Learn on PengiSaxon Math, Course 1Chapter 7: Fractions and Geometric Concepts

Lesson 67: Using Prime Factorization to Reduce Fractions

In this Grade 6 Saxon Math Course 1 lesson, students learn how to reduce fractions with large terms by writing the prime factorization of both the numerator and denominator, then canceling common prime factors. The lesson demonstrates this process step by step using examples like reducing 375/1000 by identifying and eliminating shared factors of 5. Practice problems reinforce the skill of applying prime factorization as a reliable strategy for simplifying fractions to lowest terms.

Section 1

📘 Using Prime Factorization to Reduce Fractions

New Concept

One way to reduce fractions with large terms is to factor the terms and then reduce the common factors. To reduce a fraction like 150750\frac{150}{750}, we can begin by writing the prime factorizations of 150 and 750.

150750=2⋅3⋅5⋅52⋅3⋅5⋅5⋅5\frac{150}{750} = \frac{2 \cdot 3 \cdot 5 \cdot 5}{2 \cdot 3 \cdot 5 \cdot 5 \cdot 5}

What’s next

This is just the foundation. Next, you'll apply this process in worked examples and practice problems to build your skill and confidence.

Section 2

Reducing Fractions With Prime Factors

Property

To reduce fractions with large terms, factor the terms into their prime factorizations. Then, reduce the common factors that appear in both the numerator and the denominator.

Examples

1251000=5⋅5⋅52⋅2⋅2⋅5⋅5⋅5=18 \frac{125}{1000} = \frac{5 \cdot 5 \cdot 5}{2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{1}{8}
3751000=3⋅5⋅5⋅52⋅2⋅2⋅5⋅5⋅5=38 \frac{375}{1000} = \frac{3 \cdot 5 \cdot 5 \cdot 5}{2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{3}{8}
3681=2⋅2⋅3⋅33⋅3⋅3⋅3=49 \frac{36}{81} = \frac{2 \cdot 2 \cdot 3 \cdot 3}{3 \cdot 3 \cdot 3 \cdot 3} = \frac{4}{9}

Explanation

Think of it like a puzzle! Breaking down big, scary numbers into their prime building blocks lets you spot matching pairs. Once you find a match on the top and bottom, poof! They cancel each other out, leaving you with a much simpler, tidier fraction. It’s the ultimate cleanup trick for numbers.

Section 3

Canceling Common Factors

Property

Any prime factor that appears in both the numerator and the denominator can be canceled. This is because any number divided by itself, such as 55 \frac{5}{5} , is equal to 1.

Examples

48400=2⋅2⋅2⋅2⋅32⋅2⋅2⋅2⋅5⋅5=2⋅2⋅2⋅2⋅32⋅2⋅2⋅2⋅5⋅5=325 \frac{48}{400} = \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5} = \frac{\cancel{2\cdot2\cdot2\cdot2} \cdot 3}{\cancel{2\cdot2\cdot2\cdot2} \cdot 5 \cdot 5} = \frac{3}{25}
125500=5⋅5⋅52⋅2⋅5⋅5⋅5=5⋅5⋅52⋅2⋅5⋅5⋅5=14 \frac{125}{500} = \frac{5 \cdot 5 \cdot 5}{2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{\cancel{5\cdot5\cdot5}}{2 \cdot 2 \cdot \cancel{5\cdot5\cdot5}} = \frac{1}{4}
3⋅5⋅5⋅52⋅2⋅2⋅5⋅5⋅5=32⋅2⋅2=38 \frac{3 \cdot \cancel{5} \cdot \cancel{5} \cdot \cancel{5}}{2 \cdot 2 \cdot 2 \cdot \cancel{5} \cdot \cancel{5} \cdot \cancel{5}} = \frac{3}{2 \cdot 2 \cdot 2} = \frac{3}{8}

Explanation

Think of common factors in a fraction as a tug-of-war. A '5' on top pulls against a '5' on the bottom, and they cancel each other out! This works because 55 \frac{5}{5} is just 1. Wiping out these matching pairs makes big fractions simple without changing their overall value.

Section 4

Taming Large Fractions

Property

For fractions with large terms, like 6251000 \frac{625}{1000} , prime factorization provides a systematic way to find all common factors and reduce the fraction to its simplest form, ensuring no common factors are missed.

Examples

6251000=5⋅5⋅5⋅52⋅2⋅2⋅5⋅5⋅5=58 \frac{625}{1000} = \frac{5 \cdot 5 \cdot 5 \cdot 5}{2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{5}{8}
8751000=5⋅5⋅5⋅72⋅2⋅2⋅5⋅5⋅5=78 \frac{875}{1000} = \frac{5 \cdot 5 \cdot 5 \cdot 7}{2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{7}{8}
144600=2⋅2⋅2⋅2⋅3⋅32⋅2⋅2⋅3⋅5⋅5=625 \frac{144}{600} = \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3}{2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5} = \frac{6}{25}

Explanation

Got a giant fraction? Prime factorization is a foolproof map. It breaks huge numbers into their prime parts, revealing all hidden common factors. This method ensures intimidating fractions like 8751000 \frac{875}{1000} are simplified perfectly to their smallest form, making tough problems much easier to solve with confidence.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

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Chapter 7: Fractions and Geometric Concepts

  1. Lesson 1

    Lesson 61: Adding Three or More Fractions

    LearnPractice
  2. Lesson 2

    Lesson 62: Writing Mixed Numbers as Improper Fractions

    LearnPractice
  3. Lesson 3

    Lesson 63: Subtracting Mixed Numbers with Regrouping, Part 2

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  4. Lesson 4

    Lesson 64: Classifying Quadrilaterals

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  5. Lesson 5

    Lesson 65: Prime Factorization

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  6. Lesson 6

    Lesson 66: Multiplying Mixed Numbers

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  7. Lesson 7Current

    Lesson 67: Using Prime Factorization to Reduce Fractions

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  8. Lesson 8

    Lesson 68: Dividing Mixed Numbers

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  9. Lesson 9

    Lesson 69: Lengths of Segments

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  10. Lesson 10

    Lesson 70: Reducing Fractions Before Multiplying

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  11. Lesson 11

    Investigation 7: The Coordinate Plane

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Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Using Prime Factorization to Reduce Fractions

New Concept

One way to reduce fractions with large terms is to factor the terms and then reduce the common factors. To reduce a fraction like 150750\frac{150}{750}, we can begin by writing the prime factorizations of 150 and 750.

150750=2⋅3⋅5⋅52⋅3⋅5⋅5⋅5\frac{150}{750} = \frac{2 \cdot 3 \cdot 5 \cdot 5}{2 \cdot 3 \cdot 5 \cdot 5 \cdot 5}

What’s next

This is just the foundation. Next, you'll apply this process in worked examples and practice problems to build your skill and confidence.

Section 2

Reducing Fractions With Prime Factors

Property

To reduce fractions with large terms, factor the terms into their prime factorizations. Then, reduce the common factors that appear in both the numerator and the denominator.

Examples

1251000=5⋅5⋅52⋅2⋅2⋅5⋅5⋅5=18 \frac{125}{1000} = \frac{5 \cdot 5 \cdot 5}{2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{1}{8}
3751000=3⋅5⋅5⋅52⋅2⋅2⋅5⋅5⋅5=38 \frac{375}{1000} = \frac{3 \cdot 5 \cdot 5 \cdot 5}{2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{3}{8}
3681=2⋅2⋅3⋅33⋅3⋅3⋅3=49 \frac{36}{81} = \frac{2 \cdot 2 \cdot 3 \cdot 3}{3 \cdot 3 \cdot 3 \cdot 3} = \frac{4}{9}

Explanation

Think of it like a puzzle! Breaking down big, scary numbers into their prime building blocks lets you spot matching pairs. Once you find a match on the top and bottom, poof! They cancel each other out, leaving you with a much simpler, tidier fraction. It’s the ultimate cleanup trick for numbers.

Section 3

Canceling Common Factors

Property

Any prime factor that appears in both the numerator and the denominator can be canceled. This is because any number divided by itself, such as 55 \frac{5}{5} , is equal to 1.

Examples

48400=2⋅2⋅2⋅2⋅32⋅2⋅2⋅2⋅5⋅5=2⋅2⋅2⋅2⋅32⋅2⋅2⋅2⋅5⋅5=325 \frac{48}{400} = \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5} = \frac{\cancel{2\cdot2\cdot2\cdot2} \cdot 3}{\cancel{2\cdot2\cdot2\cdot2} \cdot 5 \cdot 5} = \frac{3}{25}
125500=5⋅5⋅52⋅2⋅5⋅5⋅5=5⋅5⋅52⋅2⋅5⋅5⋅5=14 \frac{125}{500} = \frac{5 \cdot 5 \cdot 5}{2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{\cancel{5\cdot5\cdot5}}{2 \cdot 2 \cdot \cancel{5\cdot5\cdot5}} = \frac{1}{4}
3⋅5⋅5⋅52⋅2⋅2⋅5⋅5⋅5=32⋅2⋅2=38 \frac{3 \cdot \cancel{5} \cdot \cancel{5} \cdot \cancel{5}}{2 \cdot 2 \cdot 2 \cdot \cancel{5} \cdot \cancel{5} \cdot \cancel{5}} = \frac{3}{2 \cdot 2 \cdot 2} = \frac{3}{8}

Explanation

Think of common factors in a fraction as a tug-of-war. A '5' on top pulls against a '5' on the bottom, and they cancel each other out! This works because 55 \frac{5}{5} is just 1. Wiping out these matching pairs makes big fractions simple without changing their overall value.

Section 4

Taming Large Fractions

Property

For fractions with large terms, like 6251000 \frac{625}{1000} , prime factorization provides a systematic way to find all common factors and reduce the fraction to its simplest form, ensuring no common factors are missed.

Examples

6251000=5⋅5⋅5⋅52⋅2⋅2⋅5⋅5⋅5=58 \frac{625}{1000} = \frac{5 \cdot 5 \cdot 5 \cdot 5}{2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{5}{8}
8751000=5⋅5⋅5⋅72⋅2⋅2⋅5⋅5⋅5=78 \frac{875}{1000} = \frac{5 \cdot 5 \cdot 5 \cdot 7}{2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{7}{8}
144600=2⋅2⋅2⋅2⋅3⋅32⋅2⋅2⋅3⋅5⋅5=625 \frac{144}{600} = \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3}{2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5} = \frac{6}{25}

Explanation

Got a giant fraction? Prime factorization is a foolproof map. It breaks huge numbers into their prime parts, revealing all hidden common factors. This method ensures intimidating fractions like 8751000 \frac{875}{1000} are simplified perfectly to their smallest form, making tough problems much easier to solve with confidence.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 7: Fractions and Geometric Concepts

  1. Lesson 1

    Lesson 61: Adding Three or More Fractions

    LearnPractice
  2. Lesson 2

    Lesson 62: Writing Mixed Numbers as Improper Fractions

    LearnPractice
  3. Lesson 3

    Lesson 63: Subtracting Mixed Numbers with Regrouping, Part 2

    LearnPractice
  4. Lesson 4

    Lesson 64: Classifying Quadrilaterals

    LearnPractice
  5. Lesson 5

    Lesson 65: Prime Factorization

    LearnPractice
  6. Lesson 6

    Lesson 66: Multiplying Mixed Numbers

    LearnPractice
  7. Lesson 7Current

    Lesson 67: Using Prime Factorization to Reduce Fractions

    LearnPractice
  8. Lesson 8

    Lesson 68: Dividing Mixed Numbers

    LearnPractice
  9. Lesson 9

    Lesson 69: Lengths of Segments

    LearnPractice
  10. Lesson 10

    Lesson 70: Reducing Fractions Before Multiplying

    LearnPractice
  11. Lesson 11

    Investigation 7: The Coordinate Plane

    LearnPractice