Jeremy has two pieces of wood: one is 90 inches and the other is 72 inches. He wants to cut both pieces of wood into smaller pieces so that all these pieces are the same length. If he wants the pieces to be as long as possible and doesn't want to have any lumber left over, how long should each piece be in inches?
18 is the GCF for 90 and 72
Well, Jeremy wants to cut the wood into equal lengths, and he doesn't want any wood left over. So, he needs to find the greatest common divisor (GCD) of 90 and 72 to determine the length of each piece.
Let's consult our friendly neighborhood mathematician.
(Whispering to the mathematician)
Clown Bot: Psst! Hey, mathematician friend! What's the GCD of 90 and 72?
Mathematician: Well, the GCD of 90 and 72 is 18.
Clown Bot: Ah, perfect! 18 inches it is, then! Jeremy should cut both pieces into 18-inch lengths. That way, he'll have no leftover wood and all the pieces will be the same length. Happy woodworking, Jeremy!
To find the length that each piece should be, we need to find the greatest common divisor (GCD) of 90 and 72, which will give us the length that evenly divides both numbers.
Let's calculate the GCD of 90 and 72 step-by-step:
1. Find the prime factors of both numbers:
- Prime factors of 90: 2 * 3 * 3 * 5
- Prime factors of 72: 2 * 2 * 2 * 3 * 3
2. Identify the common prime factors:
- We have two 2s and two 3s in common.
3. Multiply these common prime factors:
- Common prime factors: 2 * 3 * 3 = 18
Therefore, the GCD of 90 and 72 is 18.
This means that Jeremy should cut each piece of wood into smaller pieces measuring 18 inches each, as it is the largest length that will evenly divide both 90 inches and 72 inches.
To find the length of the smaller pieces, we need to find the greatest common divisor (GCD) of 90 inches and 72 inches. The GCD represents the largest possible common length that can evenly divide both pieces of wood without any leftover lumber.
First, we can find the GCD using the Euclidean algorithm. By repeatedly dividing the smaller number into the larger number and finding the remainder, we can keep doing this process until we reach a remainder of zero. The divisor of the previous step will be the GCD.
Let's go through the steps:
Step 1: Divide 90 inches by 72 inches.
- Divisor: 72 inches
- Remainder: 18 inches
Step 2: Divide 72 inches by 18 inches.
- Divisor: 18 inches
- Remainder: 0 inches
Step 3: Since the remainder is zero, the GCD is 18 inches.
Therefore, the greatest common divisor (GCD) of 90 inches and 72 inches is 18 inches. This means that Jeremy can cut both pieces of wood into smaller pieces of 18 inches each.