If some amount becomes `(216)/(125)` of itself in 3 years then find the rate of interest?

Options

A. `28%`
B. `30%`
C. `20% `
D. `25%`

Answer

To solve the problem of finding the rate of interest when an amount becomes \( \frac{216}{125} \) of itself in 3 years, we can follow these steps:

### Step 1: Understand the relationship between principal and amount
Let the principal amount be \( P \). According to the problem, the amount after 3 years becomes \( \frac{216}{125} P \).

### Step 2: Use the formula for compound interest
The formula for the amount \( A \) after \( t \) years at a rate \( r \) is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Here, \( A = \frac{216}{125} P \), \( P = P \), and \( t = 3 \).

### Step 3: Set up the equation
Substituting the values into the formula, we have:
\[ \frac{216}{125} P = P \left(1 + \frac{r}{100}\right)^3 \]

### Step 4: Simplify the equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{216}{125} = \left(1 + \frac{r}{100}\right)^3 \]

### Step 5: Take the cube root
To isolate \( 1 + \frac{r}{100} \), we take the cube root of both sides:
\[ 1 + \frac{r}{100} = \sqrt[3]{\frac{216}{125}} \]

### Step 6: Simplify the cube root
Calculating the cube root:
\[ \frac{216}{125} = \left(\frac{6}{5}\right)^3 \]
Thus,
\[ \sqrt[3]{\frac{216}{125}} = \frac{6}{5} \]

### Step 7: Solve for \( r \)
Now we can set up the equation:
\[ 1 + \frac{r}{100} = \frac{6}{5} \]
Subtracting 1 from both sides:
\[ \frac{r}{100} = \frac{6}{5} - 1 = \frac{6}{5} - \frac{5}{5} = \frac{1}{5} \]
Multiplying both sides by 100:
\[ r = 100 \times \frac{1}{5} = 20 \]

### Step 8: Conclusion
The rate of interest \( r \) is \( 20\% \).