A certain sum of money becomes 4 times in 12 years when invested at simple interest. In how many years will it become 10 times of itself at the same rate?

Options

A. 60
B. 48
C. 24
D. 36

Answer

To solve the problem, we need to determine how long it will take for a sum of money to become 10 times itself when invested at simple interest, given that it becomes 4 times in 12 years.

### Step-by-Step Solution:

1. **Understanding Simple Interest**:
The formula for simple interest is given by:
\[ A = P + SI \]
where \( A \) is the total amount after time \( t \), \( P \) is the principal amount (initial investment), and \( SI \) is the simple interest earned.

2. **Identifying the Initial Condition**:
We know that the sum of money becomes 4 times itself in 12 years. This means:
\[ A = 4P \]
Therefore, the simple interest earned in 12 years can be expressed as:
\[ SI = A - P = 4P - P = 3P \]

3. **Using the Simple Interest Formula**:
We can express the simple interest in terms of the principal, rate, and time:
\[ SI = \frac{P \times r \times t}{100} \]
where \( r \) is the rate of interest and \( t \) is the time in years. For our case:
\[ SI = \frac{P \times r \times 12}{100} \]
Setting the two expressions for simple interest equal gives:
\[ 3P = \frac{P \times r \times 12}{100} \]

4. **Solving for the Rate of Interest**:
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = \frac{r \times 12}{100} \]
Rearranging this gives:
\[ r = \frac{3 \times 100}{12} = 25 \]
So, the rate of interest \( r \) is 25%.

5. **Finding Time to Become 10 Times**:
Now we want to find out how long it will take for the sum to become 10 times itself:
\[ A = 10P \]
The simple interest in this case would be:
\[ SI = A - P = 10P - P = 9P \]
Using the simple interest formula again:
\[ 9P = \frac{P \times r \times t}{100} \]
Substituting \( r = 25 \):
\[ 9P = \frac{P \times 25 \times t}{100} \]
Dividing both sides by \( P \):
\[ 9 = \frac{25t}{100} \]
Rearranging gives:
\[ t = \frac{9 \times 100}{25} = 36 \]

### Final Answer:
Thus, it will take **36 years** for the sum of money to become 10 times itself at the same rate of interest.