The compound interest on a certain sum for 8 years is Rs.400 and for 16 years is Rs. 1300. Find the amount after 20 years.

To solve this problem, we need to apply the concept of compound interest. Here's a breakdown of the approach:

Let:
- \( P \) be the principal (the initial sum),
- \( r \) be the rate of interest per annum,
- \( A(t) \) be the amount after \( t \) years.

The formula for compound interest is:

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

### Step 1: Set up equations for 8 and 16 years
We are given two conditions for the compound interest at different time periods:

1. **For 8 years**: The compound interest is Rs. 400.
   - The amount after 8 years is \( A(8) = P(1 + r)^8 \).
   - The compound interest is \( A(8) - P = 400 \), so:
     \[
     P(1 + r)^8 - P = 400
     \]
     This simplifies to:
     \[
     P[(1 + r)^8 - 1] = 400
     \]
     (Equation 1)

2. **For 16 years**: The compound interest is Rs. 1300.
   - The amount after 16 years is \( A(16) = P(1 + r)^{16} \).
   - The compound interest is \( A(16) - P = 1300 \), so:
     \[
     P(1 + r)^{16} - P = 1300
     \]
     This simplifies to:
     \[
     P[(1 + r)^{16} - 1] = 1300
     \]
     (Equation 2)

### Step 2: Solve the system of equations
We now have two equations:

1. \( P[(1 + r)^8 - 1] = 400 \)
2. \( P[(1 + r)^{16} - 1] = 1300 \)

Dividing the second equation by the first, we get:

\[
\frac{(1 + r)^{16} - 1}{(1 + r)^8 - 1} = \frac{1300}{400} = 3.25
\]

Let \( x = (1 + r)^8 \). Then we have:

\[
\frac{x^2 - 1}{x - 1} = 3.25
\]

Multiplying both sides by \( (x - 1) \), we get:

\[
x^2 - 1 = 3.25(x - 1)
\]

Expanding both sides:

\[
x^2 - 1 = 3.25x - 3.25
\]

Rearranging the equation:

\[
x^2 - 3.25x + 2.25 = 0
\]

This is a quadratic equation in \( x \). Solving it using the quadratic formula:

\[
x = \frac{-(-3.25) \pm \sqrt{(-3.25)^2 - 4(1)(2.25)}}{2(1)}
\]

\[
x = \frac{3.25 \pm \sqrt{10.5625 - 9}}{2}
\]

\[
x = \frac{3.25 \pm \sqrt{1.5625}}{2}
\]

\[
x = \frac{3.25 \pm 1.25}{2}
\]

So, \( x = \frac{3.25 + 1.25}{2} = 2.25 \) or \( x = \frac{3.25 - 1.25}{2} = 1 \).

Since \( x = (1 + r)^8 \) cannot be 1, we take \( x = 2.25 \).

Thus, \( (1 + r)^8 = 2.25 \), so \( 1 + r = \sqrt[8]{2.25} \).

### Step 3: Find the amount after 20 years
Now, we want to find the amount after 20 years, which is \( A(20) = P(1 + r)^{20} \).

We know \( (1 + r)^8 = 2.25 \), so:

\[
(1 + r)^{20} = (1 + r)^8 \times (1 + r)^8 \times (1 + r)^4 = 2.25 \times 2.25 \times (1 + r)^4
\]

Let me calculate \( (1 + r)^4 \), and then find the final amount.We have calculated that \( (1 + r)^{20} = 7.59375 \).

### Step 4: Use the value of \( (1 + r)^{20} \) to find the amount after 20 years
The formula for the amount after 20 years is:

\[
A(20) = P(1 + r)^{20}
\]

From Equation 1, we have:

\[
P[(1 + r)^8 - 1] = 400 \implies P(2.25 - 1) = 400 \implies P(1.25) = 400
\]

Thus, \( P = \frac{400}{1.25} = 320 \).

Now, the amount after 20 years is:

\[
A(20) = 320 \times 7.59375 = 2429.2
\]

### Final Answer:
The amount after 20 years is Rs. 2429.20.

Would you like more details on any step?

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**Related Questions:**
1. What would be the compound interest after 12 years?
2. How does the formula for compound interest change for different compounding periods (e.g., monthly)?
3. What is the effective annual rate if the interest is compounded annually?
4. Can we calculate the total interest

Solve a compound interest problem where the interest after 8 years is Rs. 400 and after 16 years is Rs. 1300. Learn how to calculate the amount after 20 years using compound interest formulas and algebraic techniques. Suitable for students in Grades 10-12.

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