A sum of money at compound interest (compound annually) doubles itself in 4 years. In how many years will it amount to eight times of itself?

Answer

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

### Step 1: Understand the Problem
We are given 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 amount to eight times itself.

### Step 2: Set Up the Formula
The formula for the amount \( A \) in compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^N \]
where:
- \( P \) = Principal amount (initial sum of money)
- \( R \) = Rate of interest per annum
- \( N \) = Number of years

### Step 3: Case 1 - Doubling in 4 Years
Since the money doubles in 4 years, we can set up the equation:
\[ 2P = P \left(1 + \frac{R}{100}\right)^4 \]
Dividing both sides by \( P \):
\[ 2 = \left(1 + \frac{R}{100}\right)^4 \]

### Step 4: Solve for \( 1 + \frac{R}{100} \)
To isolate \( 1 + \frac{R}{100} \), we take the fourth root of both sides:
\[ 1 + \frac{R}{100} = 2^{1/4} \]

### Step 5: Case 2 - Finding When Amount is Eight Times
Now we need to find out how many years \( N \) it will take for the amount to be eight times the principal:
\[ 8P = P \left(1 + \frac{R}{100}\right)^N \]
Dividing both sides by \( P \):
\[ 8 = \left(1 + \frac{R}{100}\right)^N \]

### Step 6: Express 8 in Terms of 2
We know that \( 8 = 2^3 \). Therefore, we can rewrite the equation:
\[ 2^3 = \left(1 + \frac{R}{100}\right)^N \]

### Step 7: Substitute \( 1 + \frac{R}{100} \)
From Step 4, we know that \( 1 + \frac{R}{100} = 2^{1/4} \). Substitute this into the equation:
\[ 2^3 = \left(2^{1/4}\right)^N \]

### Step 8: Simplify the Equation
Using the power of a power property:
\[ 2^3 = 2^{N/4} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 3 = \frac{N}{4} \]

### Step 9: Solve for \( N \)
To find \( N \), multiply both sides by 4:
\[ N = 3 \times 4 = 12 \]

### Conclusion
It will take **12 years** for the sum of money to amount to eight times itself.

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