GCF and LCM Problem Solving How can you tell if a word problem requires you to use Greatest Common Factor or Least Common Multiple to solve?

If it is a GCF Problem If it is an LCM Problem GCF and LCM Examples and Solutions

If it is a GCF Problem What is the question asking us? Do we have to split things into smaller sections? Are we trying to figure out the maximum number of people we can invite? Are we trying to arrange something into rows or groups?

GCF Example: Applying what we have learned… Samantha has two pieces of cloth. One piece is 72 inches wide and the other piece is 90 inches wide. She wants to cut both pieces into strips of equal width that are as wide as possible. How wide should she cut the strips?

K: The pieces of cloth are 72 and 90 inches wide. W: How wide should she cut the strips so that they are the largest possible equal lengths. L: This problem can be solved using Greatest Common Factor because we are cutting or “dividing” the strips of cloth into smaller pieces (factor) of 72 and 90. Find the GCF of 72 and 90

GCF Word Problem Solution 72 90 1x72 1x90 2x36 2x45 3x24 3x30 4x18 5x18 6x12 6x15 8x9 9x10 72 and 90 have common factors of 1, 2, 3, 6, 9, and 18 The GCF = 18 Samantha should cut each piece to be 18 inches wide

If it is an LCM Problem What is the question asking us? Do we have an event that is or will be repeating over and over? Will we have to purchase or get multiple items in order to have enough? Are we trying to figure out when something will happen again at the same time?

LCM Example: Applying what we have learned… Ben exercises every 12 days and Isabel every 8 days. Ben and Isabel both exercised today. How many days will it be until they exercise together again?

K: Ben exercises every 12 days and Isabel every 8 days and they both exercised today. W: How many days is it until they will both exercise on the same day again. L: This problem can be solved using Least Common Multiple. We are trying to figure out when will be the next time they are exercising together. Find the LCM of 12 and 8.

LCM Word Problem Solution 12 :12, 24, 36, 48, 60, 72, 84 8 :8, 16, 24, 32, 40, 48, 56, 64, 72 Common Factors of 12 and 8 are 24, 48, 72 The LCM = 24 Ben and Isabel would exercise on the same day every 24 days.

Question #1 You made 63 wheat dinner rolls, 45 rye dinner rolls, and 54 sourdough dinner rolls for a family picnic. You want to make up plates of rolls to set on the picnic table. If each plate is to contain the same amount of each type of roll, and there are no left over rolls, what is the greatest number of plates that can be made? How many wheat dinner rolls, rye dinner rolls, and sourdough dinner rolls are there on each plate? Answer We are finding the greatest number of groups so this is a GCF problem. Factors of 63: 1, 3, 7, 9, 21, 63 Factors of 45: 1, 3, 5, 9, 15, 45 Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54 We would need 9 plates and each plate would have 7 wheat rolls, 5 rye rolls, and 6 sourdough rolls.

Question #2 Rebecca has 20 table tennis balls and 16 table tennis paddles. She wants to sell packages of balls and paddles bundled together. What is the maximum number of packages she can sell with no leftover balls or paddles? Answer We are making equal groups so we use GCF. Factors of 20: 1, 2, 4, 5, 10, 20 Factors of 16: 1, 2, 4, 8, 16 The maximum number of packages will be 4.

Question 3 Eggs come in packages of 12 and English muffins come in packages of 10. What is the fewest number of packages of each that can be bought to be able to make egg sandwiches with no muffins or eggs left over? Answer We are finding the number needed of each in order to have the same amount of both so this is LCM. Multiples of 12: 12, 24, 36, 48, 60 Multiples of 10: 10, 20, 30, 40, 50, 60 We need 60 packages of each.

Question #4 Ming has 15 quarters, 30 dimes and 48 nickels. He wants to group his money so that each group has the same number of each coin. What is the greatest number of groups he can make? How many of each coin will be in each group? How much money will each group be worth? Answer We have to divide into smaller groups so we are using GCF. Factors of 15: 1, 3, 5, 15 Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Factors of 48: 1, 2, 4, 6, 8, 12, 24, 48 There will be 3 groups. Each group has 5 quarters, 10 dimes, 16 nickels. The value of each group is $3.05

Question #5 Three clocks ring once at the same time. After that, the first clock rings after every 90 minutes, the second after every 30 minutes, and third after every 60 minutes. After how many minutes will they again ring together? Answer Multiples of 90: 90, 180, 270, 360 Multiples of 30: 30, 60, 90, 120, 150, 180 Multiples of 60: 60, 120, 180 We have to determine when events will repeats so we use LCM. They will repeat again after 180 minutes.

Question #6 Mary wears a jacket every four days and her hat every five days. If she wears her jacket and hat on March 8th, what is the next day she will wear both her jacket and hat? Answer Multiples of 4: 4, 8, 12, 16, 20, 24 Multiples of 5: 5, 10, 15, 20, 25 LCM = 20 We need to determine when an event will repeat so we use LCM which is 20. Since she first wore it on March 8 we add 20 to this and the next time she will wear both the hat and the jacket will be March 28.