Finding the Time to Triple Money with Simple Interest

This question asks us to determine the number of years it takes for an initial sum of money (principal) to become three times its original value when earning simple interest at a rate of 5% per annum.

Understanding Simple Interest

Simple interest is a type of interest calculation where the interest earned is only on the initial principal amount. It does not compound, meaning interest earned in previous periods does not earn interest in subsequent periods.

The formula for simple interest is:

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

Where:

• \( I \) is the Simple Interest earned.
• \( P \) is the Principal amount (the initial sum).
• \( R \) is the Rate of Interest per annum (in percentage).
• \( T \) is the Time period in years.

Setting up the Problem

Let the initial Principal amount be \( P \).

According to the question, the sum will become thrice of itself. This means the final Amount (\( A \)) will be \( 3 \times P \), or \( A = 3P \).

The total interest earned (\( I \)) is the difference between the final Amount and the Principal:

\( I = A - P \)

\( I = 3P - P \)

\( I = 2P \)

So, the total interest earned must be equal to twice the original principal.

The given Rate of Interest (\( R \)) is 5% per annum.

We need to find the Time (\( T \)) in years.

Calculating the Time Period

Now we can substitute the values we know into the simple interest formula:

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

Substitute \( I = 2P \) and \( R = 5 \):

\( 2P = \frac{P \times 5 \times T}{100} \)

To solve for \( T \), we can simplify the equation. Assuming the principal \( P \) is not zero, we can divide both sides by \( P \):

\( 2 = \frac{5 \times T}{100} \)

Now, multiply both sides by 100 to isolate the term with \( T \):

\( 2 \times 100 = 5 \times T \)

\( 200 = 5T \)

Finally, divide by 5 to find the value of \( T \):

\( T = \frac{200}{5} \)

\( T = 40 \)

So, it will take 40 years for the sum to become thrice of itself at a simple interest rate of 5% per annum.

Let's verify this with an example. Assume Principal \( P = 100 \). The amount needs to be \( 3 \times 100 = 300 \). The interest needed is \( 300 - 100 = 200 \). Using the formula with \( T = 40 \):

\( I = \frac{100 \times 5 \times 40}{100} = \frac{20000}{100} = 200 \)

This confirms our calculation.

Conclusion

Based on the simple interest calculation, a sum will become thrice of itself in 40 years at a 5% annual interest rate.

Revision Table: Simple Interest Calculation

Review the key concepts and formulas used in this problem.

Additional Information: Comparing Simple and Compound Interest

It's important to understand that simple interest and compound interest work differently. This problem specifically uses simple interest.

• Simple Interest: Interest is calculated only on the original principal. The interest amount is the same for each period (assuming constant rate and time unit).
• Compound Interest: Interest is calculated on the initial principal as well as the accumulated interest from previous periods. This leads to faster growth of the principal over time compared to simple interest.

The formula for the Amount (\( A \)) with compound interest is \( A = P(1 + \frac{R}{100})^T \). If this problem involved compound interest, the time taken to triple the money would be significantly less than 40 years at a 5% rate.