Sure! Let's solve the problem step-by-step.
1. The friend gives the neighbor [tex]\(13 \frac{2}{3}\)[/tex] feet of twine.
2. Let's convert this mixed number into an improper fraction to make calculations easier:
[tex]\[
13 \frac{2}{3} = 13 + \frac{2}{3} = \frac{39}{3} + \frac{2}{3} = \frac{41}{3}
\][/tex]
3. The remaining quantity of twine on the spool is [tex]\(38 \frac{2}{5}\)[/tex] feet.
4. Convert this mixed number into an improper fraction:
[tex]\[
38 \frac{2}{5} = 38 + \frac{2}{5} = \frac{190}{5} + \frac{2}{5} = \frac{192}{5}
\][/tex]
5. To find the total original amount of twine on the spool, add the given twine and the remaining twine.
6. First, we need to add the fractions [tex]\(\frac{41}{3}\)[/tex] and [tex]\(\frac{192}{5}\)[/tex]. We find a common denominator to add these fractions. For 3 and 5, the least common multiple is 15:
[tex]\[
\frac{41}{3} = \frac{41 \cdot 5}{3 \cdot 5} = \frac{205}{15}
\][/tex]
[tex]\[
\frac{192}{5} = \frac{192 \cdot 3}{5 \cdot 3} = \frac{576}{15}
\][/tex]
7. Now, adding these fractions:
[tex]\[
\frac{205}{15} + \frac{576}{15} = \frac{205 + 576}{15} = \frac{781}{15}
\][/tex]
8. Converting the improper fraction back to a mixed number:
[tex]\[
\frac{781}{15} = 52 \frac{11}{15}
\][/tex]
9. Thus, the total original amount of twine on the spool is:
[tex]\[
52 \frac{11}{15} \text{ feet}
\][/tex]
None of the provided answer choices directly match this form. Given that the numerical calculations yielded approximately:
[tex]\[
52.06666666666666 \text{ feet} \text{ (same as } 52 + \frac{1}{15})
\][/tex]
This most closely aligns with choice [tex]\( 52 \frac{1}{15} \)[/tex].
Therefore, the correct answer is:
C) [tex]\(52 \frac{1}{15}\)[/tex]
