A certain sum becomes 5 times in 3 years, at simple interest, then in how many years it will become 13 times?

Options

A. 6
B. 5
C. 9
D. 12

Answer

To solve the problem step by step, we will use the concept of simple interest.

### Step 1: Understand the given information
We know that a certain sum becomes 5 times in 3 years at simple interest. This means if the principal amount (P) is 100, then after 3 years, the amount (A) will be 500.

**Hint:** Identify the principal amount and the total amount after the specified time.

### Step 2: Calculate the interest earned in 3 years
If the principal is 100, and the amount after 3 years is 500, then the interest earned (I) in 3 years is:
\[ I = A - P = 500 - 100 = 400 \]

**Hint:** Use the formula for interest, which is the total amount minus the principal.

### Step 3: Calculate the annual interest
Since the interest earned in 3 years is 400, the annual interest (I) can be calculated as:
\[ \text{Annual Interest} = \frac{I}{\text{Number of years}} = \frac{400}{3} \approx 133.33 \]

**Hint:** Divide the total interest by the number of years to find the annual interest.

### Step 4: Determine the interest needed to become 13 times
To find out how many years it will take for the principal to become 13 times, we first calculate the amount when it becomes 13 times:
If the principal is 100, then 13 times the principal is:
\[ A = 13 \times 100 = 1300 \]
The interest needed to reach this amount is:
\[ I = A - P = 1300 - 100 = 1200 \]

**Hint:** Calculate the total amount when it becomes 13 times and find the interest needed.

### Step 5: Calculate the time required to earn the interest of 1200
Now, we know that the annual interest is approximately 133.33. To find the number of years (T) required to earn 1200 in interest, we use the formula:
\[ I = \text{Annual Interest} \times T \]
Rearranging gives us:
\[ T = \frac{I}{\text{Annual Interest}} = \frac{1200}{133.33} \approx 9 \]

**Hint:** Use the interest formula to find the time required to accumulate the desired interest.

### Conclusion
Thus, it will take approximately 9 years for the principal to become 13 times.

**Final Answer:** 9 years (Option C).