A certain sum becomes 4 times in 7 years, at simple interest, then in how many years it will become 16 times?

Options

A. 16
B. 21
C. 28
D. 35

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We need to find out how many years it will take for a certain sum of money to become 16 times its original amount at simple interest, given that it becomes 4 times in 7 years.

### Step 2: Define the Principal
Let's assume the principal amount (P) is 100. Therefore, if it becomes 4 times in 7 years, the amount (A) after 7 years is:
\[ A = 4 \times P = 4 \times 100 = 400 \]

### Step 3: Calculate the Simple Interest
The simple interest (SI) earned in 7 years can be calculated as:
\[ SI = A - P = 400 - 100 = 300 \]

### Step 4: Use the Simple Interest Formula
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest
- P = Principal
- R = Rate of interest per annum
- T = Time in years

Using the values we have:
\[ 300 = \frac{100 \times R \times 7}{100} \]

### Step 5: Solve for the Rate (R)
From the equation, we can simplify:
\[ 300 = R \times 7 \]
\[ R = \frac{300}{7} \]

### Step 6: Determine the Simple Interest for 16 Times the Principal
Now we need to find out how long it will take for the principal to become 16 times:
\[ A = 16 \times P = 16 \times 100 = 1600 \]
The simple interest required to reach this amount is:
\[ SI = A - P = 1600 - 100 = 1500 \]

### Step 7: Set Up the Equation for Time (T)
Using the simple interest formula again:
\[ 1500 = \frac{100 \times \left(\frac{300}{7}\right) \times T}{100} \]

### Step 8: Simplify and Solve for T
This simplifies to:
\[ 1500 = \frac{300}{7} \times T \]
To find T, we rearrange the equation:
\[ T = \frac{1500 \times 7}{300} \]

### Step 9: Calculate T
\[ T = \frac{10500}{300} = 35 \]

### Conclusion
Thus, it will take **35 years** for the principal to become 16 times at simple interest.