Understanding the Compound Interest Problem

The question asks us to find the original principal sum that was invested. We are given the amounts received after two different time periods under compound interest, compounded annually. This is a common type of problem where we can use the compound interest formula to find both the interest rate and the principal sum.

The formula for compound interest when compounded annually is:

$$A = P(1 + r)^t$$

Where:

• \(A\) is the amount after \(t\) years
• \(P\) is the principal sum
• \(r\) is the annual interest rate (as a decimal)
• \(t\) is the number of years

Setting up Equations from the Given Information

We are given two pieces of information:

• Amount after 3 years = Rs. 20,736
• Amount after 5 years = Rs. 29,859.84

Using the compound interest formula, we can write two equations:

1. For \(t = 3\) years: $$20736 = P(1 + r)^3$$
2. For \(t = 5\) years: $$29859.84 = P(1 + r)^5$$

Finding the Annual Interest Rate

We have two equations with two unknowns (\(P\) and \(r\)). To find the interest rate (\(r\)), we can divide the second equation by the first equation:

$$\frac{29859.84}{20736} = \frac{P(1 + r)^5}{P(1 + r)^3}$$

The \(P\) term cancels out, and we are left with:

$$\frac{29859.84}{20736} = (1 + r)^{5-3}$$

$$\frac{29859.84}{20736} = (1 + r)^2$$

Let's calculate the ratio on the left side:

$$\frac{29859.84}{20736} = 1.44$$

So, we have:

$$(1 + r)^2 = 1.44$$

To find \(1 + r\), take the square root of both sides:

$$1 + r = \sqrt{1.44}$$

$$1 + r = 1.2$$

Now, solve for \(r\):

$$r = 1.2 - 1$$

$$r = 0.2$$

This means the annual interest rate is 0.2 or 20%.

Calculating the Principal Sum

Now that we know the interest rate \(r = 0.2\), we can substitute this value back into either of the original equations to find the principal sum \(P\). Let's use the equation for 3 years:

$$20736 = P(1 + 0.2)^3$$

$$20736 = P(1.2)^3$$

Calculate \((1.2)^3\):

$$(1.2)^3 = 1.2 \times 1.2 \times 1.2 = 1.44 \times 1.2 = 1.728$$

So, the equation becomes:

$$20736 = P(1.728)$$

Now, solve for \(P\):

$$P = \frac{20736}{1.728}$$

$$P = 12000$$

The original principal sum is Rs. 12000.

Verification

We can verify this by calculating the amount after 5 years with P = 12000 and r = 0.2:

$$A_5 = 12000(1 + 0.2)^5$$

$$A_5 = 12000(1.2)^5$$

Calculate \((1.2)^5\):

$$(1.2)^5 = 1.2^3 \times 1.2^2 = 1.728 \times 1.44 = 2.48832$$

$$A_5 = 12000 \times 2.48832$$

$$A_5 = 29859.84$$

This matches the amount given for 5 years, confirming our calculation is correct.

Final Answer Summary

The principal sum invested is Rs. 12000.

Revision Table: Compound Interest Calculation Steps

Additional Information on Compound Interest

Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. This makes it grow faster than simple interest.

Key aspects of compound interest:

• Compounding Frequency: The number of times interest is calculated and added to the principal within a year (e.g., annually, semi-annually, quarterly, monthly). In this problem, it's annual.
• Growth Factor: The term \((1 + r)\) is often called the growth factor per period. In this case, it's the annual growth factor, which we found to be 1.2. This means the amount increases by 20% each year.
• Power of Time: The exponent \(t\) in the formula \(P(1+r)^t\) shows how compounding works over time. The longer the time, the greater the effect of compounding.
• Solving for unknowns: If you have amounts at different time points, dividing the equations helps eliminate the principal \(P\) and allows you to solve for the rate \((1+r)\) or \(r\).