A sum of money doubles itself at compound interest in 4 years. In how many years will it become 8 times?

Answer

To solve the problem step by step, we will follow the logic used in the video transcript.

### Step 1: Understand the Problem
We know that a sum of money doubles itself in 4 years at compound interest. We need to find out how many years it will take for the same sum to become 8 times.

### Step 2: Set Up the Initial Equation
Let the principal amount be \( P \) and the rate of interest be \( R \). According to the formula for compound interest, the amount \( A \) after time \( T \) is given by:

\[ A = P \left(1 + \frac{R}{100}\right)^T \]

Since the amount doubles in 4 years, we can write:

\[ 2P = P \left(1 + \frac{R}{100}\right)^4 \]

### Step 3: Simplify the Equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):

\[ 2 = \left(1 + \frac{R}{100}\right)^4 \]

### Step 4: Solve for \( R \)
Taking the fourth root of both sides, we get:

\[ 1 + \frac{R}{100} = 2^{1/4} \]

Calculating \( 2^{1/4} \):

\[ 2^{1/4} = \sqrt[4]{2} \approx 1.189207 \]

Now, we can express \( R \):

\[ \frac{R}{100} = 2^{1/4} - 1 \approx 1.189207 - 1 = 0.189207 \]

Thus,

\[ R \approx 0.189207 \times 100 \approx 18.92 \]

### Step 5: Set Up the Second Equation for 8 Times
Now we need to find out how long it takes for the amount to become 8 times the principal:

\[ 8P = P \left(1 + \frac{R}{100}\right)^T \]

Again, we can cancel \( P \):

\[ 8 = \left(1 + \frac{R}{100}\right)^T \]

### Step 6: Substitute the Value of \( 1 + \frac{R}{100} \)
From our earlier calculation, we know:

\[ 1 + \frac{R}{100} = 2^{1/4} \]

Thus, we can rewrite the equation as:

\[ 8 = \left(2^{1/4}\right)^T \]

### Step 7: Rewrite 8 in Terms of Powers of 2
We know that \( 8 = 2^3 \). Therefore, we can write:

\[ 2^3 = \left(2^{1/4}\right)^T \]

### Step 8: Equate the Exponents
Since the bases are the same, we can equate the exponents:

\[ 3 = \frac{T}{4} \]

### Step 9: Solve for \( T \)
Multiplying both sides by 4 gives:

\[ T = 3 \times 4 = 12 \]

### Conclusion
Thus, it will take **12 years** for the sum to become 8 times.

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