A sum of money invested at compound interest doubles itself in six years. In how many years will it become 64 times itself at the same rate of compound interest?

Options

A. 30
B. 36
C. 42
D. 48

Explanation

To solve the problem step-by-step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the Problem
We know that a sum of money doubles itself in 6 years at compound interest. We need to find out how many years it will take for the same sum to become 64 times itself.

### Step 2: Set Up the Initial Conditions
Let's assume the principal amount (P) is 100 units (this could be in rupees, dollars, etc.).

- After 6 years, the amount (A) becomes:
\[ A = 2P = 200 \text{ units} \]

### Step 3: Use the Compound Interest Formula
The formula for the amount in compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( A \) = final amount
- \( P \) = principal amount
- \( R \) = rate of interest
- \( T \) = time in years

Substituting the values we have:
\[ 200 = 100 \left(1 + \frac{R}{100}\right)^6 \]

### Step 4: Simplify the Equation
Dividing both sides by 100:
\[ 2 = \left(1 + \frac{R}{100}\right)^6 \]

### Step 5: Take the Sixth Root
To isolate \( 1 + \frac{R}{100} \), we take the sixth root of both sides:
\[ 1 + \frac{R}{100} = 2^{\frac{1}{6}} \]

### Step 6: Find the Rate of Interest
Now, we can express \( \frac{R}{100} \):
\[ \frac{R}{100} = 2^{\frac{1}{6}} - 1 \]
Thus,
\[ R = 100 \left(2^{\frac{1}{6}} - 1\right) \]

### Step 7: Determine When the Amount Becomes 64 Times
Now, we want to find out how long it takes for the amount to become 64 times the principal:
\[ A = 64P = 6400 \text{ units} \]
Using the compound interest formula again:
\[ 6400 = 100 \left(1 + \frac{R}{100}\right)^T \]
Dividing both sides by 100:
\[ 64 = \left(1 + \frac{R}{100}\right)^T \]

### Step 8: Substitute the Value of \( 1 + \frac{R}{100} \)
We know that:
\[ 1 + \frac{R}{100} = 2^{\frac{1}{6}} \]
Substituting this into the equation:
\[ 64 = \left(2^{\frac{1}{6}}\right)^T \]

### Step 9: Express 64 as a Power of 2
We know that:
\[ 64 = 2^6 \]
So we can rewrite the equation:
\[ 2^6 = \left(2^{\frac{1}{6}}\right)^T \]

### Step 10: Equate the Exponents
Since the bases are the same, we can equate the exponents:
\[ 6 = \frac{T}{6} \]
Multiplying both sides by 6:
\[ T = 36 \]

### Conclusion
Thus, it will take **36 years** for the sum to become 64 times itself at the same rate of compound interest.

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