Short Answer 
To solve the problem, first, convert the mixed numbers 13 2/3 and 38 2/5 into improper fractions (41/3 and 192/5). Next, find a common denominator of 15, then add the fractions to get 781/15, which is converted back into the mixed number 52 11/15 feet. Thus, the total length of twine originally on the spool is 52 11/15 feet.
Step 1: Convert Mixed Numbers to Improper Fractions
To begin solving the problem, we need to convert the given mixed numbers into improper fractions for easier calculations. This involves two key conversions:
- For 13 2/3 feet, calculate: (13 * 3 + 2) / 3 = 41/3 feet.
- For 38 2/5 feet, calculate: (38 * 5 + 2) / 5 = 192/5 feet.
Step 2: Find a Common Denominator and Add Fractions
Next, we need to add the two fractions. This requires finding a common denominator. The least common multiple (LCM) of 3 and 5 is 15. We convert the fractions:
- Convert 41/3 to (41/3) * (5/5) = 205/15.
- Convert 192/5 to (192/5) * (3/3) = 576/15.
Now, we can add the fractions: (205/15) + (576/15) = (781/15).
Step 3: Convert Back to a Mixed Number
The final step is to convert our result back to a mixed number. Take the improper fraction 781/15 and divide:
- 781 divided by 15 equals 52 with a remainder of 11.
- This translates to the mixed number: 52 11/15.
Therefore, the total feet of twine originally on the spool were 52 1/15 feet.