Calculate the compound interest on Rs 6750 for 3yr at `6(2)/(3)%` per annum.

Options

A. Rs 1645
B. Rs 1442
C. Rs 2415
D. Rs 2142

Answer

To calculate the compound interest on Rs 6750 for 3 years at a rate of \(6 \frac{2}{3}\%\) per annum, we will follow these steps:

### Step 1: Convert the rate into a fraction
The rate given is \(6 \frac{2}{3}\%\). We can convert this mixed number into an improper fraction:
\[ 6 \frac{2}{3} = \frac{20}{3}\% \]

### Step 2: Identify the principal, rate, and time
- Principal (P) = Rs 6750
- Rate (R) = \(\frac{20}{3}\%\)
- Time (n) = 3 years

### Step 3: Convert the percentage rate to a decimal
To use the formula for compound interest, we need to convert the percentage into a decimal:
\[ R = \frac{20}{3} \div 100 = \frac{20}{300} = \frac{1}{15} \]

### Step 4: Use the compound interest formula
The formula for the amount (A) after n years is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Substituting the values we have:
\[ A = 6750 \left(1 + \frac{20}{300}\right)^3 \]
This simplifies to:
\[ A = 6750 \left(1 + \frac{1}{15}\right)^3 \]
\[ A = 6750 \left(\frac{16}{15}\right)^3 \]

### Step 5: Calculate \(\left(\frac{16}{15}\right)^3\)
Calculating \(\left(\frac{16}{15}\right)^3\):
\[ \left(\frac{16}{15}\right)^3 = \frac{16^3}{15^3} = \frac{4096}{3375} \]

### Step 6: Calculate the amount (A)
Now substitute this back into the equation for A:
\[ A = 6750 \times \frac{4096}{3375} \]
Calculating this gives:
\[ A = \frac{6750 \times 4096}{3375} \]
Calculating the numerator:
\[ 6750 \times 4096 = 27648000 \]
Now divide by the denominator:
\[ A = \frac{27648000}{3375} \approx 8192 \]

### Step 7: Calculate the compound interest (CI)
The compound interest is given by:
\[ CI = A - P \]
Substituting the values we have:
\[ CI = 8192 - 6750 = 1442 \]

### Final Result
The compound interest on Rs 6750 for 3 years at \(6 \frac{2}{3}\%\) per annum is Rs 1442.

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