The perimeter of a rectangle is `13 cm` and its width is `2 3/4 cm .` Find its length.

Options

A. `3 11/4`
B. `3 3/4`
C. `3 15/4`
D. `7 31/7`

Explanation

To find the length of the rectangle given its perimeter and width, we can follow these steps:

### Step 1: Understand the formula for the perimeter of a rectangle.
The perimeter \( P \) of a rectangle can be calculated using the formula:
\[ P = 2 \times (L + W) \]
where \( L \) is the length and \( W \) is the width.

### Step 2: Substitute the given values into the formula.
We know the perimeter \( P = 13 \, \text{cm} \) and the width \( W = 2 \frac{3}{4} \, \text{cm} \). First, we need to convert the mixed fraction for the width into an improper fraction:
\[ W = 2 \frac{3}{4} = 2 + \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4} \, \text{cm} \]

### Step 3: Substitute the values into the perimeter formula.
Now, we substitute the values into the perimeter formula:
\[ 13 = 2 \times (L + \frac{11}{4}) \]

### Step 4: Simplify the equation.
Divide both sides by 2 to isolate \( L + \frac{11}{4} \):
\[ \frac{13}{2} = L + \frac{11}{4} \]

### Step 5: Convert \( \frac{13}{2} \) to a fraction with a common denominator.
To solve for \( L \), we need a common denominator. The least common multiple of 2 and 4 is 4. Convert \( \frac{13}{2} \) to have a denominator of 4:
\[ \frac{13}{2} = \frac{13 \times 2}{2 \times 2} = \frac{26}{4} \]

### Step 6: Substitute back into the equation.
Now we have:
\[ \frac{26}{4} = L + \frac{11}{4} \]

### Step 7: Solve for \( L \).
Subtract \( \frac{11}{4} \) from both sides:
\[ L = \frac{26}{4} - \frac{11}{4} = \frac{26 - 11}{4} = \frac{15}{4} \]

### Step 8: Convert \( \frac{15}{4} \) to a mixed number.
To convert \( \frac{15}{4} \) into a mixed number:
\[ \frac{15}{4} = 3 \frac{3}{4} \, \text{cm} \]

### Final Answer:
The length of the rectangle is \( 3 \frac{3}{4} \, \text{cm} \).

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