Which of the following schemes of computing interest yields the maximum interest for a year?

Options

A. Interest compounded every year at 24% p.a.
B. Interest compounded monthly at 2% per month.
C. Interest compounded per qaurter 6% per quarter.
D. Interest compounded every year at 12% for every-6 months

Answer

To determine which scheme of computing interest yields the maximum interest for a year, we'll evaluate each option using the formula for compound interest:

**Formula:**
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) = Amount after time \( t \)
- \( P \) = Principal amount (initial investment)
- \( r \) = Annual interest rate (in decimal)
- \( n \) = Number of times interest is compounded per year
- \( t \) = Time in years

Let's analyze each option step by step.

### Step 1: Calculate for the first option
**Option 1:** Interest compounded annually at 24% per annum.
- \( P = 1 \)
- \( r = 0.24 \)
- \( n = 1 \)
- \( t = 1 \)

Using the formula:
\[ A = 1 \left(1 + \frac{0.24}{1}\right)^{1 \cdot 1} = 1 \left(1 + 0.24\right)^{1} = 1.24 \]

### Step 2: Calculate for the second option
**Option 2:** Interest compounded monthly at 2% per month.
- \( P = 1 \)
- \( r = 0.02 \times 12 = 0.24 \) (annual rate)
- \( n = 12 \)
- \( t = 1 \)

Using the formula:
\[ A = 1 \left(1 + \frac{0.02}{1}\right)^{12 \cdot 1} = 1 \left(1 + 0.02\right)^{12} \]

Calculating \( (1.02)^{12} \):
\[ A \approx 1.2682 \]

### Step 3: Calculate for the third option
**Option 3:** Interest compounded quarterly at 6% per quarter.
- \( P = 1 \)
- \( r = 0.06 \times 4 = 0.24 \) (annual rate)
- \( n = 4 \)
- \( t = 1 \)

Using the formula:
\[ A = 1 \left(1 + \frac{0.06}{1}\right)^{4 \cdot 1} = 1 \left(1 + 0.06\right)^{4} \]

Calculating \( (1.06)^{4} \):
\[ A \approx 1.2625 \]

### Step 4: Calculate for the fourth option
**Option 4:** Interest compounded semi-annually at 12% every 6 months.
- \( P = 1 \)
- \( r = 0.12 \times 2 = 0.24 \) (annual rate)
- \( n = 2 \)
- \( t = 1 \)

Using the formula:
\[ A = 1 \left(1 + \frac{0.12}{1}\right)^{2 \cdot 1} = 1 \left(1 + 0.12\right)^{2} \]

Calculating \( (1.12)^{2} \):
\[ A \approx 1.2544 \]

### Step 5: Compare the amounts
- Option 1: \( A \approx 1.24 \)
- Option 2: \( A \approx 1.2682 \)
- Option 3: \( A \approx 1.2625 \)
- Option 4: \( A \approx 1.2544 \)

### Conclusion
The maximum amount is from **Option 2**, where interest is compounded monthly at 2% per month, yielding approximately \( 1.2682 \).

### Final Answer
**Option 2** is the correct answer.

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