Understanding how to solve word problems involving the Greatest Common Factor (GCF) is a vital math skill that enhances your logical thinking and problem-solving abilities. GCF problems often appear in various contexts from basic arithmetic to algebra, offering a robust workout for your mental faculties. Let's dive into seven GCF word problems designed to boost your math skills, providing detailed step-by-step solutions to help you understand the process better.
Problem 1: Classroom Arrangements
Teacher Mia needs to arrange her class into groups with the largest possible size where all students are equally divided. If there are 24 students in the class, how many students should be in each group?
- Step 1: Identify the numbers: We need the GCF of the number of students, which is 24.
- Step 2: List all factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
- Step 3: The GCF is the largest factor, so Mia should place 12 students in each group.
📝 Note: The GCF is always 1 unless there is a common factor other than 1.
Problem 2: Manufacturing Chocolate Bars
A chocolate factory needs to produce chocolate bars from large slabs that measure 24 inches by 36 inches. What is the largest possible square-sized bar they can make without any waste?
- Step 1: Find the GCF of 24 and 36:
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- The largest common factor is 12.
- Step 2: The largest possible square bar size is 12 inches by 12 inches.
🍫 Note: This ensures no chocolate goes to waste, optimizing production.
Problem 3: Gold Coins Distribution
A treasure chest contains 18 large gold coins and 12 small gold coins. How many bags can you make with the same number of coins in each?
- Step 1: Identify the numbers: 18 and 12.
- Step 2: Find the GCF of 18 and 12:
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 12: 1, 2, 3, 4, 6, 12
- The GCF is 6.
- Step 3: Each bag will have 6 coins, making 3 bags from 18 coins and 2 bags from 12 coins.
Problem 4: River Crossing
A river-crossing puzzle involves using 15 oranges, 25 limes, and 30 cherries to make packs. Each pack must contain the same number of items. What is the largest number of items that can go into each pack?
- Step 1: Determine the GCF of 15, 25, and 30:
- Factors of 15: 1, 3, 5, 15
- Factors of 25: 1, 5, 25
- Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
- The GCF is 5.
- Step 2: Therefore, each pack can contain 5 items, making 3 packs of oranges, 5 packs of limes, and 6 packs of cherries.
Problem 5: Land Division
A farmer has two rectangular plots of land, one measuring 16 acres by 24 acres and another 32 acres by 48 acres. What is the largest possible equal-sized plot he can create?
- Step 1: Find the GCF of 16 and 24 for the first plot:
- Factors of 16: 1, 2, 4, 8, 16
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- The GCF is 8.
- Step 2: Repeat for the second plot: GCF of 32 and 48 is also 16.
- Step 3: The largest possible plot size is 8 acres by 8 acres.
Problem 6: Electrical Circuits
An electrical engineer needs to fit exactly 42 resistors and 63 capacitors onto a circuit board. How many identical circuits can he create?
- Step 1: Find the GCF of 42 and 63:
- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
- Factors of 63: 1, 3, 7, 9, 21, 63
- The GCF is 21.
- Step 2: Each circuit can have 21 components, making 2 identical circuits.
Problem 7: Lego Bricks
A child has 48 red, 72 blue, and 36 yellow Lego bricks to make identical towers. What is the largest number of bricks per tower?
- Step 1: Calculate the GCF of 48, 72, and 36:
- Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- The GCF is 12.
- Step 2: Therefore, each tower can have 12 bricks, making 4 red, 6 blue, and 3 yellow towers.
Mastering these GCF word problems teaches you to break down complex scenarios into more manageable parts. These skills not only enhance your math prowess but also provide a structured approach to solving real-life problems where resource allocation is key. Remember, GCF is not just a mathematical concept; it's a tool for understanding how things are grouped or divided efficiently.
Why is it important to find the GCF?
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Finding the GCF helps in problems where resources need to be divided equally, ensuring no excess or waste.
Can GCF be used in everyday life?
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Yes, GCF can be applied in scenarios like dividing a pizza among friends, scheduling, manufacturing, and budgeting.
What if numbers do not share a common factor?
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If numbers do not share a common factor, the GCF is 1, meaning they are relatively prime.
How does understanding GCF help in algebra?
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Understanding GCF simplifies expressions, helps in factoring, solving polynomial equations, and simplifying fractions.
