A sum of Rs. 1600 gives a simple interest of Rs. 252 in 2 years and 4 months. The rate of interest per annum is:

Options

A. 0.06
B. `6(1)/(4)`%
C. `6(1)/(2)%`
D. `6(3)/(4)%`

Answer

To find the rate of interest per annum for the given problem, we can follow these steps:

### Step 1: Convert the time into years
The time given is 2 years and 4 months. We need to convert the months into years.

\[ \text{Time (T)} = 2 \text{ years} + \frac{4 \text{ months}}{12} = 2 + \frac{1}{3} = \frac{7}{3} \text{ years} \]

### Step 2: Identify the given values
From the problem, we have:
- Principal (P) = Rs. 1600
- Simple Interest (SI) = Rs. 252
- Time (T) = \(\frac{7}{3}\) years

### Step 3: Use the formula for rate of interest
The formula for calculating the rate of interest (R) is:

\[ R = \frac{100 \times SI}{P \times T} \]

### Step 4: Substitute the values into the formula
Now, substituting the known values into the formula:

\[ R = \frac{100 \times 252}{1600 \times \frac{7}{3}} \]

### Step 5: Simplify the expression
First, calculate the denominator:

\[ 1600 \times \frac{7}{3} = \frac{11200}{3} \]

Now substituting back into the formula:

\[ R = \frac{100 \times 252}{\frac{11200}{3}} = \frac{100 \times 252 \times 3}{11200} \]

### Step 6: Calculate the numerator
Calculating the numerator:

\[ 100 \times 252 \times 3 = 75600 \]

### Step 7: Divide the numerator by the denominator
Now, we divide:

\[ R = \frac{75600}{11200} \]

### Step 8: Simplify the fraction
Now simplify:

\[ R = \frac{75600 \div 11200}{11200 \div 11200} = \frac{27}{4} \]

### Step 9: Convert to percentage
To convert to percentage:

\[ R = 6.75 \text{ or } 6 \frac{3}{4} \text{ percent} \]

### Final Answer
Thus, the rate of interest per annum is:

\[ \boxed{6 \frac{3}{4} \text{ percent}} \]

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