A sum becomes 5 times of itself in 3 years at compound interest (interest is compounded annually). In how many years will the sum becomes 125 times of itself?

Options

A. 9
B. 6
C. 8
D. 12

Answer

To solve the problem step by step, we will use the formula for compound interest and the information provided in the question.

### Step 1: Understand the Problem
We need to find out how many years it will take for a sum of money (principal) to become 125 times itself at compound interest, given that it becomes 5 times itself in 3 years.

### Step 2: Set Up the Initial Equation
Let the principal amount be \( P \) and the rate of interest be \( r\% \). The formula for the amount \( A \) after \( n \) years at compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
From the problem, we know that after 3 years, the amount becomes 5 times the principal:
\[ 5P = P \left(1 + \frac{r}{100}\right)^3 \]

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

### Step 4: Solve for \( 1 + \frac{r}{100} \)
To isolate \( 1 + \frac{r}{100} \), we take the cube root of both sides:
\[ 1 + \frac{r}{100} = 5^{1/3} \]

### Step 5: Calculate \( 5^{1/3} \)
Calculating \( 5^{1/3} \) gives us approximately \( 1.710 \). Therefore:
\[ \frac{r}{100} = 5^{1/3} - 1 \]

### Step 6: Set Up the Second Equation
Now we need to find out how many years \( n \) it will take for the sum to become 125 times itself:
\[ 125P = P \left(1 + \frac{r}{100}\right)^n \]
Again, we can cancel \( P \):
\[ 125 = \left(1 + \frac{r}{100}\right)^n \]

### Step 7: Substitute \( 1 + \frac{r}{100} \)
Substituting \( 1 + \frac{r}{100} = 5^{1/3} \) into the equation:
\[ 125 = \left(5^{1/3}\right)^n \]

### Step 8: Rewrite 125 as a Power of 5
We know that \( 125 = 5^3 \), so we can rewrite the equation as:
\[ 5^3 = \left(5^{1/3}\right)^n \]

### Step 9: Equate the Exponents
Since the bases are the same, we can equate the exponents:
\[ 3 = \frac{n}{3} \]

### Step 10: Solve for \( n \)
Multiplying both sides by 3 gives:
\[ n = 9 \]

### Conclusion
Thus, the number of years it will take for the sum to become 125 times itself is **9 years**.