A certain amount double in 5 years, when invested at simple interest. In how many years will it become 8 times ?

Options

A. 35 years
B. 40 years
C. 30 years
D. 45 years

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the Problem
We are given that a certain amount doubles in 5 years at simple interest. We need to find out how many years it will take for the same amount to become 8 times.

### Step 2: Define the Variables
Let the principal amount (P) be \( x \). According to the problem, the amount (A) after 5 years is:
\[ A = 2x \]

### Step 3: Calculate the Simple Interest
The simple interest (SI) can be calculated as:
\[ SI = A - P \]
Substituting the values we have:
\[ SI = 2x - x = x \]

### Step 4: Use the Simple Interest Formula
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) is the principal
- \( R \) is the rate of interest
- \( T \) is the time in years

We know:
- \( SI = x \)
- \( P = x \)
- \( T = 5 \) years

Substituting these values into the formula gives:
\[ x = \frac{x \times R \times 5}{100} \]

### Step 5: Solve for the Rate of Interest (R)
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 1 = \frac{R \times 5}{100} \]
Multiplying both sides by 100:
\[ 100 = R \times 5 \]
Dividing both sides by 5:
\[ R = 20\% \]

### Step 6: Determine the Simple Interest Needed to Become 8 Times
Now, we need to find out how long it will take for the amount to become 8 times the principal. If the principal is \( x \), then the amount (A) should be:
\[ A = 8x \]

The simple interest needed to achieve this is:
\[ SI = A - P = 8x - x = 7x \]

### Step 7: Use the Simple Interest Formula Again
Using the simple interest formula again:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 7x = \frac{x \times 20 \times T}{100} \]

### Step 8: Solve for Time (T)
Cancel \( x \) from both sides:
\[ 7 = \frac{20T}{100} \]
Multiplying both sides by 100:
\[ 700 = 20T \]
Dividing both sides by 20:
\[ T = 35 \]

### Conclusion
Thus, it will take **35 years** for the amount to become 8 times the principal.