A sum was lent at a certain rate of compound interest. It amounted to ₹15,600 at the end of 4 years and to ₹18,720 at the end of 8 years. Find the sum (in).

The problem involves compound interest, and we are given the amount after 4 years and after 8 years. Let's denote the following:

- \( A_4 = 15,600 \) (the amount after 4 years),
- \( A_8 = 18,720 \) (the amount after 8 years),
- \( P \) is the principal (the sum lent),
- \( r \) is the rate of compound interest per annum.

### Step-by-step solution:

#### Step 1: Using the formula for compound interest

The formula for the compound interest amount after \( t \) years is:
\[
A_t = P \times \left(1 + \frac{r}{100}\right)^t
\]

For the first scenario:
\[
A_4 = P \times \left(1 + \frac{r}{100}\right)^4
\]
Substituting \( A_4 = 15,600 \):
\[
15,600 = P \times \left(1 + \frac{r}{100}\right)^4 \tag{1}
\]

For the second scenario:
\[
A_8 = P \times \left(1 + \frac{r}{100}\right)^8
\]
Substituting \( A_8 = 18,720 \):
\[
18,720 = P \times \left(1 + \frac{r}{100}\right)^8 \tag{2}
\]

#### Step 2: Dividing the equations to eliminate \( P \)

Dividing equation (2) by equation (1):
\[
\frac{18,720}{15,600} = \frac{P \times \left(1 + \frac{r}{100}\right)^8}{P \times \left(1 + \frac{r}{100}\right)^4}
\]
Simplifying:
\[
\frac{18,720}{15,600} = \left(1 + \frac{r}{100}\right)^4
\]
\[
1.2 = \left(1 + \frac{r}{100}\right)^4
\]

#### Step 3: Solving for \( r \)

Taking the fourth root of both sides:
\[
1 + \frac{r}{100} = \sqrt[4]{1.2}
\]
Now, calculate the fourth root of 1.2:
\[
1 + \frac{r}{100} \approx 1.046
\]
\[
\frac{r}{100} \approx 0.046
\]
\[
r \approx 4.6\%
\]

#### Step 4: Finding the principal \( P \)

Now, using the value of \( r \), we substitute back into equation (1) to find \( P \).

\[
15,600 = P \times \left(1 + \frac{r}{100}\right)^4
\]
\[
15,600 = P \times (1.046)^4
\]
Calculating \( (1.046)^4 \):
\[
15,600 = P \times 1.2
\]
\[
P = \frac{15,600}{1.2}
\]
\[
P = 13,000
\]

Thus, the principal sum lent was **₹13,000**.

Would you like further details or have any questions? 

Here are five related questions:
1. How does compound interest differ from simple interest?
2. What is the formula for compound interest if the interest is compounded monthly?
3. How would the principal change if the rate was 5% instead of 4.6%?
4. What happens to the amount if the interest is compounded continuously?
5. How can you calculate the interest rate if only the initial and final amounts are given?

**Tip:** When solving compound interest problems, using logarithms can simplify finding time or rates.

Learn how to find the principal amount lent at compound interest, given the amounts ₹15,600 after 4 years and ₹18,720 after 8 years. This step-by-step solution uses the compound interest formula and is suitable for Grades 9-12.

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