Let's solve the problem step-by-step.
First, we need to add the lengths of the two pieces of carpet runner that Arliss has:
1. Convert the mixed numbers to improper fractions:
- The first piece is [tex]\(2 \frac{1}{3}\)[/tex] yards. To convert this to an improper fraction:
[tex]\[
2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}
\][/tex]
- The second piece is [tex]\(3 \frac{1}{3}\)[/tex] yards. To convert this to an improper fraction:
[tex]\[
3 \frac{1}{3} = 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3}
\][/tex]
2. Add the improper fractions:
[tex]\[
\frac{7}{3} + \frac{10}{3} = \frac{7 + 10}{3} = \frac{17}{3}
\][/tex]
3. Convert the improper fraction back to a mixed number:
[tex]\[
\frac{17}{3} = 5 \frac{2}{3} \text{ yards}
\][/tex]
Therefore, the total length of the two pieces of carpet that Arliss has is:
[tex]\[
2 \frac{1}{3} + 3 \frac{1}{3} = 5 \frac{2}{3} \text{ yards}
\][/tex]
Now, we need to determine how much more carpet Arliss needs to buy. We know that she needs a total of 10 yards of carpet runner:
4. Subtract the total length she currently has from the total length she needs:
[tex]\[
10 - 5 \frac{2}{3}
\][/tex]
First, convert [tex]\(10\)[/tex] to a fraction with the same denominator:
[tex]\[
10 = \frac{30}{3}
\][/tex]
Next, subtract [tex]\(5 \frac{2}{3}\)[/tex]:
[tex]\[
\frac{30}{3} - \frac{17}{3} = \frac{30 - 17}{3} = \frac{13}{3}
\][/tex]
Convert [tex]\(\frac{13}{3}\)[/tex] back to a mixed number:
[tex]\[
\frac{13}{3} = 4 \frac{1}{3} \text{ yards}
\][/tex]
Therefore, Arliss needs to buy:
[tex]\[
4 \frac{1}{3} \text{ yards}
\][/tex]
So, the answer to the problem is:
Arliss needs to buy [tex]\(4 \frac{1}{3}\)[/tex] more yards of carpet runner.
