Maximizing Rope Pieces with GCD and LCM: A Comprehensive Guide

Andy has two ropes, one 120 feet long and the other 96 feet long. He wants to cut these ropes into equal pieces, with each piece being as long as possible. This article will walk you through the process of finding the maximum number of equal-length pieces he can get, using both the GCD (Greatest Common Divisor) and LCM (Least Common Multiple) methods.

Using the GCD Method

Step-by-Step Calculation Using the Euclidean Algorithm

To find the maximum number of equal-length pieces, we first need to determine the GCD of 120 and 96. We use the Euclidean algorithm for this purpose.

First iteration: GCD(120, 96)  GCD(96, 120 mod 96)  GCD(96, 24)Second iteration: GCD(96, 24)  GCD(24, 96 mod 24)  GCD(24, 0)

The GCD of 120 and 96 is 24. Now, we divide each rope length by this GCD to find the number of pieces.

For the 120-foot rope:

(frac{120}{24} 5 ) pieces

For the 96-foot rope:

(frac{96}{24} 4 ) pieces

Therefore, Andy can get a total of 5 4 9 equal-length pieces by cutting the two ropes in this way.

Alternative Calculation for Equal Pieces

Alternatively, we can calculate the GCF (Greatest Common Factor) between 120 and 96. We break down the numbers into their prime factors:

120 4 x 3 x 5 x 2

96 4 x 4 x 2

The GCF is 4 x 2 8. Now we calculate the total length of both ropes combined:

120 96 216

216 ÷ 8 27 pieces, each 8 feet long.

Using the LCM Method

Calculating LCM to Find the Maximum Equal Length

Alternatively, we can use the LCM (Least Common Multiple) method to find the maximum length of the equal pieces.

First, we find the LCM of 120 and 96:

LCM of 120 and 96 is 240.

Next, we find how many pieces each rope can be divided into using this length:

120 ÷ 24 5 pieces

96 ÷ 24 4 pieces

Total pieces 5 4 9 pieces

Method to Find the LCM

The LCM can be found using the prime factorization method as well:

The LCM of 120 and 96 is 240. Therefore, the rope pieces will be 240 feet long, and the total number of pieces is 9.

Conclusion

By either method, Andy can get 9 equal-length pieces, with each piece being 24 feet long. This ensures that all pieces are as long as possible, maximizing the efficiency of the cutting process.