A sum of money, lent out at simple interest, doubles itself in 8 years. Find :
(i) the rate of interest.
(ii) In how many years will the sum become triple (three times) of itself at the same rate percent ?

Answer

To solve the problem step by step, we will break it down into two parts as given in the question.

### Part (i): Finding the Rate of Interest

1. **Understand the Problem**: We know that a sum of money doubles itself in 8 years. Let's denote the principal amount as \( P = x \). Therefore, the amount after 8 years will be \( A = 2x \).

2. **Calculate the Interest**: The interest earned over 8 years can be calculated as:
\[ \text{Interest} = \text{Amount} - \text{Principal} = A - P = 2x - x = x \]

3. **Use the Simple Interest Formula**: The formula for simple interest is:
\[ I = \frac{P \cdot r \cdot t}{100} \]
where \( I \) is the interest, \( P \) is the principal, \( r \) is the rate of interest, and \( t \) is the time in years.

4. **Substituting Known Values**: We know:
- \( I = x \)
- \( P = x \)
- \( t = 8 \) years

Substituting these values into the formula gives:
\[ x = \frac{x \cdot r \cdot 8}{100} \]

5. **Simplifying the Equation**: We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 1 = \frac{r \cdot 8}{100} \]

6. **Solving for \( r \)**: Rearranging the equation to find \( r \):
\[ r = \frac{100}{8} = 12.5\% \]

### Part (ii): Finding the Time to Triple the Amount

1. **Understand the New Scenario**: Now, we want to find out how long it will take for the principal \( P = x \) to become triple \( A = 3x \).

2. **Calculate the New Interest**: The interest earned when the amount becomes triple is:
\[ \text{Interest} = A - P = 3x - x = 2x \]

3. **Using the Simple Interest Formula Again**: We will use the same formula:
\[ I = \frac{P \cdot r \cdot t}{100} \]
where now:
- \( I = 2x \)
- \( P = x \)
- \( r = 12.5\% \)

4. **Substituting Known Values**: Plugging in the values:
\[ 2x = \frac{x \cdot 12.5 \cdot t}{100} \]

5. **Simplifying the Equation**: Cancel \( x \) from both sides:
\[ 2 = \frac{12.5 \cdot t}{100} \]

6. **Solving for \( t \)**: Rearranging gives:
\[ t = \frac{2 \cdot 100}{12.5} = \frac{200}{12.5} = 16 \text{ years} \]

### Final Answers:
(i) The rate of interest is **12.5% per annum**.
(ii) The time taken for the sum to become triple is **16 years**.