At simple interest, a sum becomes 3 times in 20 yr. Find the time, in which the sum will be double at the same rate of interest.

Options

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

Answer

To solve the problem step by step, we will use the concept of simple interest and the relationship between the time taken for a sum to become a certain multiple of itself.

### Step-by-Step Solution:

1. **Understand the Problem**: We know that a sum of money becomes three times itself in 20 years at simple interest. We need to find out how long it will take for the same sum to double at the same rate of interest.

2. **Define Variables**:
- Let the original sum be \( P \).
- After 20 years, the amount becomes \( 3P \).
- We need to find the time \( t_2 \) when the amount becomes \( 2P \).

3. **Use the Simple Interest Formula**:
The formula for the amount \( A \) in simple interest is given by:
\[ A = P + SI \]
where \( SI \) is the simple interest calculated as:
\[ SI = P \times r \times t \]
Here, \( r \) is the rate of interest and \( t \) is the time in years.

4. **Calculate the Interest Rate**:
From the information given, we know:
\[ 3P = P + P \times r \times 20 \]
Simplifying this, we get:
\[ 3P - P = P \times r \times 20 \]
\[ 2P = P \times r \times 20 \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 2 = r \times 20 \]
Therefore, the rate of interest \( r \) is:
\[ r = \frac{2}{20} = \frac{1}{10} = 0.1 \text{ or } 10\% \]

5. **Find Time for the Sum to Double**:
Now, we need to find the time \( t_2 \) when the amount becomes \( 2P \):
\[ 2P = P + P \times r \times t_2 \]
Simplifying this gives:
\[ 2P - P = P \times r \times t_2 \]
\[ P = P \times r \times t_2 \]
Dividing both sides by \( P \):
\[ 1 = r \times t_2 \]
Substituting \( r = 0.1 \):
\[ 1 = 0.1 \times t_2 \]
Therefore:
\[ t_2 = \frac{1}{0.1} = 10 \text{ years} \]

6. **Conclusion**: The time in which the sum will double at the same rate of interest is **10 years**.