To solve this problem, we need to determine the correct fraction of the total horizontal distance between points [tex]\( C \)[/tex] and [tex]\( D \)[/tex] that corresponds to the portion [tex]\( CE \)[/tex].
Given:
- The coordinates of point [tex]\( C \)[/tex] are [tex]\( (3, 4) \)[/tex].
- The coordinates of point [tex]\( D \)[/tex] are [tex]\( (11, 3) \)[/tex].
- The ratio [tex]\( CE : DE \)[/tex] is [tex]\( 3 : 5 \)[/tex].
First, let's find the total ratio. The total ratio is obtained by adding the individual parts of the ratio:
[tex]\[ 3 + 5 = 8 \][/tex]
Next, we determine the fraction of the total horizontal distance between [tex]\( C \)[/tex] and [tex]\( D \)[/tex] that corresponds to [tex]\( CE \)[/tex]. Since [tex]\( CE \)[/tex] is the portion that we are interested in, its fraction is calculated as follows:
[tex]\[ \text{Fraction of } CE = \frac{3}{8} \][/tex]
Therefore, the fraction that Grace should use to find the [tex]\( x \)[/tex]-coordinate of point [tex]\( E \)[/tex] is:
[tex]\[ \frac{3}{8} \][/tex]
Thus, the correct answer is:
[tex]\[ \frac{3}{8} \][/tex]
