Find the compound interest on Rs 21600 for 3 years at the rate `16 (2)/(3) %` ?

Options

A. ₹ 12600
B. ₹ 17200
C. ₹ 16200
D. ₹ 12700

Answer

To find the compound interest on Rs. 21,600 for 3 years at the rate of 16⅔%, we can follow these steps:

### Step 1: Convert the percentage to a decimal
The rate of interest is given as 16⅔%. We can convert this to a fraction:
\[ 16 \frac{2}{3} \% = \frac{50}{3} \% \]
To convert this to a decimal, we divide by 100:
\[ \frac{50}{3} \div 100 = \frac{50}{300} = \frac{1}{6} \]

### Step 2: Calculate the compound interest for each year
The formula for compound interest is:
\[ A = P \left(1 + r\right)^n \]
where:
- \( A \) is the amount after time \( n \),
- \( P \) is the principal amount (Rs. 21,600),
- \( r \) is the rate of interest (as a decimal, \( \frac{1}{6} \)),
- \( n \) is the number of years (3 years).

Substituting the values:
\[ A = 21600 \left(1 + \frac{1}{6}\right)^3 \]
\[ A = 21600 \left(\frac{7}{6}\right)^3 \]

### Step 3: Calculate \( \left(\frac{7}{6}\right)^3 \)
Calculating \( \left(\frac{7}{6}\right)^3 \):
\[ \left(\frac{7}{6}\right)^3 = \frac{7^3}{6^3} = \frac{343}{216} \]

### Step 4: Calculate the total amount \( A \)
Now substituting back into the formula:
\[ A = 21600 \times \frac{343}{216} \]
\[ A = 21600 \div 216 \times 343 = 100 \times 343 = 34300 \]

### Step 5: Calculate the compound interest
The compound interest (CI) is given by:
\[ CI = A - P \]
Substituting the values:
\[ CI = 34300 - 21600 = 12700 \]

### Final Answer
The compound interest on Rs. 21,600 for 3 years at the rate of 16⅔% is Rs. 12,700.

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