A sum of money invested at a certain rate of interest doubles itself in `8` years. In how much time will it treble itself at the same interest rate ?

Answer

To solve the problem step by step, we will follow the process of calculating the rate of interest first and then use that rate to find out how long it will take for the initial amount to triple itself.

### Step 1: Understand the problem
We know that a sum of money doubles itself in 8 years. We need to find out how long it will take for the same amount to triple itself at the same rate of interest.

### Step 2: Define the variables
Let:
- Principal amount (P) = x (initial amount)
- After 8 years, the amount becomes double, which is 2x.

### Step 3: Calculate the simple interest
Using the formula for simple interest:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) = Simple Interest
- \( P \) = Principal
- \( R \) = Rate of interest
- \( T \) = Time in years

From the problem, we know:
- After 8 years, the interest earned is \( SI = 2x - x = x \) (since the amount doubles).

Substituting the values into the formula:
\[ x = \frac{x \times R \times 8}{100} \]

### Step 4: Simplify the equation
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 1 = \frac{R \times 8}{100} \]

### Step 5: Solve for R
Rearranging the equation gives:
\[ R = \frac{100}{8} = 12.5\% \]

### Step 6: Find the time to triple the amount
Now we need to find out how long it will take for the amount to become three times the principal (3x). The interest earned in this case will be:
\[ SI = 3x - x = 2x \]

Using the same formula for simple interest:
\[ 2x = \frac{x \times 12.5 \times T}{100} \]

### Step 7: Simplify the equation again
Cancel \( x \) from both sides:
\[ 2 = \frac{12.5 \times T}{100} \]

### Step 8: Solve for T
Rearranging gives:
\[ T = \frac{2 \times 100}{12.5} \]
\[ T = \frac{200}{12.5} \]
\[ T = 16 \text{ years} \]

### Conclusion
Thus, it will take **16 years** for the initial amount to triple itself at the same rate of interest.

---