Learn on PengiSaxon Algebra 2Chapter 5: Lessons 41-50, Investigation 5

Lesson 48: Understanding Complex Fractions

In Lesson 48 of Saxon Algebra 2, Grade 10 students learn to simplify complex fractions — fractions containing fractions in their numerator or denominator — using two methods: rewriting the numerator and denominator as single terms before dividing, or multiplying through by the least common denominator (LCD) of all fractions present. The lesson covers both numerical and algebraic complex fractions, including expressions with variables, and applies simplification techniques to a real-world volume formula involving a frustum of a cone.

Section 1

📘 Understanding Complex Fractions

New Concept

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Why it matters

Mastering complex fractions teaches a core principle of advanced mathematics: simplifying intricate structures into elegant, solvable forms. This skill is foundational for tackling multi-step problems in fields ranging from engineering to economics.

What’s next

Next, you'll learn two powerful methods to simplify these fractions and solve problems.

Section 2

Complex fraction

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

x3x+1\frac{\frac{x}{3}}{x+1}
2+1a2b+5\frac{2+\frac{1}{a}}{2b+5}
y+3yy−14y\frac{y+\frac{3}{y}}{y-\frac{1}{4y}}

Imagine a fraction throwing a party, and its guests are other fractions! That's a complex fraction—a fraction inside another fraction. It looks like a multi-level puzzle, but don't worry. Our goal is to flatten this 'fraction-ception' into a single, simple fraction. It’s really just a fancy division problem waiting to be solved and simplified.

Section 3

Simplifying using division

Simplify as needed to write the numerator and the denominator each as a single term. Then divide.

14+132+15=3+41210+15=712115=712⋅511=35132\frac{\frac{1}{4} + \frac{1}{3}}{2 + \frac{1}{5}} = \frac{\frac{3+4}{12}}{\frac{10+1}{5}} = \frac{\frac{7}{12}}{\frac{11}{5}} = \frac{7}{12} \cdot \frac{5}{11} = \frac{35}{132}
xy−2xy+2=x−2yyx+2yy=x−2yy⋅yx+2y=x−2yx+2y\frac{\frac{x}{y} - 2}{\frac{x}{y} + 2} = \frac{\frac{x-2y}{y}}{\frac{x+2y}{y}} = \frac{x-2y}{y} \cdot \frac{y}{x+2y} = \frac{x-2y}{x+2y}

Section 4

Simplifying using the LCD

Multiply the numerator and the denominator by the Least Common Denominator (LCD) of all the fractions in the complex fraction. Then simplify.

5+2x4−1y=(5+2x)⋅xy(4−1y)⋅xy=5xy+2y4xy−x\frac{5 + \frac{2}{x}}{4 - \frac{1}{y}} = \frac{(5 + \frac{2}{x}) \cdot xy}{(4 - \frac{1}{y}) \cdot xy} = \frac{5xy + 2y}{4xy - x}
4a+1a−12aa−1=(4a+1a−1)⋅a(a−1)(2aa−1)⋅a(a−1)=4(a−1)+a2a⋅a=5a−42a2\frac{\frac{4}{a} + \frac{1}{a-1}}{\frac{2a}{a-1}} = \frac{(\frac{4}{a} + \frac{1}{a-1}) \cdot a(a-1)}{(\frac{2a}{a-1}) \cdot a(a-1)} = \frac{4(a-1) + a}{2a \cdot a} = \frac{5a-4}{2a^2}

Section 5

Do not divide out factors of individual terms

You cannot simplify by canceling parts of terms that are being added or subtracted. For example,

4ax+3x2ax−x≠4+3x2−x\frac{4ax + 3x}{2ax - x} \neq \frac{4+3x}{2-x}

Wrong:

6x+126≠x+12\frac{6x+12}{6} \neq x+12
. Correct:
6(x+2)6=x+2\frac{6(x+2)}{6} = x+2
Wrong:
x2−16x−4≠x−4\frac{x^2-16}{x-4} \neq x-4
. Correct:
(x−4)(x+4)x−4=x+4\frac{(x-4)(x+4)}{x-4} = x+4

Hold on! You can't just slash parts of terms connected by addition or subtraction. It’s like trying to cancel the 'utter' out of 'butterfly' and 'mutter' – it makes no sense! To simplify correctly, you must first factor the entire numerator and denominator. Only then can you cancel out common factors that are being multiplied.

Book overview

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Chapter 5: Lessons 41-50, Investigation 5

  1. Lesson 1

    Lesson 41: Using the Pythagorean Theorem and the Distance Formula (Exploration: Visualizing the Pythagorean Theorem)

    LearnPractice
  2. Lesson 2

    LAB 7: Graphing Calculator: Calculating Permutations and Combinations

    LearnPractice
  3. Lesson 3

    Lesson 42: Finding Permutations and Combinations (Exploration: Pascal's Triangle and Combinations)

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  4. Lesson 4

    Lesson 43: Solving Systems of Linear Inequalities

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  5. Lesson 5

    Lesson 44: Rationalizing Denominators

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  6. Lesson 6

    LAB 8: Graphing Calculator: Applying Linear and Median Regression

    LearnPractice
  7. Lesson 7

    Lesson 45: Finding the Line of Best Fit (Exploration: Collecting and Analyzing Data)

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  8. Lesson 8

    Lesson 46: Finding Trigonometric Functions and their Reciprocals

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  9. Lesson 9

    Lesson 47: Graphing Exponential Functions

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  10. Lesson 10Current

    Lesson 48: Understanding Complex Fractions

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  11. Lesson 11

    Lesson 49: Using the Binomial Theorem

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  12. Lesson 12

    Lesson 50: Finding Inverses of Relations and Functions (Exploration: Graphing a Function and its Inverse)

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  13. Lesson 13

    Investigation 5: Finding the Binomial Distribution

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Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Understanding Complex Fractions

New Concept

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Why it matters

Mastering complex fractions teaches a core principle of advanced mathematics: simplifying intricate structures into elegant, solvable forms. This skill is foundational for tackling multi-step problems in fields ranging from engineering to economics.

What’s next

Next, you'll learn two powerful methods to simplify these fractions and solve problems.

Section 2

Complex fraction

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

x3x+1\frac{\frac{x}{3}}{x+1}
2+1a2b+5\frac{2+\frac{1}{a}}{2b+5}
y+3yy−14y\frac{y+\frac{3}{y}}{y-\frac{1}{4y}}

Imagine a fraction throwing a party, and its guests are other fractions! That's a complex fraction—a fraction inside another fraction. It looks like a multi-level puzzle, but don't worry. Our goal is to flatten this 'fraction-ception' into a single, simple fraction. It’s really just a fancy division problem waiting to be solved and simplified.

Section 3

Simplifying using division

Simplify as needed to write the numerator and the denominator each as a single term. Then divide.

14+132+15=3+41210+15=712115=712⋅511=35132\frac{\frac{1}{4} + \frac{1}{3}}{2 + \frac{1}{5}} = \frac{\frac{3+4}{12}}{\frac{10+1}{5}} = \frac{\frac{7}{12}}{\frac{11}{5}} = \frac{7}{12} \cdot \frac{5}{11} = \frac{35}{132}
xy−2xy+2=x−2yyx+2yy=x−2yy⋅yx+2y=x−2yx+2y\frac{\frac{x}{y} - 2}{\frac{x}{y} + 2} = \frac{\frac{x-2y}{y}}{\frac{x+2y}{y}} = \frac{x-2y}{y} \cdot \frac{y}{x+2y} = \frac{x-2y}{x+2y}

Section 4

Simplifying using the LCD

Multiply the numerator and the denominator by the Least Common Denominator (LCD) of all the fractions in the complex fraction. Then simplify.

5+2x4−1y=(5+2x)⋅xy(4−1y)⋅xy=5xy+2y4xy−x\frac{5 + \frac{2}{x}}{4 - \frac{1}{y}} = \frac{(5 + \frac{2}{x}) \cdot xy}{(4 - \frac{1}{y}) \cdot xy} = \frac{5xy + 2y}{4xy - x}
4a+1a−12aa−1=(4a+1a−1)⋅a(a−1)(2aa−1)⋅a(a−1)=4(a−1)+a2a⋅a=5a−42a2\frac{\frac{4}{a} + \frac{1}{a-1}}{\frac{2a}{a-1}} = \frac{(\frac{4}{a} + \frac{1}{a-1}) \cdot a(a-1)}{(\frac{2a}{a-1}) \cdot a(a-1)} = \frac{4(a-1) + a}{2a \cdot a} = \frac{5a-4}{2a^2}

Section 5

Do not divide out factors of individual terms

You cannot simplify by canceling parts of terms that are being added or subtracted. For example,

4ax+3x2ax−x≠4+3x2−x\frac{4ax + 3x}{2ax - x} \neq \frac{4+3x}{2-x}

Wrong:

6x+126≠x+12\frac{6x+12}{6} \neq x+12
. Correct:
6(x+2)6=x+2\frac{6(x+2)}{6} = x+2
Wrong:
x2−16x−4≠x−4\frac{x^2-16}{x-4} \neq x-4
. Correct:
(x−4)(x+4)x−4=x+4\frac{(x-4)(x+4)}{x-4} = x+4

Hold on! You can't just slash parts of terms connected by addition or subtraction. It’s like trying to cancel the 'utter' out of 'butterfly' and 'mutter' – it makes no sense! To simplify correctly, you must first factor the entire numerator and denominator. Only then can you cancel out common factors that are being multiplied.

Book overview

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

Continue this chapter

Chapter 5: Lessons 41-50, Investigation 5

  1. Lesson 1

    Lesson 41: Using the Pythagorean Theorem and the Distance Formula (Exploration: Visualizing the Pythagorean Theorem)

    LearnPractice
  2. Lesson 2

    LAB 7: Graphing Calculator: Calculating Permutations and Combinations

    LearnPractice
  3. Lesson 3

    Lesson 42: Finding Permutations and Combinations (Exploration: Pascal's Triangle and Combinations)

    LearnPractice
  4. Lesson 4

    Lesson 43: Solving Systems of Linear Inequalities

    LearnPractice
  5. Lesson 5

    Lesson 44: Rationalizing Denominators

    LearnPractice
  6. Lesson 6

    LAB 8: Graphing Calculator: Applying Linear and Median Regression

    LearnPractice
  7. Lesson 7

    Lesson 45: Finding the Line of Best Fit (Exploration: Collecting and Analyzing Data)

    LearnPractice
  8. Lesson 8

    Lesson 46: Finding Trigonometric Functions and their Reciprocals

    LearnPractice
  9. Lesson 9

    Lesson 47: Graphing Exponential Functions

    LearnPractice
  10. Lesson 10Current

    Lesson 48: Understanding Complex Fractions

    LearnPractice
  11. Lesson 11

    Lesson 49: Using the Binomial Theorem

    LearnPractice
  12. Lesson 12

    Lesson 50: Finding Inverses of Relations and Functions (Exploration: Graphing a Function and its Inverse)

    LearnPractice
  13. Lesson 13

    Investigation 5: Finding the Binomial Distribution

    LearnPractice