Compound Interest Problem Solving: Finding the Original Sum

This problem involves understanding how money grows under compound interest over different time periods. We are given the amounts after 4 years and 6 years and need to find the original principal amount (the sum) and then half of it.

Understanding Compound Interest

Compound interest means that the interest earned in each period is added to the principal sum for the next period, leading to exponential growth. The formula for the amount (A) after 't' years with principal (P) and annual interest rate (r) is:

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

Here, the interest is compounded annually, which simplifies the formula.

Setting Up the Equations based on Amounts

We are given two pieces of information, which we can translate into two equations using the compound interest formula:

1. Amount after 4 years (A₄) = Rs. 11,880
2. Amount after 6 years (A₆) = Rs. 17,820

So, we have:

• Equation 1: \(11880 = P(1 + r)^4\)
• Equation 2: \(17820 = P(1 + r)^6\)

Finding the Growth Factor Between Time Periods

To find the annual growth factor \((1 + r)\), we can divide the equation for the later time period by the equation for the earlier time period. This eliminates the principal (P).

Divide Equation 2 by Equation 1:

\(\frac{17820}{11880} = \frac{P(1 + r)^6}{P(1 + r)^4}\)

Simplify the fraction on the left and the terms on the right:

\(\frac{1782}{1188} = (1 + r)^{6-4}\)

\(\frac{1782}{1188} = (1 + r)^2\)

Now, let's simplify the fraction \(\frac{1782}{1188}\):

• Both numbers are divisible by 10, resulting in \(\frac{1782}{1188}\).
• Both are even, divide by 2: \(\frac{891}{594}\).
• Sum of digits of 891 (8+9+1=18) and 594 (5+9+4=18) are divisible by 9, so both numbers are divisible by 9: \(\frac{891 \div 9}{594 \div 9} = \frac{99}{66}\).
• Both 99 and 66 are divisible by 33: \(\frac{99 \div 33}{66 \div 33} = \frac{3}{2}\).

So, we have:

\((1 + r)^2 = \frac{3}{2} = 1.5\)

This means the amount grows by a factor of 1.5 over every two years.

Calculating the Original Principal (Sum)

We know \((1 + r)^2 = 1.5\). We can use Equation 1, which is \(11880 = P(1 + r)^4\).

We can rewrite \((1 + r)^4\) as \(((1 + r)^2)^2\).

Substitute the value of \((1 + r)^2\):

\((1 + r)^4 = (1.5)^2 = 2.25\)

Now, substitute this back into Equation 1:

\(11880 = P \times 2.25\)

Solve for P:

\(P = \frac{11880}{2.25}\)

To make the division easier, we can multiply the numerator and denominator by 100 to remove the decimal:

\(P = \frac{11880 \times 100}{2.25 \times 100} = \frac{1188000}{225}\)

We can simplify this fraction:

• Both numbers end in 0 or 5, so they are divisible by 5: \(\frac{1188000 \div 5}{225 \div 5} = \frac{237600}{45}\).
• Again, divisible by 5: \(\frac{237600 \div 5}{45 \div 5} = \frac{47520}{9}\).
• Now, divide 47520 by 9: \(\frac{47520}{9} = 5280\).

So, the original principal sum (P) is Rs. 5,280.

Finding Half of the Sum

The question asks for half of the sum (half of the principal P).

Half of the sum = \(\frac{P}{2} = \frac{5280}{2}\)

Half of the sum = Rs. 2,640.

Conclusion

Half of the original sum of money is Rs. 2,640.

Revision Table: Key Concepts for Compound Interest

Additional Information on Compound Interest Calculations

When solving compound interest problems where amounts are given at different time intervals, dividing the later amount's equation by the earlier amount's equation is a common and effective technique. This method helps in finding the growth factor \((1+r)\) or a power of the growth factor, simplifying the process of finding the principal or the rate.

In this specific problem, the time difference is \(6 - 4 = 2\) years. Dividing the equations gave us \((1+r)^2\). If the time difference was 3 years, dividing would give \((1+r)^3\), and so on. The resulting power of \((1+r)\) corresponds to the difference in time periods.

Once \((1+r)\) or a power of \((1+r)\) is found, it can be substituted back into one of the original equations to solve for the principal (P). The rate (r) itself can also be calculated from \((1+r)\) if needed, although it wasn't required to find the principal in this problem.