Rs. 12000 amounts to Rs. 20736 in 3 years at r% p.a. of compound interest . What is the value of r ?

Options

A. `10%`
B. `25%`
C. `12%`
D. `20%`

Answer

To solve the problem step by step, we will use the formula for compound interest. The formula for the amount \( A \) after \( n \) years is given by:

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

Where:
- \( A \) = Amount after \( n \) years
- \( P \) = Principal amount (initial investment)
- \( r \) = Rate of interest per annum
- \( n \) = Number of years

Given:
- \( A = 20736 \)
- \( P = 12000 \)
- \( n = 3 \)

### Step 1: Substitute the values into the formula
We substitute the known values into the compound interest formula:

\[ 20736 = 12000 \left(1 + \frac{r}{100}\right)^3 \]

### Step 2: Divide both sides by 12000
To isolate the term with \( r \), we divide both sides by 12000:

\[ \frac{20736}{12000} = \left(1 + \frac{r}{100}\right)^3 \]

Calculating the left side:

\[ \frac{20736}{12000} = 1.728 \]

So, we have:

\[ 1.728 = \left(1 + \frac{r}{100}\right)^3 \]

### Step 3: Take the cube root of both sides
Next, we take the cube root of both sides to solve for \( 1 + \frac{r}{100} \):

\[ \sqrt[3]{1.728} = 1 + \frac{r}{100} \]

Calculating the cube root:

\[ \sqrt[3]{1.728} = 1.2 \]

So, we have:

\[ 1.2 = 1 + \frac{r}{100} \]

### Step 4: Isolate \( r \)
Now, we isolate \( r \) by subtracting 1 from both sides:

\[ 1.2 - 1 = \frac{r}{100} \]

This simplifies to:

\[ 0.2 = \frac{r}{100} \]

### Step 5: Multiply both sides by 100
To find \( r \), we multiply both sides by 100:

\[ r = 0.2 \times 100 \]

Calculating this gives:

\[ r = 20 \]

### Conclusion
The value of \( r \) is \( 20\% \).

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