In how many years will a certain sum becomes five times itself at 20% per annum simple interest?

Options

A. 20
B. 18
C. 16
D. 12

Explanation

To solve the problem of how many years it will take for a certain sum to become five times itself at a simple interest rate of 20% per annum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find out how many years (t) it will take for a principal amount (P) to grow to five times itself (5P) at a simple interest rate of 20% per annum.

2. **Define the Variables**:
- Let the principal amount (P) be represented as \( x \).
- The amount after time \( t \) will be \( 5x \).

3. **Use the Simple Interest Formula**:
The formula for the amount (A) in simple interest is given by:
\[ A = P + I \]
where \( I \) is the simple interest. The simple interest can be calculated as:
\[ I = \frac{P \times R \times T}{100} \]
Here, \( R \) is the rate of interest (20%), and \( T \) is the time in years.

4. **Set Up the Equation**:
Since the amount becomes five times the principal, we can write:
\[ 5x = x + I \]
Substituting the formula for simple interest:
\[ 5x = x + \frac{x \times 20 \times t}{100} \]

5. **Simplify the Equation**:
Rearranging gives:
\[ 5x - x = \frac{x \times 20 \times t}{100} \]
This simplifies to:
\[ 4x = \frac{x \times 20 \times t}{100} \]

6. **Eliminate x**:
Since \( x \) is common on both sides, we can divide both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 4 = \frac{20t}{100} \]

7. **Solve for t**:
Multiply both sides by 100 to eliminate the fraction:
\[ 400 = 20t \]
Now divide both sides by 20:
\[ t = \frac{400}{20} = 20 \]

8. **Conclusion**:
The time it will take for the sum to become five times itself at a simple interest rate of 20% per annum is **20 years**.