Final answer:
The Greatest Common Divisor of the rope lengths 140 cm, 168 cm, and 210 cm is 14 cm, resulting in the greatest possible length for each smaller piece. By dividing each original rope length by the GCD, we can find that Ali can get a total of 37 smaller pieces of rope altogether.
Step-by-step explanation:
The student's question deals with finding the greatest possible length of smaller rope pieces (the Greatest Common Divisor) and the number of such pieces that can be obtained from the given lengths of ropes. To find the Greatest Common Divisor (GCD), various methods such as prime factorization, the Euclidean algorithm, or by listing out the divisors can be used. However, for this problem, we can start by inspecting the smallest rope to find potential smaller lengths and then check them against the other two to find the largest common divisor.
- First, consider the factors of 140, which are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140.
- Next, check which of these factors divide both 168 and 210 without leaving any remainder.
- The greatest of these common factors is 14. Therefore, the greatest possible length of each smaller piece is 14 cm.
- To calculate the total number of smaller pieces, divide each rope length by the GCD:
- 140 cm / 14 cm = 10 pieces
- 168 cm / 14 cm = 12 pieces
- 210 cm / 14 cm = 15 pieces
Adding up these numbers gives us a total of 10 + 12 + 15 = 37 smaller pieces.