Learn on PengiYoshiwara Intermediate AlgebraChapter 8: Polynomial and Rational Functions

Lesson 4: More Operations on Fractions

In this Grade 7 lesson from Yoshiwara Intermediate Algebra, Chapter 8, students learn how to simplify complex fractions using two methods: rewriting them as division and multiplying by the LCD of all embedded fractions. The lesson also covers expressing algebraic expressions with negative exponents, such as x⁻¹ and x⁻², as standard algebraic fractions by applying reciprocal and LCD techniques.

Section 1

📘 More Operations on Fractions

New Concept

This lesson introduces complex fractions—fractions containing other fractions, like x+12x1\frac{x + \frac{1}{2}}{x - 1}. You'll learn powerful techniques, such as using the LCD and polynomial division, to simplify these expressions into a single, manageable fraction.

What’s next

You'll start by simplifying complex fractions using our interactive examples. Then, you will test your understanding with a series of practice cards.

Section 2

Complex Fractions

Property

A fraction that contains one or more fractions in either its numerator or its denominator or both is called a complex fraction. Like simple fractions, complex fractions represent quotients. For example, 2356=23÷56\frac{\frac{2}{3}}{\frac{5}{6}} = \frac{2}{3} \div \frac{5}{6}.

To simplify a complex fraction:

  1. Find the LCD of all the fractions contained in the complex fraction.
  2. Multiply the numerator and the denominator of the complex fraction by the LCD.
  3. Reduce the resulting simple fraction, if possible.

Examples

  • To simplify 3478\frac{\frac{3}{4}}{\frac{7}{8}}, we can treat it as a division: 34÷78=3487=2428=67\frac{3}{4} \div \frac{7}{8} = \frac{3}{4} \cdot \frac{8}{7} = \frac{24}{28} = \frac{6}{7}.

Section 3

Negative Exponents in Fractions

Property

Algebraic fractions are sometimes written using negative exponents. To simplify, first convert the negative exponents into fractions. For example, x1y1=1x1yx^{-1} - y^{-1} = \frac{1}{x} - \frac{1}{y}. Then combine the fractions using a common denominator.

Caution:

  1. A difference of fractions is not the fraction of the difference: 1x1y1xy\frac{1}{x} - \frac{1}{y} \neq \frac{1}{x - y}.
  2. The power of a sum is not the sum of the powers: (a+b)nan+bn(a + b)^n \neq a^n + b^n. Therefore, (x2+y2)1x2+y2(x^{-2} + y^{-2})^{-1} \neq x^2 + y^2.

Examples

  • To simplify a1+3b1a^{-1} + 3b^{-1}, first rewrite as fractions: 1a+3b\frac{1}{a} + \frac{3}{b}. The LCD is abab, so we get bab+3aab=b+3aab\frac{b}{ab} + \frac{3a}{ab} = \frac{b + 3a}{ab}.

Section 4

Average Speed on a Two-Part Trip

Property

The average speed for a trip is the total distance divided by the total time. For a two-part trip with distances d1d_1 and d2d_2 traveled at rates r1r_1 and r2r_2, the times for each part are t1=d1r1t_1 = \frac{d_1}{r_1} and t2=d2r2t_2 = \frac{d_2}{r_2}.

The average speed for the entire trip is given by the formula:

Average speed=Total distanceTotal time=d1+d2d1r1+d2r2\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{d_1 + d_2}{\frac{d_1}{r_1} + \frac{d_2}{r_2}}

Examples

  • A car travels 150 miles at 50 mph and returns the same distance at 30 mph. Total distance is 300 miles. Total time is 15050+15030=3+5=8\frac{150}{50} + \frac{150}{30} = 3 + 5 = 8 hours. Average speed is 3008=37.5\frac{300}{8} = 37.5 mph.

Section 5

Polynomial Division

Property

An algebraic fraction is 'improper' if the degree of the numerator is greater than or equal to the degree of the denominator. We can simplify it by division. The result is the sum of a polynomial and a simpler fraction.

If the divisor is a monomial, divide it into each term of the numerator:

9x36x2+43x=9x33x6x23x+43x=3x22x+43x\frac{9x^3 - 6x^2 + 4}{3x} = \frac{9x^3}{3x} - \frac{6x^2}{3x} + \frac{4}{3x} = 3x^2 - 2x + \frac{4}{3x}

If the divisor is a polynomial, use a method similar to long division. The final answer is expressed as:

Quotient+RemainderDivisor\text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}

Book overview

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

Continue this chapter

Chapter 8: Polynomial and Rational Functions

  1. Lesson 1

    Lesson 1: Polynomial Functions

  2. Lesson 2

    Lesson 2: Algebraic Fractions

  3. Lesson 3

    Lesson 3: Operations on Algebraic Fractions

  4. Lesson 4Current

    Lesson 4: More Operations on Fractions

  5. Lesson 5

    Lesson 5: Equations with Fractions

  6. Lesson 6

    Lesson 6: Chapter Summary and Review

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 More Operations on Fractions

New Concept

This lesson introduces complex fractions—fractions containing other fractions, like x+12x1\frac{x + \frac{1}{2}}{x - 1}. You'll learn powerful techniques, such as using the LCD and polynomial division, to simplify these expressions into a single, manageable fraction.

What’s next

You'll start by simplifying complex fractions using our interactive examples. Then, you will test your understanding with a series of practice cards.

Section 2

Complex Fractions

Property

A fraction that contains one or more fractions in either its numerator or its denominator or both is called a complex fraction. Like simple fractions, complex fractions represent quotients. For example, 2356=23÷56\frac{\frac{2}{3}}{\frac{5}{6}} = \frac{2}{3} \div \frac{5}{6}.

To simplify a complex fraction:

  1. Find the LCD of all the fractions contained in the complex fraction.
  2. Multiply the numerator and the denominator of the complex fraction by the LCD.
  3. Reduce the resulting simple fraction, if possible.

Examples

  • To simplify 3478\frac{\frac{3}{4}}{\frac{7}{8}}, we can treat it as a division: 34÷78=3487=2428=67\frac{3}{4} \div \frac{7}{8} = \frac{3}{4} \cdot \frac{8}{7} = \frac{24}{28} = \frac{6}{7}.

Section 3

Negative Exponents in Fractions

Property

Algebraic fractions are sometimes written using negative exponents. To simplify, first convert the negative exponents into fractions. For example, x1y1=1x1yx^{-1} - y^{-1} = \frac{1}{x} - \frac{1}{y}. Then combine the fractions using a common denominator.

Caution:

  1. A difference of fractions is not the fraction of the difference: 1x1y1xy\frac{1}{x} - \frac{1}{y} \neq \frac{1}{x - y}.
  2. The power of a sum is not the sum of the powers: (a+b)nan+bn(a + b)^n \neq a^n + b^n. Therefore, (x2+y2)1x2+y2(x^{-2} + y^{-2})^{-1} \neq x^2 + y^2.

Examples

  • To simplify a1+3b1a^{-1} + 3b^{-1}, first rewrite as fractions: 1a+3b\frac{1}{a} + \frac{3}{b}. The LCD is abab, so we get bab+3aab=b+3aab\frac{b}{ab} + \frac{3a}{ab} = \frac{b + 3a}{ab}.

Section 4

Average Speed on a Two-Part Trip

Property

The average speed for a trip is the total distance divided by the total time. For a two-part trip with distances d1d_1 and d2d_2 traveled at rates r1r_1 and r2r_2, the times for each part are t1=d1r1t_1 = \frac{d_1}{r_1} and t2=d2r2t_2 = \frac{d_2}{r_2}.

The average speed for the entire trip is given by the formula:

Average speed=Total distanceTotal time=d1+d2d1r1+d2r2\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{d_1 + d_2}{\frac{d_1}{r_1} + \frac{d_2}{r_2}}

Examples

  • A car travels 150 miles at 50 mph and returns the same distance at 30 mph. Total distance is 300 miles. Total time is 15050+15030=3+5=8\frac{150}{50} + \frac{150}{30} = 3 + 5 = 8 hours. Average speed is 3008=37.5\frac{300}{8} = 37.5 mph.

Section 5

Polynomial Division

Property

An algebraic fraction is 'improper' if the degree of the numerator is greater than or equal to the degree of the denominator. We can simplify it by division. The result is the sum of a polynomial and a simpler fraction.

If the divisor is a monomial, divide it into each term of the numerator:

9x36x2+43x=9x33x6x23x+43x=3x22x+43x\frac{9x^3 - 6x^2 + 4}{3x} = \frac{9x^3}{3x} - \frac{6x^2}{3x} + \frac{4}{3x} = 3x^2 - 2x + \frac{4}{3x}

If the divisor is a polynomial, use a method similar to long division. The final answer is expressed as:

Quotient+RemainderDivisor\text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}

Book overview

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

Continue this chapter

Chapter 8: Polynomial and Rational Functions

  1. Lesson 1

    Lesson 1: Polynomial Functions

  2. Lesson 2

    Lesson 2: Algebraic Fractions

  3. Lesson 3

    Lesson 3: Operations on Algebraic Fractions

  4. Lesson 4Current

    Lesson 4: More Operations on Fractions

  5. Lesson 5

    Lesson 5: Equations with Fractions

  6. Lesson 6

    Lesson 6: Chapter Summary and Review