Understanding the Compound Interest Problem

This problem asks us to calculate the compound interest earned specifically during the fifth year, given the total amount accumulated after 4 years and 6 years at a certain compound interest rate.

In compound interest, the interest earned each year is added to the principal for calculating the interest in the next year. This means the amount grows exponentially. If the interest rate is constant, the ratio of amounts at two different times depends only on the time difference and the interest rate.

Setting up the Equations

Let the principal amount be $P$.

Let the annual compound interest rate be $r$ (as a decimal).

The amount after $t$ years is given by the formula: $A(t) = P(1+r)^t$.

We are given:

• Amount after 4 years, $A(4) = 64800$
• Amount after 6 years, $A(6) = 93312$

Using the formula, we can write:

• $P(1+r)^4 = 64800$ (Equation 1)
• $P(1+r)^6 = 93312$ (Equation 2)

Calculating the Annual Growth Factor

To find the growth factor $(1+r)$, we can divide Equation 2 by Equation 1:

$\frac{P(1+r)^6}{P(1+r)^4} = \frac{93312}{64800}$

This simplifies to:

$(1+r)^{6-4} = \frac{93312}{64800}$

$(1+r)^2 = \frac{93312}{64800}$

Now, let's calculate the value of the ratio:

$\frac{93312}{64800} = 1.44$

So, $(1+r)^2 = 1.44$.

Taking the square root of both sides:

$1+r = \sqrt{1.44}$

$1+r = 1.2$

The annual growth factor is 1.2. This means the amount increases by 20% each year ($r = 1.2 - 1 = 0.2$).

Calculating the Amount at the End of the Fifth Year

The amount at the end of the fourth year is $A(4) = 64800$.

The amount at the end of the fifth year, $A(5)$, is the amount at the end of the fourth year multiplied by the annual growth factor $(1+r)$.

$A(5) = A(4) \times (1+r)$

$A(5) = 64800 \times 1.2$

Let's calculate $A(5)$:

$64800 \times 1.2 = 64800 \times \frac{12}{10} = 6480 \times 12$

$6480 \times 12 = 77760$

So, the amount at the end of the fifth year is Rs. 77760.

Calculating the Compound Interest Earned in the Fifth Year

The compound interest earned in the fifth year is the difference between the amount at the end of the fifth year and the amount at the end of the fourth year.

Interest in 5th year = $A(5) - A(4)$

Interest in 5th year = $77760 - 64800$

Calculating the difference:

$77760 - 64800 = 12960$

The compound interest earned in the fifth year is Rs. 12960.

Summary of Steps

1. Used the given amounts at two different times (4 years and 6 years) to find the growth factor over the time difference (2 years).
2. Calculated the annual growth factor by taking the appropriate root (square root in this case).
3. Used the annual growth factor to calculate the amount at the end of the fifth year from the amount at the end of the fourth year.
4. Subtracted the amount at the end of the fourth year from the amount at the end of the fifth year to find the interest earned during the fifth year.

Revision Table: Compound Interest Concepts

Additional Information: Compound Growth

The power of compound interest lies in the fact that the base for interest calculation increases each period. This leads to faster growth compared to simple interest where interest is only calculated on the initial principal.

If you know the amount at time $t_1$, say $A(t_1)$, and the amount at time $t_2$, say $A(t_2)$, where $t_2 > t_1$, you can find the growth factor for the period $(t_2 - t_1)$ years by calculating $\frac{A(t_2)}{A(t_1)}$. This ratio is equal to $(1+r)^{t_2-t_1}$. From this, you can determine the annual growth factor $(1+r)$ or the annual interest rate $r$.

For example, in this problem, the time difference is $6 - 4 = 2$ years. The ratio of amounts is $\frac{93312}{64800} = 1.44$. So, the growth factor over 2 years is 1.44. The annual growth factor is $\sqrt{1.44} = 1.2$.

The interest earned in any specific year (say, the $n$-th year) is the difference between the amount at the end of year $n$ and the amount at the end of year $n-1$.

Interest in $n$-th year = $A(n) - A(n-1)$

$A(n) = A(n-1) \times (1+r)$

So, Interest in $n$-th year = $A(n-1) \times (1+r) - A(n-1) = A(n-1) \times ((1+r) - 1) = A(n-1) \times r$.

Using this formula for the 5th year:

Interest in 5th year = $A(4) \times r$

Since $A(4) = 64800$ and $r = 0.2$ (or 20%),

Interest in 5th year = $64800 \times 0.2 = 64800 \times \frac{2}{10} = 6480 \times 2 = 12960$.

Both methods give the same result, confirming the calculation.