If a certain sum becomes two times in 7 years at compound interest, then in how many years, it will become eight times ?

Options

A. 14
B. `21`
C. `28`
D. `35`

Answer

To solve the problem of how many years it will take for a certain sum to become eight times at compound interest, given that it becomes two times in 7 years, we can follow these steps:

### Step 1: Understand the Compound Interest Formula
The formula for the amount \( A \) in compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]
where:
- \( P \) is the principal amount (initial sum),
- \( R \) is the rate of interest,
- \( t \) is the time in years.

### Step 2: Set Up the First Condition
From the problem, we know that the sum becomes two times in 7 years. Therefore, we can set up the equation:
\[ 2P = P \left(1 + \frac{R}{100}\right)^7 \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 2 = \left(1 + \frac{R}{100}\right)^7 \]

### Step 3: Set Up the Second Condition
Now, we want to find out how long it will take for the sum to become eight times. We set up the equation:
\[ 8P = P \left(1 + \frac{R}{100}\right)^x \]
Again, dividing both sides by \( P \):
\[ 8 = \left(1 + \frac{R}{100}\right)^x \]

### Step 4: Express 8 in Terms of 2
We know that \( 8 \) can be expressed as \( 2^3 \). Therefore, we rewrite the equation:
\[ 2^3 = \left(1 + \frac{R}{100}\right)^x \]

### Step 5: Relate the Two Equations
From Step 2, we have:
\[ 2 = \left(1 + \frac{R}{100}\right)^7 \]
Now, we can substitute this into our equation for \( 8 \):
\[ (2)^3 = \left(1 + \frac{R}{100}\right)^x \]
This means:
\[ \left(\left(1 + \frac{R}{100}\right)^7\right)^3 = \left(1 + \frac{R}{100}\right)^x \]
Using the property of exponents, we can simplify this to:
\[ \left(1 + \frac{R}{100}\right)^{21} = \left(1 + \frac{R}{100}\right)^x \]

### Step 6: Equate the Exponents
Since the bases are the same, we can equate the exponents:
\[ 21 = x \]

### Conclusion
Thus, it will take **21 years** for the sum to become eight times.