Question

a border to surround a picture is to be cut from a 13\frac{1}{2} in. by 13\frac{1}{2} in. mat board. if the picture is 9\frac{3}{4} in. by 9\frac{3}{4} in., what is the width (in inches) of the border? (enter an exact number as an integer, fraction, or decimal.)

Explanation:

Step1: Let the width of the border be $x$ inches.

The side - length of the mat board is $13\frac{1}{2}=\frac{27}{2}$ inches and the side - length of the picture is $9\frac{3}{4}=\frac{39}{4}$ inches.
The relationship between the side - length of the mat board and the side - length of the picture considering the border width is $\frac{27}{2}-2x=\frac{39}{4}$.

Step2: Solve the equation for $x$.

First, rewrite the equation:
\[

$$\begin{align*} \frac{27}{2}-2x&=\frac{39}{4}\\ - 2x&=\frac{39}{4}-\frac{27}{2}\\ -2x&=\frac{39}{4}-\frac{54}{4}\\ -2x&=\frac{39 - 54}{4}\\ -2x&=-\frac{15}{4}\\ x&=\frac{15}{8} \end{align*}$$

\]

Answer:

$\frac{15}{8}$