Understanding the Simple Interest Problem

This question involves calculating the total time required for an initial investment to grow to a specific amount under simple interest, with the special condition that the earned interest is added back to the principal after a fixed period (10 years in this case).

Let's break down the problem:

• Initial Principal (\(P_1\)) = ₹1500
• Rate of Interest (\(R\)) = 5% per annum (p.a.) simple interest
• Interest is added to the principal every 10 years.
• Target Amount = ₹3000

We need to find the total time it takes for the initial ₹1500 to become ₹3000.

Step-by-Step Calculation of Simple Interest and Time

Calculation for the First 10 Years

First, let's calculate the simple interest earned in the initial 10 years on the principal of ₹1500 at 5% p.a.

The formula for simple interest is:

\( \text{Simple Interest (SI)} = \frac{\text{Principal (P)} \times \text{Rate (R)} \times \text{Time (T)}}{100} \)

For the first 10 years:

• \( P_1 = ₹1500 \)
• \( R = 5\% \)
• \( T_1 = 10 \text{ years} \)

Simple Interest earned in the first 10 years (\( SI_1 \)) is:

\( SI_1 = \frac{1500 \times 5 \times 10}{100} \)

\( SI_1 = \frac{1500 \times 50}{100} \)

\( SI_1 = 15 \times 50 \)

\( SI_1 = ₹750 \)

Principal After 10 Years

According to the problem, the interest earned after 10 years is added to the principal. So, the new principal for the subsequent period (\( P_2 \)) will be:

\( P_2 = P_1 + SI_1 \)

\( P_2 = 1500 + 750 \)

\( P_2 = ₹2250 \)

Calculating Time to Reach the Target Amount

The target amount is ₹3000. After 10 years, the amount has become ₹2250. The additional amount needed to reach the target is ₹3000 - ₹2250 = ₹750.

This additional ₹750 needs to be earned as simple interest on the new principal \( P_2 = ₹2250 \) at the same rate of 5% p.a.

Let \( T_2 \) be the time required to earn this additional ₹750.

Using the simple interest formula again, with \( SI_2 = ₹750 \), \( P_2 = ₹2250 \), and \( R = 5\% \):

\( SI_2 = \frac{P_2 \times R \times T_2}{100} \)

\( 750 = \frac{2250 \times 5 \times T_2}{100} \)

\( 750 = \frac{225 \times 5 \times T_2}{10} \)

Multiply both sides by 10:

\( 7500 = 225 \times 5 \times T_2 \)

\( 7500 = 1125 \times T_2 \)

Now, solve for \( T_2 \):

\( T_2 = \frac{7500}{1125} \)

To simplify the fraction \( \frac{7500}{1125} \):

Divide numerator and denominator by 25:

\( T_2 = \frac{7500 \div 25}{1125 \div 25} = \frac{300}{45} \)

Divide numerator and denominator by 5:

\( T_2 = \frac{300 \div 5}{45 \div 5} = \frac{60}{9} \)

Divide numerator and denominator by 3:

\( T_2 = \frac{60 \div 3}{9 \div 3} = \frac{20}{3} \)

So, \( T_2 = \frac{20}{3} \) years.

Converting the fraction to a mixed number:

\( \frac{20}{3} = 6 \text{ with a remainder of } 2 \)

So, \( T_2 = 6 \frac{2}{3} \) years.

Total Time Taken

The total time taken to reach the target amount of ₹3000 is the sum of the first 10 years and the additional time \( T_2 \).

\( \text{Total Time (T)} = T_1 + T_2 \)

\( T = 10 \text{ years} + 6 \frac{2}{3} \text{ years} \)

\( T = 16 \frac{2}{3} \text{ years} \)

Thus, the amount will become ₹3000 after \( 16 \frac{2}{3} \) years.

Summary of Calculation Steps

We found that \( T_2 = 6 \frac{2}{3} \) years were needed in the subsequent period.

Total Time = 10 years + \( 6 \frac{2}{3} \) years = \( 16 \frac{2}{3} \) years.

Revision Table: Key Concepts

Additional Information: Simple vs. Compound Interest

It's important to distinguish this problem from standard compound interest:

• Simple Interest: Interest is calculated only on the initial principal throughout the investment period. \( A = P(1 + \frac{RT}{100}) \).
• Compound Interest: Interest is calculated on the initial principal plus any accumulated interest from previous periods. The interest gets added to the principal, and the next interest calculation is on this new, larger principal. \( A = P(1 + \frac{R}{n})^{nt} \), where n is the number of times interest is compounded per year.

In this problem, the interest is added back only after a fixed interval (10 years), not compounded annually or more frequently as is typical in standard compound interest scenarios. This requires a step-by-step simple interest calculation for each period with the adjusted principal.