Final answer:
The greatest length you can cut both ropes into so all pieces will be the same length with none left over is 14 feet, as it is the greatest common divisor of 98 feet and 56 feet.
Step-by-step explanation:
The problem you're describing involves finding the greatest common divisor (GCD) for the two lengths of rope. This will determine the longest possible length that you can cut both ropes into, with none left over.
To find the GCD, you can use the Euclidean algorithm which involves a series of divisions:
Divide the longer length (98 feet) by the shorter length (56 feet).
If there's no remainder, the divisor (56 feet) is the GCD.
If there is a remainder, divide the divisor (56 feet) by this remainder.
Continue this process, with the remainder each time becoming the new divisor until you get a remainder of 0. The last non-zero remainder is the GCD.
In this case:
98 ÷ 56 = 1 remainder 42
56 ÷ 42 = 1 remainder 14
42 ÷ 14 = 3 remainder 0
Since 14 divides into 42 without a remainder, 14 feet is the longest piece of rope you can cut both ropes into so all pieces will be the same length with none left over.