137e. Multiplying and Dividing Algebraic Fractions

Learning Intentions

• To understand that the rules of multiplying and dividing fractions extend to algebraic fractions
• multiply algebraic fractions and simplify the result
• divide algebraic fractions and simplify the result

Pre-requisite Summary

• Know that an algebraic fraction is a fraction containing pronumerals in the numerator, denominator, or both
• Understand that multiplying fractions involves multiplying numerators together and denominators together
• Understand that dividing fractions involves multiplying by the reciprocal
• Be able to simplify numerical fractions
• Be able to simplify algebraic expressions such as $ab÷a=b$
• Know that common factors can be cancelled only when they are factors of the whole numerator and denominator

Worked Examples

Worked Example 1

Use the same rule as numerical fractions to multiply:

a) ${\displaystyle \frac{x}{3}}\times {\displaystyle \frac{2}{5}}$
b) ${\displaystyle \frac{a}{4}}\times {\displaystyle \frac{3b}{7}}$

Worked Example 2

Multiply the algebraic fractions and simplify:

a) ${\displaystyle \frac{2x}{3}}\times {\displaystyle \frac{9}{4}}$
b) ${\displaystyle \frac{5a}{6}}\times {\displaystyle \frac{3b}{10}}$
c) ${\displaystyle \frac{4m}{15}}\times {\displaystyle \frac{5}{2m}}$

Worked Example 3

Multiply the algebraic fractions and simplify:

a) ${\displaystyle \frac{x}{8}}\times {\displaystyle \frac{4x}{3}}$
b) ${\displaystyle \frac{7p}{9q}}\times {\displaystyle \frac{3q}{14}}$
c) ${\displaystyle \frac{2a}{5b}}\times {\displaystyle \frac{15b}{4a}}$

Worked Example 4

Use the reciprocal to divide:

a) ${\displaystyle \frac{x}{3}}÷{\displaystyle \frac{2}{5}}$
b) ${\displaystyle \frac{a}{4}}÷{\displaystyle \frac{b}{7}}$

Worked Example 5

Divide the algebraic fractions and simplify:

a) ${\displaystyle \frac{3x}{4}}÷{\displaystyle \frac{9}{8}}$
b) ${\displaystyle \frac{5a}{6}}÷{\displaystyle \frac{10b}{3}}$
c) ${\displaystyle \frac{7m}{12n}}÷{\displaystyle \frac{14m}{3n}}$

Worked Example 6

Divide the algebraic fractions and simplify:

a) ${\displaystyle \frac{x}{5}}÷{\displaystyle \frac{x}{10}}$
b) ${\displaystyle \frac{4p}{9q}}÷{\displaystyle \frac{2p}{3q}}$
c) ${\displaystyle \frac{3a}{8b}}÷{\displaystyle \frac{9a}{16b}}$

Problems

Problem 1

Use the same rule as numerical fractions to multiply:

a) ${\displaystyle \frac{y}{2}}\times {\displaystyle \frac{3}{4}}$
b) ${\displaystyle \frac{m}{5}}\times {\displaystyle \frac{2n}{7}}$

Problem 2

Multiply the algebraic fractions and simplify:

a) ${\displaystyle \frac{3x}{4}}\times {\displaystyle \frac{8}{9}}$
b) ${\displaystyle \frac{4a}{5}}\times {\displaystyle \frac{10b}{12}}$
c) ${\displaystyle \frac{6m}{7}}\times {\displaystyle \frac{14}{9m}}$

Problem 3

Multiply the algebraic fractions and simplify:

a) ${\displaystyle \frac{y}{6}}\times {\displaystyle \frac{3y}{2}}$
b) ${\displaystyle \frac{5p}{8q}}\times {\displaystyle \frac{4q}{15}}$
c) ${\displaystyle \frac{3a}{7b}}\times {\displaystyle \frac{14b}{9a}}$

Problem 4

Use the reciprocal to divide:

a) ${\displaystyle \frac{y}{2}}÷{\displaystyle \frac{3}{4}}$
b) ${\displaystyle \frac{m}{5}}÷{\displaystyle \frac{n}{7}}$

Problem 5

Divide the algebraic fractions and simplify:

a) ${\displaystyle \frac{5x}{6}}÷{\displaystyle \frac{15}{12}}$
b) ${\displaystyle \frac{3a}{4}}÷{\displaystyle \frac{9b}{2}}$
c) ${\displaystyle \frac{8m}{15n}}÷{\displaystyle \frac{4m}{3n}}$

Problem 6

Divide the algebraic fractions and simplify:

a) ${\displaystyle \frac{y}{4}}÷{\displaystyle \frac{y}{8}}$
b) ${\displaystyle \frac{6p}{11q}}÷{\displaystyle \frac{3p}{2q}}$
c) ${\displaystyle \frac{5a}{12b}}÷{\displaystyle \frac{15a}{24b}}$

Exercises

Understanding and Fluency

1. Complete each statement:
   a) To multiply fractions, multiply the ______ and multiply the ______
   b) To divide by a fraction, multiply by its ______
   c) The same multiplication and division rules for numerical fractions also apply to ______ fractions

2. Multiply and simplify:
   a) ${\displaystyle \frac{x}{2}}\times {\displaystyle \frac{3}{5}}$
   b) ${\displaystyle \frac{2a}{3}}\times {\displaystyle \frac{9b}{4}}$
   c) ${\displaystyle \frac{5m}{8}}\times {\displaystyle \frac{4}{15m}}$

3. Multiply and simplify:
   a) ${\displaystyle \frac{y}{7}}\times {\displaystyle \frac{14y}{3}}$
   b) ${\displaystyle \frac{6p}{11q}}\times {\displaystyle \frac{22q}{9}}$
   c) ${\displaystyle \frac{4a}{9b}}\times {\displaystyle \frac{3b}{8a}}$

4. Multiply and simplify:
   a) ${\displaystyle \frac{3x}{10}}\times {\displaystyle \frac{5x}{6}}$
   b) ${\displaystyle \frac{7m}{12n}}\times {\displaystyle \frac{8n}{21m}}$
   c) ${\displaystyle \frac{2p}{5}}\times {\displaystyle \frac{15}{4p}}$

5. Divide and simplify:
   a) ${\displaystyle \frac{x}{3}}÷{\displaystyle \frac{2}{7}}$
   b) ${\displaystyle \frac{4a}{5}}÷{\displaystyle \frac{8b}{15}}$
   c) ${\displaystyle \frac{9m}{14n}}÷{\displaystyle \frac{3m}{7n}}$

6. Divide and simplify:
   a) ${\displaystyle \frac{y}{6}}÷{\displaystyle \frac{y}{12}}$
   b) ${\displaystyle \frac{5p}{8q}}÷{\displaystyle \frac{15p}{16q}}$
   c) ${\displaystyle \frac{7a}{9b}}÷{\displaystyle \frac{14a}{27b}}$

7. Simplify each result:
   a) ${\displaystyle \frac{2x}{3}}\times {\displaystyle \frac{12}{5x}}$
   b) ${\displaystyle \frac{3a}{4b}}\times {\displaystyle \frac{8b}{9a}}$
   c) ${\displaystyle \frac{6m}{11n}}÷{\displaystyle \frac{3m}{22n}}$
   d) ${\displaystyle \frac{4p}{7}}÷{\displaystyle \frac{2p}{21}}$

8. Decide whether to multiply directly or use a reciprocal, then simplify:
   a) ${\displaystyle \frac{x}{4}}\times {\displaystyle \frac{6}{x}}$
   b) ${\displaystyle \frac{3a}{10}}÷{\displaystyle \frac{9a}{20}}$
   c) ${\displaystyle \frac{5m}{12n}}\times {\displaystyle \frac{18n}{25m}}$
   d) ${\displaystyle \frac{7y}{8}}÷{\displaystyle \frac{14y}{3}}$

Reasoning

9. Explain why ${\displaystyle \frac{2a}{5}}\times {\displaystyle \frac{3b}{4}}$ can be multiplied in the same way as ${\displaystyle \frac{2}{5}}\times {\displaystyle \frac{3}{4}}$.

10. A student says that to divide algebraic fractions, you divide the numerators and divide the denominators separately. Explain the mistake.

11. Noah says that ${\displaystyle \frac{x}{3}}\times {\displaystyle \frac{6}{x}}={\displaystyle \frac{6x}{3x}}$ cannot be simplified because it contains pronumerals. Is he correct? Explain.

12. Explain why dividing by ${\displaystyle \frac{2a}{5}}$ is the same as multiplying by ${\displaystyle \frac{5}{2a}}$.

13. A student simplifies ${\displaystyle \frac{3a+2}{a}}$ by cancelling the $a$. Describe the error.

Problem-solving

14. A formula contains the product ${\displaystyle \frac{3x}{4}}\times {\displaystyle \frac{8}{9x}}$. Simplify the expression.

15. A scale factor in a design is written as ${\displaystyle \frac{5a}{6b}}÷{\displaystyle \frac{10a}{9b}}$. Simplify the scale factor.

16. A student combines two rates using ${\displaystyle \frac{2m}{7}}\times {\displaystyle \frac{21}{4m}}$. Write the simplified result.

17. A science formula uses ${\displaystyle \frac{4p}{9q}}÷{\displaystyle \frac{2p}{3q}}$. Simplify the expression.

18. A pattern rule is changed by multiplying ${\displaystyle \frac{x}{5}}$ by ${\displaystyle \frac{15x}{2}}$. Simplify the result.

19. A quantity is adjusted by dividing ${\displaystyle \frac{7a}{12b}}$ by ${\displaystyle \frac{14a}{18b}}$. Simplify the final expression.

Potential Misunderstandings

• Students may think algebraic fractions follow different multiplication and division rules from numerical fractions
• Students may forget to multiply by the reciprocal when dividing
• Students may cancel terms that are not common factors of the whole numerator and denominator
• Students may cancel across addition, for example in expressions like ${\displaystyle \frac{x+2}{x}}$
• Students may multiply the numerators correctly but forget to multiply the denominators
• Students may simplify the numerical part but not the pronumeral part, or vice versa
• Students may think pronumerals cannot be cancelled
• Students may reverse the wrong fraction when dividing
• Students may make sign errors when simplifying