A certain sum of money becomes three times of itself in 20 yr at simple interest. In how many years, does it become double of itself at the same rate of simple interest ?

Options

A. 8 yr
B. 10 yr
C. 12 yr
D. 14 yr

Answer

To solve the problem, we need to determine how many years it takes for a certain sum of money to double itself at the same rate of simple interest, given that it becomes three times itself in 20 years.

### Step-by-Step Solution:

1. **Understand the Problem**:
- We know that a principal amount (let's denote it as \( P \)) becomes three times itself in 20 years. Therefore, the amount after 20 years is \( 3P \).

2. **Set Up the Simple Interest Formula**:
- The formula for the amount \( A \) in simple interest is:
\[ A = P + \text{SI} \]
- Where SI (Simple Interest) is given by:
\[ \text{SI} = P \times r \times t / 100 \]
- Here, \( r \) is the rate of interest and \( t \) is the time in years.

3. **Calculate the Simple Interest for 20 Years**:
- From the problem, we have:
\[ 3P = P + \text{SI} \]
- Therefore:
\[ 3P - P = \text{SI} \]
- This simplifies to:
\[ 2P = \text{SI} \]

4. **Substituting SI**:
- Substitute the SI formula into the equation:
\[ 2P = P \times r \times 20 / 100 \]

5. **Canceling \( P \)**:
- We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ 2 = r \times 20 / 100 \]

6. **Solving for \( r \)**:
- Rearranging gives:
\[ r = \frac{2 \times 100}{20} = 10\% \]
- So, the rate of interest is \( 10\% \).

7. **Determine Time to Double the Principal**:
- Now, we want to find out how long it takes for the amount to double, which means:
\[ 2P = P + \text{SI} \]
- This simplifies to:
\[ 2P - P = \text{SI} \]
- Thus:
\[ P = \text{SI} \]

8. **Substituting SI Again**:
- Using the SI formula:
\[ P = P \times r \times t / 100 \]
- Substitute \( r = 10\% \):
\[ P = P \times 10 \times t / 100 \]

9. **Canceling \( P \)**:
- Cancel \( P \) from both sides:
\[ 1 = 10t / 100 \]

10. **Solving for \( t \)**:
- Rearranging gives:
\[ t = \frac{100}{10} = 10 \text{ years} \]

### Final Answer:
The sum of money will become double of itself in **10 years**.