per annum.

Calculating the Simple Interest Earned

Using the formula SI = A - P, we can find the simple interest earned over the 2 years:

SI = \( \frac{8}{7} P - P \)

To subtract \(P\) from \( \frac{8}{7} P \), we can write \(P\) as \( \frac{7}{7} P \):

SI = \( \frac{8}{7} P - \frac{7}{7} P \)

SI = \( \left(\frac{8}{7} - \frac{7}{7}\right) P \)

SI = \( \frac{8-7}{7} P \)

SI = \( \frac{1}{7} P \)

So, the simple interest earned in 2 years is \( \frac{1}{7} \) of the original Principal.

Finding the Rate Per Annum

Now we use the main simple interest formula: \( \text{SI} = \frac{P \times R \times T}{100} \). We know SI = \( \frac{1}{7} P \), P is the principal, and T = 2 years. We need to solve for R.

\( \frac{1}{7} P = \frac{P \times R \times 2}{100} \)

We can cancel \(P\) from both sides of the equation (assuming \(P \neq 0\)):

\( \frac{1}{7} = \frac{R \times 2}{100} \)

\( \frac{1}{7} = \frac{2R}{100} \)

Simplify the fraction on the right side:

\( \frac{1}{7} = \frac{R}{50} \)

Now, solve for R by cross-multiplying or multiplying both sides by 50:

\( R = 50 \times \frac{1}{7} \)

\( R = \frac{50}{7} \)

Converting to a Mixed Number

The rate is \( \frac{50}{7} \% \). To express this as a mixed number, we divide 50 by 7.

\( 50 \div 7 \)

• 7 goes into 50, 7 times (since \( 7 \times 7 = 49 \)).
• The remainder is \( 50 - 49 = 1 \).

So, \( \frac{50}{7} \) can be written as \( 7 \frac{1}{7} \).

The rate per annum is \( 7 \frac{1}{7} \% \).

Summary of Steps

1. Identify Principal, Amount, and Time from the problem.
2. Calculate Simple Interest (SI) using the formula SI = Amount - Principal.
3. Use the Simple Interest formula SI = \( \frac{P \times R \times T}{100} \) to set up an equation.
4. Solve the equation for the rate (R).
5. Convert the result to a mixed fraction if necessary.

The calculated rate matches option 3.

Revision Table: Simple Interest Concepts

Additional Information: Simple vs Compound Interest

It's important to distinguish between simple interest and compound interest, although this problem specifically deals with simple interest.

• Simple Interest: Interest is calculated solely on the initial principal amount. The interest earned in previous periods is not added to the principal for calculating the next period's interest. This results in a fixed amount of interest earned each period.
• Compound Interest: Interest is calculated on the principal amount plus any accumulated interest from previous periods. This means the interest earned in each period is added to the principal, and the next period's interest is calculated on this new, larger principal. This leads to exponential growth of the investment or loan.

The formula used in this problem is specifically for simple interest. For compound interest, a different formula is used: \( A = P \left(1 + \frac{R}{100}\right)^T \).