A sum of money placed at com pound interest doubles itself in 4 years. In how many years will it amount to four times itself ?

Options

A. 12 years
B. 13 years
C. 8 years
D. 16 years

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 sum of money doubles in 4 years under compound interest. We need to find out how many years it will take for the same sum to quadruple (i.e., become four times itself).

2. **Let the Principal Amount be \( P \)**:
For simplicity, let's assume the principal amount \( P = x \). Therefore, the amount after 4 years when it doubles will be \( 2x \).

3. **Using the Compound Interest Formula**:
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) = Amount after \( n \) years
- \( P \) = Principal amount
- \( r \) = Rate of interest
- \( n \) = Number of years

4. **Setting Up the Equation for Doubling**:
From the problem, we have:
\[ 2x = x \left(1 + \frac{r}{100}\right)^4 \]
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 2 = \left(1 + \frac{r}{100}\right)^4 \]

5. **Taking the Fourth Root**:
To find \( 1 + \frac{r}{100} \), we take the fourth root of both sides:
\[ 1 + \frac{r}{100} = 2^{1/4} \]

6. **Finding the Rate of Interest**:
We know that \( 2^{1/4} = \sqrt{2} \), so:
\[ 1 + \frac{r}{100} = \sqrt{2} \]
Thus:
\[ \frac{r}{100} = \sqrt{2} - 1 \]
Therefore:
\[ r = 100(\sqrt{2} - 1) \]

7. **Setting Up the Equation for Quadrupling**:
Now we need to find out how many years \( n \) it will take for the amount to become \( 4x \):
\[ 4x = x \left(1 + \frac{r}{100}\right)^n \]
Dividing both sides by \( x \):
\[ 4 = \left(1 + \frac{r}{100}\right)^n \]

8. **Substituting the Value of \( 1 + \frac{r}{100} \)**:
We already found that \( 1 + \frac{r}{100} = \sqrt{2} \):
\[ 4 = (\sqrt{2})^n \]

9. **Expressing 4 as a Power of 2**:
We know that \( 4 = 2^2 \) and \( \sqrt{2} = 2^{1/2} \):
\[ 2^2 = (2^{1/2})^n \]

10. **Equating the Exponents**:
From the equation, we have:
\[ 2 = \frac{n}{2} \]
Therefore:
\[ n = 4 \]

11. **Finding the Total Years**:
Since we already know it takes 4 years to double, and it takes another 4 years to double the doubled amount (to quadruple), we add the two periods:
\[ \text{Total years} = 4 + 4 = 8 \text{ years} \]

### Final Answer:
The sum of money will amount to four times itself in **8 years**.