To find the width of the backyard in the scale drawing, we can use the scale ratio provided, which is 7 inches : 4 feet.
First, we need to convert the actual width of the backyard from feet to inches because the scale drawing measurements are in inches. There are 12 inches in a foot, so:
\[ 48 \text{ feet} \times 12 \text{ inches/foot} = 576 \text{ inches} \]
Next, we need to set up a proportion using the scale to find the width of the backyard in the drawing (let's call it \( x \) inches):
\[ \frac{x \text{ inches}}{576 \text{ inches}} = \frac{7 \text{ inches}}{48 \text{ feet}} \]
To solve for \( x \), we can cross-multiply:
\[ x \times 48 = 7 \times 576 \]
Calculating the right side:
\[ 7 \times 576 = 4032 \]
Now we have:
\[ 48x = 4032 \]
Now, divide both sides by 48 to solve for \( x \):
\[ x = \frac{4032}{48} = 84 \]
Thus, the width of the backyard in the scale drawing is \( \boxed{84} \) inches.