Sudershan deposited `Rs 32,000` in a bank where the interest is credited quarterly. If the rate of interest be `5%` per annum, what amount will he receive after `6` months?

Options

A. `"Rs 32,805 "`
B. `"Rs 32,000 "`
C. `"Rs 32,430 "`
D. `"Rs 32,555 "`

Answer

To solve the problem step by step, we need to calculate the amount Sudershan will receive after 6 months when he deposits Rs 32,000 at an interest rate of 5% per annum, compounded quarterly.

### Step 1: Identify the given values
- Principal (P) = Rs 32,000
- Rate of interest (R) = 5% per annum
- Time (T) = 6 months = 0.5 years

### Step 2: Convert the annual interest rate to a quarterly rate
Since the interest is compounded quarterly, we need to divide the annual interest rate by 4:
\[ R_{quarterly} = \frac{5\%}{4} = 1.25\% \]

### Step 3: Convert the quarterly rate to a decimal
To use the formula for compound interest, we convert the percentage to a decimal:
\[ R_{quarterly} = \frac{1.25}{100} = 0.0125 \]

### Step 4: Calculate the number of compounding periods
Since we are calculating for 6 months and the interest is compounded quarterly, there will be 2 compounding periods in 6 months:
\[ n = 2 \]

### Step 5: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{n}\right)^{nt} \]
Where:
- \(A\) = Amount after time \(t\)
- \(P\) = Principal
- \(R\) = Rate of interest (in decimal)
- \(n\) = Number of times interest is compounded per year
- \(t\) = Time in years

Substituting the values:
\[ A = 32000 \left(1 + 0.0125\right)^{2} \]

### Step 6: Simplify the expression
Calculate \(1 + 0.0125\):
\[ A = 32000 \left(1.0125\right)^{2} \]

### Step 7: Calculate \((1.0125)^{2}\)
\[ (1.0125)^{2} = 1.02515625 \]

### Step 8: Calculate the final amount
Now substitute back into the equation:
\[ A = 32000 \times 1.02515625 \]
\[ A \approx 32864.50 \]

### Final Answer
After rounding, Sudershan will receive approximately Rs 32,864.50 after 6 months.

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