To determine how many feet of twine originally were on the spool, we need to add the amount given to the neighbor and the amount remaining on the spool.
1. First, convert the mixed numbers to improper fractions:
[tex]\(13 \frac{2}{3}\)[/tex]:
[tex]\[
13 \frac{2}{3} = 13 + \frac{2}{3} = \frac{13 \times 3}{3} + \frac{2}{3} = \frac{39}{3} + \frac{2}{3} = \frac{41}{3}
\][/tex]
[tex]\(38 \frac{2}{5}\)[/tex]:
[tex]\[
38 \frac{2}{5} = 38 + \frac{2}{5} = \frac{38 \times 5}{5} + \frac{2}{5} = \frac{190}{5} + \frac{2}{5} = \frac{192}{5}
\][/tex]
2. Convert these improper fractions back to decimal form:
[tex]\[
\frac{41}{3} \approx 13.6667
\][/tex]
[tex]\[
\frac{192}{5} = 38.4
\][/tex]
3. Add these decimal values:
[tex]\[
13.6667 + 38.4 = 52.0667
\][/tex]
4. Now, let's determine which option matches the calculated total. Convert each given option to a decimal to compare:
Option A: [tex]\(51 \frac{1}{15}\)[/tex]:
[tex]\[
51 + \frac{1}{15} = 51 + 0.0667 \approx 51.0667
\][/tex]
Option B: [tex]\(51 \frac{4}{5}\)[/tex]:
[tex]\[
51 + \frac{4}{5} = 51 + 0.8 = 51.8
\][/tex]
Option C: [tex]\(52 \frac{1}{15}\)[/tex]:
[tex]\[
52 + \frac{1}{15} = 52 + 0.0667 \approx 52.0667
\][/tex]
Option D: [tex]\(52 \frac{1}{5}\)[/tex]:
[tex]\[
52 + \frac{1}{5} = 52 + 0.2 = 52.2
\][/tex]
5. The closest match to our calculated result of 52.0667 is Option C [tex]\(52 \frac{1}{15}\)[/tex].
Therefore, the correct answer is:
C [tex]\( \quad 52 \frac{1}{15} \)[/tex].
