Sure, let's solve this step by step.
1. Convert the Mixed Number to an Improper Fraction:
Erica initially bought [tex]\(3 \frac{1}{2}\)[/tex] yards of fabric. First, we convert this mixed number to an improper fraction.
[tex]\[
3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}
\][/tex]
2. Calculate the Fabric Used:
Erica uses [tex]\(\frac{2}{3}\)[/tex] of this fabric to make a curtain. To find out how much fabric she used, we multiply:
[tex]\[
\frac{2}{3} \times \frac{7}{2}
\][/tex]
When multiplying fractions, multiply the numerators together and the denominators together:
[tex]\[
\frac{2 \times 7}{3 \times 2} = \frac{14}{6}
\][/tex]
Simplify [tex]\(\frac{14}{6}\)[/tex]:
[tex]\[
\frac{14}{6} = \frac{7}{3}
\][/tex]
3. Calculate the Fabric Left:
To find out how much fabric is left, we subtract the used fabric from the initial fabric:
[tex]\[
\frac{7}{2} - \frac{7}{3}
\][/tex]
To subtract these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6. Convert both fractions:
[tex]\[
\frac{7}{2} = \frac{7 \times 3}{2 \times 3} = \frac{21}{6}
\][/tex]
[tex]\[
\frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6}
\][/tex]
Now we subtract:
[tex]\[
\frac{21}{6} - \frac{14}{6} = \frac{21 - 14}{6} = \frac{7}{6}
\][/tex]
4. Convert Improper Fraction to a Mixed Number:
We convert [tex]\(\frac{7}{6}\)[/tex] back to a mixed number:
[tex]\[
\frac{7}{6} = 1 \frac{1}{6}
\][/tex]
5. Determine the Correct Option:
The correct amount of fabric left is [tex]\(1 \frac{1}{6}\)[/tex] yards.
So, the correct answer is:
[tex]\[
\boxed{1 \frac{1}{6} \text{ yard}}
\][/tex]
