A certain amount becomes 3 times in 4 years on simple interest. In what time it will become 7 times?

Options

A. 8 years
B. 12 years
C. 15 years
D. None of these

Answer

To solve the problem step by step, we need to find out how long it will take for a certain amount to become 7 times its original value when it has been established that it becomes 3 times in 4 years under simple interest.

### Step 1: Understand the relationship between principal, interest, and amount.
The total amount (A) after a certain time can be expressed as:
\[ A = P + SI \]
where \( P \) is the principal and \( SI \) is the simple interest.

### Step 2: Define the principal and the simple interest.
Let the principal amount be \( P = x \). According to the problem, the amount becomes 3 times the principal in 4 years. Therefore:
\[ A = 3x \]
The simple interest (SI) can be calculated as:
\[ SI = A - P = 3x - x = 2x \]

### Step 3: Use the simple interest formula.
The formula for simple interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where \( R \) is the rate of interest and \( T \) is the time in years. From the previous step, we know that:
\[ SI = 2x \]
Substituting into the formula, we have:
\[ 2x = \frac{x \times R \times 4}{100} \]

### Step 4: Simplify the equation.
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 2 = \frac{R \times 4}{100} \]
Multiplying both sides by 100 gives:
\[ 200 = 4R \]
Now, divide by 4:
\[ R = 50 \]

### Step 5: Find the time to become 7 times the principal.
Now we need to find out how long it will take for the amount to become 7 times the principal. So, we have:
\[ A = 7x \]
The simple interest in this case will be:
\[ SI = A - P = 7x - x = 6x \]

### Step 6: Set up the equation for the new simple interest.
Using the simple interest formula again:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 6x = \frac{x \times 50 \times T}{100} \]

### Step 7: Simplify the equation to find T.
Cancel \( x \) from both sides:
\[ 6 = \frac{50T}{100} \]
Multiplying both sides by 100 gives:
\[ 600 = 50T \]
Now, divide by 50:
\[ T = 12 \]

### Conclusion:
Therefore, it will take **12 years** for the amount to become 7 times the principal.

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