Understanding the Compound Interest Problem

This question asks us to find the rate of interest per year when a sum of money invested at compound interest grows to specific amounts over different time periods. We are given the amount after 3 years and the amount after 5 years and need to determine the annual compound interest rate.

Let the principal amount be \(P\), the annual compound interest rate be \(R\) per cent, and the time period be \(n\) years. The formula for the amount \(A\) after \(n\) years at compound interest is:

\(A = P \left(1 + \frac{R}{100}\right)^n\)

Setting Up Equations from the Given Information

We are given two pieces of information based on the compound interest growth:

• Amount after 3 years (\(n=3\)) is Rs. 7,800.
• Amount after 5 years (\(n=5\)) is Rs. 11,232.

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

1. For \(n=3\): \(7800 = P \left(1 + \frac{R}{100}\right)^3\)
2. For \(n=5\): \(11232 = P \left(1 + \frac{R}{100}\right)^5\)

Calculating the Rate of Compound Interest

To find the rate \(R\), we can eliminate the principal amount \(P\) by dividing the second equation by the first equation:

\(\frac{11232}{7800} = \frac{P \left(1 + \frac{R}{100}\right)^5}{P \left(1 + \frac{R}{100}\right)^3}\)

The \(P\) term cancels out, and we use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):

\(\frac{11232}{7800} = \left(1 + \frac{R}{100}\right)^{5-3}\)

\(\frac{11232}{7800} = \left(1 + \frac{R}{100}\right)^2\)

Now, let's simplify the fraction \(\frac{11232}{7800}\). Both numbers are divisible by various factors. We can start by dividing both by 4:

\(\frac{11232 \div 4}{7800 \div 4} = \frac{2808}{1950}\)

Both are even, divide by 2:

\(\frac{2808 \div 2}{1950 \div 2} = \frac{1404}{975}\)

Sum of digits of 1404 = 1+4+0+4 = 9 (divisible by 3 and 9). Sum of digits of 975 = 9+7+5 = 21 (divisible by 3).

Divide by 3:

\(\frac{1404 \div 3}{975 \div 3} = \frac{468}{325}\)

Let's try to simplify this further. 325 ends in 5, so it's divisible by 5 (\(325 = 5 \times 65 = 5 \times 5 \times 13 = 25 \times 13\)). 468 is not divisible by 5. Let's check if 468 is divisible by 13: \(468 \div 13 = 36\). Yes, \(468 = 13 \times 36\).

So, we have:

\(\frac{468}{325} = \frac{13 \times 36}{13 \times 25} = \frac{36}{25}\)

Now substitute this back into our equation:

\(\left(1 + \frac{R}{100}\right)^2 = \frac{36}{25}\)

To find \(1 + \frac{R}{100}\), we take the square root of both sides:

\(1 + \frac{R}{100} = \sqrt{\frac{36}{25}}\)

\(1 + \frac{R}{100} = \frac{\sqrt{36}}{\sqrt{25}}\)

\(1 + \frac{R}{100} = \frac{6}{5}\)

Convert \(\frac{6}{5}\) to a decimal or keep as fraction:

\(1 + \frac{R}{100} = 1.2\)

Now, isolate \(\frac{R}{100}\):

\(\frac{R}{100} = 1.2 - 1\)

\(\frac{R}{100} = 0.2\)

Solve for \(R\):

\(R = 0.2 \times 100\)

\(R = 20\)

So, the rate of interest is 20% per annum.

Conclusion

The sum invested at compound interest grows at a rate of 20% per year. This rate explains the increase in the amount from Rs. 7,800 in 3 years to Rs. 11,232 in 5 years.

The calculation confirms that a 20% annual compound interest rate yields these amounts over the specified periods.

Revision Table: Key Concepts in Compound Interest

Additional Information on Compound Interest Calculations

When solving problems involving compound interest over different time periods, dividing the amount equations is a common and efficient method. This approach helps eliminate the unknown principal \(P\) and directly leads to an equation involving only the rate \(R\).

The difference in the exponent of \(\left(1 + \frac{R}{100}\right)\) when dividing the equations corresponds to the difference in the time periods. In this case, the time periods were 5 years and 3 years, so the difference is \(5 - 3 = 2\) years, leading to the term being raised to the power of 2.

Recognizing perfect squares or cubes in the ratio of the amounts is crucial for simplifying the calculation quickly. In this problem, simplifying \(\frac{11232}{7800}\) to \(\frac{36}{25}\) was key because 36 and 25 are perfect squares (\(6^2\) and \(5^2\)), making it easy to find the square root.