If a sum of money placed at compound interest becomes 3 times of itself in 3 year. In how many years will it be 9 times at the same rate of interest ?

Options

A. 6 years
B. 9 years
C. 12 years
D. 5 years

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We know that a sum of money becomes 3 times itself in 3 years at compound interest. We need to find out how many years it will take for the same sum to become 9 times itself at the same rate of interest.

### Step 2: Set Up the Initial Equation
Let the principal amount be \( P \). According to the problem, after 3 years, the amount becomes \( 3P \). The formula for compound interest is given by:

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

Where:
- \( A \) is the amount after time \( t \)
- \( P \) is the principal amount
- \( r \) is the rate of interest
- \( t \) is the time in years

For our case, we have:

\[ 3P = P \left(1 + \frac{r}{100}\right)^3 \]

### Step 3: Simplify the Equation
Dividing both sides by \( P \) (assuming \( P \neq 0 \)) gives:

\[ 3 = \left(1 + \frac{r}{100}\right)^3 \]

### Step 4: Solve for \( 1 + \frac{r}{100} \)
To find \( 1 + \frac{r}{100} \), we take the cube root of both sides:

\[ 1 + \frac{r}{100} = 3^{1/3} \]

### Step 5: Find the Value of \( 9P \)
Now, we need to find out how long it will take for the principal \( P \) to become \( 9P \):

\[ 9P = P \left(1 + \frac{r}{100}\right)^n \]

Dividing both sides by \( P \):

\[ 9 = \left(1 + \frac{r}{100}\right)^n \]

### Step 6: Substitute the Value of \( 1 + \frac{r}{100} \)
From Step 4, we know that \( 1 + \frac{r}{100} = 3^{1/3} \). We can substitute this into the equation:

\[ 9 = \left(3^{1/3}\right)^n \]

### Step 7: Rewrite \( 9 \) as a Power of \( 3 \)
We know that \( 9 = 3^2 \), so we can rewrite the equation:

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

### Step 8: Set the Exponents Equal
Since the bases are the same, we can set the exponents equal to each other:

\[ 2 = \frac{n}{3} \]

### Step 9: Solve for \( n \)
To find \( n \), multiply both sides by 3:

\[ n = 2 \times 3 = 6 \]

### Final Answer
Thus, it will take **6 years** for the sum to become 9 times itself at the same rate of interest.

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