At what rate per cent compound interest, does a sum of money become 1.44 times of itself in 2 years ?

Answer

To solve the problem of finding the rate of compound interest at which a sum of money becomes 1.44 times itself in 2 years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the principal amount (initial sum of money) be \( P \).
- According to the problem, the amount \( A \) after 2 years is \( 1.44P \).
- The time \( t \) is given as 2 years.

2. **Use the Compound Interest Formula**:
- The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
- Here, \( A \) is the final amount, \( P \) is the principal, \( r \) is the rate of interest per annum, and \( t \) is the time in years.

3. **Substitute the Known Values**:
- Substitute \( A = 1.44P \) and \( t = 2 \) into the formula:
\[ 1.44P = P \left(1 + \frac{r}{100}\right)^2 \]

4. **Cancel the Principal \( P \)**:
- Since \( P \) is not zero, we can divide both sides by \( P \):
\[ 1.44 = \left(1 + \frac{r}{100}\right)^2 \]

5. **Take the Square Root of Both Sides**:
- To solve for \( 1 + \frac{r}{100} \), take the square root:
\[ \sqrt{1.44} = 1 + \frac{r}{100} \]
- We know that \( \sqrt{1.44} = 1.2 \):
\[ 1.2 = 1 + \frac{r}{100} \]

6. **Isolate \( \frac{r}{100} \)**:
- Subtract 1 from both sides:
\[ 1.2 - 1 = \frac{r}{100} \]
\[ 0.2 = \frac{r}{100} \]

7. **Solve for \( r \)**:
- Multiply both sides by 100 to find \( r \):
\[ r = 0.2 \times 100 = 20 \]

8. **Final Answer**:
- The rate of compound interest is \( r = 20\% \).