To solve this problem, we need to determine the total amount of twine that was originally on the spool before any was given away. This involves adding the amount of twine given to a neighbor to the amount that remains on the spool.
1. Identify the given quantities:
- The amount of twine given to the neighbor: [tex]\( 13 \frac{2}{3} \)[/tex] feet
- The amount of twine left on the spool: [tex]\( 38 \frac{2}{5} \)[/tex] feet
2. Convert the mixed numbers to improper fractions:
- For [tex]\( 13 \frac{2}{3} \)[/tex]:
[tex]\[
13 \frac{2}{3} = 13 + \frac{2}{3} = \frac{39}{3} + \frac{2}{3} = \frac{39 + 2}{3} = \frac{41}{3}
\][/tex]
- For [tex]\( 38 \frac{2}{5} \)[/tex]:
[tex]\[
38 \frac{2}{5} = 38 + \frac{2}{5} = \frac{190}{5} + \frac{2}{5} = \frac{190 + 2}{5} = \frac{192}{5}
\][/tex]
3. Convert the improper fractions to decimal form:
- [tex]\( \frac{41}{3} \approx 13.6667 \)[/tex] (rounded to four decimal places)
- [tex]\( \frac{192}{5} = 38.4 \)[/tex]
4. Add the two quantities in decimal form to find the total amount of twine originally on the spool:
[tex]\[
13.6667 + 38.4 = 52.0667
\][/tex]
Thus, the total amount of twine on the spool originally was approximately [tex]\( 52.0667 \)[/tex] feet. Considering the options given, we can determine which option matches our calculated total most closely.
The total, [tex]\( 52.0667 \)[/tex], matches option C:
[tex]\[ 52 \frac{1}{15} \approx 52.0667 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{52 \frac{1}{15}} \][/tex]
