At what rate per cent per annum simple interest will a sum treble itself in 16 years?

Answer

To solve the problem of finding the rate per cent per annum simple interest at which a sum will treble itself in 16 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- We need to find the rate of simple interest (R) at which a principal amount (P) will become three times its original value (3P) in 16 years.

2. **Define the Variables**:
- Let the principal amount (P) be \( x \).
- Therefore, the amount after 16 years will be \( 3x \) (since it treble itself).

3. **Calculate the Simple Interest (SI)**:
- The formula for the amount (A) in terms of principal (P) and simple interest (SI) is:
\[ A = P + SI \]
- Here, we have:
\[ 3x = x + SI \]
- Rearranging gives:
\[ SI = 3x - x = 2x \]

4. **Use the Simple Interest Formula**:
- The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
- Substituting the known values:
- \( SI = 2x \)
- \( P = x \)
- \( T = 16 \) years
- We have:
\[ 2x = \frac{x \times R \times 16}{100} \]

5. **Simplify the Equation**:
- To eliminate \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 2 = \frac{R \times 16}{100} \]
- Cross-multiplying gives:
\[ 2 \times 100 = R \times 16 \]
- This simplifies to:
\[ 200 = 16R \]

6. **Solve for R**:
- Divide both sides by 16:
\[ R = \frac{200}{16} = 12.5 \]

7. **Conclusion**:
- The rate of simple interest required for the sum to treble in 16 years is \( 12.5\% \) per annum.