A certain sum becomes 3 times itself in 6 years at simple interest. In how many years will it become 9 times itself ?

Options

A. 18
B. 20
C. 24
D. 22

Explanation

To solve the problem step by step, we need to find out how long it will take for a certain sum of money (let's call it P) to become 9 times itself at simple interest, given that it becomes 3 times itself in 6 years.

### Step 1: Understand the problem
We know that the amount becomes 3 times the principal in 6 years. This means:
- Amount (A) = 3P
- Principal (P) = P
- Time (T) = 6 years

### Step 2: Calculate the interest earned in 6 years
The interest earned (I) can be calculated as:
\[ I = A - P \]
Substituting the values we have:
\[ I = 3P - P = 2P \]

### Step 3: Use the formula for simple interest
The formula for simple interest is:
\[ I = \frac{P \times R \times T}{100} \]
Where:
- I = Interest
- P = Principal
- R = Rate of interest
- T = Time in years

From the previous step, we know:
- I = 2P
- P = P
- T = 6 years

Substituting these values into the formula:
\[ 2P = \frac{P \times R \times 6}{100} \]

### Step 4: Simplify the equation
We can cancel P from both sides (assuming P ≠ 0):
\[ 2 = \frac{R \times 6}{100} \]

### Step 5: Solve for the rate of interest (R)
Multiplying both sides by 100:
\[ 200 = R \times 6 \]
Now, divide by 6:
\[ R = \frac{200}{6} = \frac{100}{3} \]

### Step 6: Calculate the interest needed to become 9 times the principal
Now we need to find out how long it will take for the amount to become 9 times the principal:
- New Amount (A) = 9P
- Interest (I) = A - P = 9P - P = 8P

### Step 7: Use the simple interest formula again
Using the same formula for simple interest:
\[ I = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 8P = \frac{P \times \frac{100}{3} \times T}{100} \]

### Step 8: Simplify the equation
Again, we can cancel P from both sides:
\[ 8 = \frac{100}{3} \times T \]

### Step 9: Solve for T
Multiplying both sides by 3:
\[ 24 = 100T \]
Now, divide by 100:
\[ T = \frac{24}{100} = 0.24 \text{ years} \]

### Step 10: Convert to years
Since we need the time in years, we can see that:
\[ T = 24 \text{ years} \]

Thus, it will take **24 years** for the sum to become 9 times itself.

### Summary
The answer is that it will take **24 years** for the principal to become 9 times itself at simple interest.