A sum of money (P) doubles in 10 years . In how many years is will treble at the same rate of simple interest ?

Answer

To solve the problem, we need to find out how many years it will take for a sum of money (P) to treble (become 3P) at the same rate of simple interest, given that it doubles (becomes 2P) in 10 years.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Principal (P)
- Amount after 10 years (A) = 2P
- Time (T) = 10 years

2. **Use the Simple Interest Formula:**
The formula for the amount in simple interest is:
\[ A = P + \text{SI} \]
where SI (Simple Interest) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Therefore, we can rewrite the amount as:
\[ A = P + \frac{P \times R \times T}{100} \]

3. **Substituting the Known Values:**
Substitute A = 2P and T = 10 years into the equation:
\[ 2P = P + \frac{P \times R \times 10}{100} \]

4. **Simplifying the Equation:**
Factor out P from both sides:
\[ 2P - P = \frac{P \times R \times 10}{100} \]
This simplifies to:
\[ P = \frac{P \times R \times 10}{100} \]

5. **Canceling P:**
Since P is not zero, we can divide both sides by P:
\[ 1 = \frac{R \times 10}{100} \]

6. **Solving for R (Rate of Interest):**
Multiply both sides by 100:
\[ 100 = R \times 10 \]
Thus, we find:
\[ R = 10\% \]

7. **Finding Time to Treble:**
Now, we need to find the time (T') it takes for the amount to become 3P. Using the same formula:
\[ A = P + \frac{P \times R \times T'}{100} \]
Substitute A = 3P:
\[ 3P = P + \frac{P \times 10 \times T'}{100} \]

8. **Simplifying Again:**
Factor out P:
\[ 3P - P = \frac{P \times 10 \times T'}{100} \]
This simplifies to:
\[ 2P = \frac{P \times 10 \times T'}{100} \]

9. **Canceling P Again:**
Divide both sides by P:
\[ 2 = \frac{10 \times T'}{100} \]

10. **Solving for T':**
Multiply both sides by 100:
\[ 200 = 10 \times T' \]
Thus:
\[ T' = \frac{200}{10} = 20 \text{ years} \]

### Final Answer:
It will take **20 years** for the sum of money to treble at the same rate of simple interest.