If a certain sum becomes 2 times in 7 years at compound interest, then in how many years, it will become 8 times?

Options

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

Answer

To solve the problem step by step, we will use the concept of compound interest.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that a certain sum (let's denote it as P) becomes 2 times itself in 7 years at compound interest. We need to find out how many years it will take for the same sum to become 8 times itself.

2. **Setting Up the First Equation**:
The amount (A) after time (t) at compound interest is given by the formula:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]
Here, when the amount becomes 2P in 7 years:
\[ 2P = P \left(1 + \frac{R}{100}\right)^7 \]
Dividing both sides by P (assuming P ≠ 0):
\[ 2 = \left(1 + \frac{R}{100}\right)^7 \quad \text{(Equation 1)} \]

3. **Setting Up the Second Equation**:
Now, we want to find out when the amount becomes 8P:
\[ 8P = P \left(1 + \frac{R}{100}\right)^x \]
Again, dividing both sides by P:
\[ 8 = \left(1 + \frac{R}{100}\right)^x \quad \text{(Equation 2)} \]

4. **Expressing 8 in terms of 2**:
We can express 8 as \(2^3\):
\[ 8 = 2^3 \]
From Equation 1, we know that \(2 = \left(1 + \frac{R}{100}\right)^7\). Therefore, we can rewrite 8 as:
\[ 2^3 = \left(\left(1 + \frac{R}{100}\right)^7\right)^3 \]
This implies:
\[ 8 = \left(1 + \frac{R}{100}\right)^{21} \quad \text{(using the property of exponents)} \]

5. **Equating the Two Expressions**:
Now we have:
\[ \left(1 + \frac{R}{100}\right)^x = \left(1 + \frac{R}{100}\right)^{21} \]
Since the bases are the same, we can equate the exponents:
\[ x = 21 \]

6. **Conclusion**:
Therefore, it will take 21 years for the sum to become 8 times itself at the same rate of compound interest.

### Final Answer:
The sum will become 8 times in **21 years**.