A sum of money lent out at C.I at a certain r% (rate of growth) doubles itself in 5 years. Find in how many years will the money become eight times of itself at the same rate of growth.

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the problem
We know that a sum of money doubles itself in 5 years at a certain rate of interest (r%). We need to find out how many years it will take for the same sum to become eight times its original value at the same rate.

### Step 2: Set up the equations
Let the principal amount be \( P \). According to the problem, the amount after 5 years is:
\[ A = 2P \]
Using the compound interest formula:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Substituting the values we have:
\[ 2P = P \left(1 + \frac{r}{100}\right)^5 \]

### Step 3: Simplify the equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ 2 = \left(1 + \frac{r}{100}\right)^5 \]

### Step 4: Solve for \( 1 + \frac{r}{100} \)
To isolate \( 1 + \frac{r}{100} \), we take the fifth root of both sides:
\[ 1 + \frac{r}{100} = 2^{\frac{1}{5}} \]

### Step 5: Find the value of \( 1 + \frac{r}{100} \)
Now, we can express \( 1 + \frac{r}{100} \) as:
\[ 1 + \frac{r}{100} = 2^{0.2} \quad \text{(since } \frac{1}{5} = 0.2\text{)} \]

### Step 6: Set up the equation for eight times the principal
Now, we want to find the time \( t \) when the amount is eight times the principal:
\[ A = 8P \]
Using the compound interest formula again:
\[ 8P = P \left(1 + \frac{r}{100}\right)^t \]
Cancel \( P \):
\[ 8 = \left(1 + \frac{r}{100}\right)^t \]

### Step 7: Substitute the value of \( 1 + \frac{r}{100} \)
Substituting \( 1 + \frac{r}{100} = 2^{0.2} \):
\[ 8 = \left(2^{0.2}\right)^t \]

### Step 8: Express 8 as a power of 2
We know that \( 8 = 2^3 \), so we can rewrite the equation:
\[ 2^3 = \left(2^{0.2}\right)^t \]

### Step 9: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ 3 = 0.2t \]

### Step 10: Solve for \( t \)
Now, solve for \( t \):
\[ t = \frac{3}{0.2} = 15 \]

### Conclusion
Thus, it will take **15 years** for the money to become eight times itself at the same rate of growth.

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