The simple interest on a sum of money for 3 yr at `6 (2)/(3)%` per annum is Rs. 6750. The compound interest on the same sum at the same rate of interest for the same period will be

Options

A. Rs. 7200
B. Rs. 7210
C. Rs. 7120
D. Rs. 7012

Answer

To solve the problem step by step, we will first determine the principal amount (P) using the given simple interest (SI) and then calculate the compound interest (CI) using the same principal and rate.

### Step 1: Understand the Simple Interest Formula
The formula for Simple Interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI\) = Simple Interest
- \(P\) = Principal amount
- \(R\) = Rate of interest per annum
- \(T\) = Time in years

### Step 2: Substitute the Known Values
From the problem, we know:
- \(SI = 6750\)
- \(R = 6 \frac{2}{3}\% = \frac{20}{3}\%\)
- \(T = 3\) years

Substituting these values into the formula:
\[ 6750 = \frac{P \times \frac{20}{3} \times 3}{100} \]

### Step 3: Simplify the Equation
Now, simplify the equation:
\[ 6750 = \frac{P \times 20}{100} \]
\[ 6750 = \frac{P \times 20}{100} \implies 6750 = \frac{P \times 20}{100} \]
Multiplying both sides by 100:
\[ 675000 = P \times 20 \]
Now, divide both sides by 20 to find \(P\):
\[ P = \frac{675000}{20} = 33750 \]

### Step 4: Calculate the Compound Interest
The formula for Compound Interest is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting the values we have:
\[ CI = 33750 \left(1 + \frac{20}{300}\right)^3 - 33750 \]
\[ = 33750 \left(1 + \frac{1}{15}\right)^3 - 33750 \]
\[ = 33750 \left(\frac{16}{15}\right)^3 - 33750 \]

### 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: Substitute Back to Find CI
Now substitute this back into the CI formula:
\[ CI = 33750 \times \frac{4096}{3375} - 33750 \]
\[ = 4096 \times 10 - 33750 \]
\[ = 40960 - 33750 = 7200 \]

### Final Answer
The compound interest on the same sum at the same rate for the same period is:
\[ \text{CI} = 7200 \]