A certain sum of money becomes double of itself in 15 years at a rate of simple interest. In how many years will it become 5 times of itself at the same rate of simple interest?

Options

A. 45
B. `37.5`
C. 75
D. 60

Answer

To solve the problem step by step, we will follow the given information and apply the formula for simple interest.

### Step 1: Understand the Problem
We know that a certain sum of money doubles in 15 years at a rate of simple interest. We need to find out how many years it will take for the same sum to become 5 times itself at the same rate of interest.

### Step 2: Define the Variables
Let the principal amount be \( P \). According to the problem:
- The amount after 15 years when it doubles is \( 2P \).
- The simple interest (SI) earned in 15 years is \( 2P - P = P \).

### Step 3: Use the Simple Interest Formula
The formula for simple interest is:
\[ SI = \frac{P \cdot R \cdot T}{100} \]
Where:
- \( SI \) is the simple interest,
- \( P \) is the principal,
- \( R \) is the rate of interest,
- \( T \) is the time in years.

From the information given, we can substitute the values:
\[ P = \frac{P \cdot R \cdot 15}{100} \]
Since \( SI = P \), we have:
\[ P = \frac{P \cdot R \cdot 15}{100} \]

### Step 4: Simplify the Equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ 1 = \frac{R \cdot 15}{100} \]
Now, multiplying both sides by 100:
\[ 100 = R \cdot 15 \]
Dividing both sides by 15 gives:
\[ R = \frac{100}{15} = \frac{20}{3} \text{ percent} \]

### Step 5: Calculate Time for 5 Times the Principal
Now, we need to find out how long it will take for the amount to become 5 times the principal:
- The amount when it becomes 5 times is \( 5P \).
- The simple interest earned in this case is \( 5P - P = 4P \).

Using the simple interest formula again:
\[ 4P = \frac{P \cdot R \cdot T}{100} \]
Substituting \( R = \frac{20}{3} \):
\[ 4P = \frac{P \cdot \frac{20}{3} \cdot T}{100} \]
Cancelling \( P \) from both sides:
\[ 4 = \frac{\frac{20}{3} \cdot T}{100} \]

### Step 6: Solve for T
Multiplying both sides by 100:
\[ 400 = \frac{20}{3} \cdot T \]
Now, multiplying both sides by \( \frac{3}{20} \):
\[ T = \frac{400 \cdot 3}{20} \]
Simplifying:
\[ T = \frac{1200}{20} = 60 \text{ years} \]

### Conclusion
Thus, it will take **60 years** for the sum of money to become 5 times itself at the same rate of simple interest.