136e. Algebraic Fractions and Common Denominators

Learning Intentions

• To understand what an algebraic fraction is
• find the lowest common denominator of two algebraic fractions
• find equivalent algebraic fractions with different denominators
• add and subtract algebraic fractions and simplify the result

Pre-requisite Summary

• Know that a fraction has a numerator and a denominator
• Understand that a pronumeral can stand for a number
• Be able to simplify numerical fractions
• Know how to find the lowest common multiple of two numbers
• Understand that equivalent fractions have the same value
• Be able to add and subtract numerical fractions with a common denominator
• Be able to simplify algebraic expressions by combining like terms

Worked Examples

Worked Example 1

State what makes each one an algebraic fraction:
a) ${\displaystyle \frac{x}{5}}$
b) ${\displaystyle \frac{3}{y}}$
c) ${\displaystyle \frac{2a+1}{4}}$

Worked Example 2

Find the lowest common denominator of each pair:
a) ${\displaystyle \frac{1}{x}}$ and ${\displaystyle \frac{1}{3x}}$
b) ${\displaystyle \frac{2}{a}}$ and ${\displaystyle \frac{5}{ab}}$
c) ${\displaystyle \frac{3}{4m}}$ and ${\displaystyle \frac{7}{6m}}$

Worked Example 3

Write an equivalent algebraic fraction with the given denominator:
a) ${\displaystyle \frac{1}{x}}$ with denominator $3x$
b) ${\displaystyle \frac{2}{a}}$ with denominator $ab$
c) ${\displaystyle \frac{3}{4m}}$ with denominator $12m$

Worked Example 4

Add the algebraic fractions and simplify:
a) ${\displaystyle \frac{2}{x}}+{\displaystyle \frac{5}{x}}$
b) ${\displaystyle \frac{3}{a}}-{\displaystyle \frac{1}{a}}$
c) ${\displaystyle \frac{4}{m}}+{\displaystyle \frac{2}{3m}}$

Worked Example 5

Add or subtract the algebraic fractions and simplify:
a) ${\displaystyle \frac{1}{x}}+{\displaystyle \frac{2}{3x}}$
b) ${\displaystyle \frac{5}{ab}}-{\displaystyle \frac{2}{a}}$
c) ${\displaystyle \frac{3}{4m}}+{\displaystyle \frac{1}{6m}}$

Worked Example 6

Add or subtract the algebraic fractions and simplify the result:
a) ${\displaystyle \frac{x}{5}}+{\displaystyle \frac{2x}{15}}$
b) ${\displaystyle \frac{3a}{4b}}-{\displaystyle \frac{a}{2b}}$
c) ${\displaystyle \frac{m+1}{6n}}+{\displaystyle \frac{2m-1}{3n}}$

Problems

Problem 1

State what makes each one an algebraic fraction:
a) ${\displaystyle \frac{y}{4}}$
b) ${\displaystyle \frac{7}{p}}$
c) ${\displaystyle \frac{3b-2}{5}}$

Problem 2

Find the lowest common denominator of each pair:
a) ${\displaystyle \frac{1}{y}}$ and ${\displaystyle \frac{1}{2y}}$
b) ${\displaystyle \frac{4}{b}}$ and ${\displaystyle \frac{3}{bc}}$
c) ${\displaystyle \frac{5}{6n}}$ and ${\displaystyle \frac{1}{4n}}$

Problem 3

Write an equivalent algebraic fraction with the given denominator:
a) ${\displaystyle \frac{1}{y}}$ with denominator $2y$
b) ${\displaystyle \frac{4}{b}}$ with denominator $bc$
c) ${\displaystyle \frac{5}{6n}}$ with denominator $12n$

Problem 4

Add the algebraic fractions and simplify:
a) ${\displaystyle \frac{3}{y}}+{\displaystyle \frac{4}{y}}$
b) ${\displaystyle \frac{7}{b}}-{\displaystyle \frac{2}{b}}$
c) ${\displaystyle \frac{5}{n}}+{\displaystyle \frac{1}{2n}}$

Problem 5

Add or subtract the algebraic fractions and simplify:
a) ${\displaystyle \frac{2}{y}}+{\displaystyle \frac{1}{2y}}$
b) ${\displaystyle \frac{6}{bc}}-{\displaystyle \frac{1}{b}}$
c) ${\displaystyle \frac{5}{6n}}+{\displaystyle \frac{1}{3n}}$

Problem 6

Add or subtract the algebraic fractions and simplify the result:
a) ${\displaystyle \frac{2x}{3}}+{\displaystyle \frac{x}{6}}$
b) ${\displaystyle \frac{5a}{6b}}-{\displaystyle \frac{a}{3b}}$
c) ${\displaystyle \frac{p+2}{4q}}+{\displaystyle \frac{p-1}{2q}}$

Exercises

Understanding and Fluency

1. Identify which of the following are algebraic fractions:
   a) ${\displaystyle \frac{x}{4}}$
   b) ${\displaystyle \frac{3}{7}}$
   c) ${\displaystyle \frac{2a+5}{9}}$
   d) ${\displaystyle \frac{5}{m}}$

2. Find the lowest common denominator of each pair:
   a) ${\displaystyle \frac{1}{x}}$ and ${\displaystyle \frac{1}{2x}}$
   b) ${\displaystyle \frac{3}{a}}$ and ${\displaystyle \frac{5}{ab}}$
   c) ${\displaystyle \frac{2}{3m}}$ and ${\displaystyle \frac{1}{6m}}$

3. Find the lowest common denominator of each pair:
   a) ${\displaystyle \frac{4}{p}}$ and ${\displaystyle \frac{7}{3p}}$
   b) ${\displaystyle \frac{1}{bc}}$ and ${\displaystyle \frac{5}{c}}$
   c) ${\displaystyle \frac{3}{8n}}$ and ${\displaystyle \frac{1}{12n}}$

4. Write an equivalent algebraic fraction with the stated denominator:
   a) ${\displaystyle \frac{1}{x}}$ with denominator $2x$
   b) ${\displaystyle \frac{3}{a}}$ with denominator $ab$
   c) ${\displaystyle \frac{2}{3m}}$ with denominator $6m$

5. Write an equivalent algebraic fraction with the stated denominator:
   a) ${\displaystyle \frac{4}{p}}$ with denominator $3p$
   b) ${\displaystyle \frac{1}{c}}$ with denominator $bc$
   c) ${\displaystyle \frac{3}{8n}}$ with denominator $24n$

6. Add or subtract and simplify:
   a) ${\displaystyle \frac{2}{x}}+{\displaystyle \frac{3}{x}}$
   b) ${\displaystyle \frac{7}{a}}-{\displaystyle \frac{4}{a}}$
   c) ${\displaystyle \frac{1}{m}}+{\displaystyle \frac{1}{2m}}$
   d) ${\displaystyle \frac{5}{6p}}+{\displaystyle \frac{1}{3p}}$

7. Add or subtract and simplify:
   a) ${\displaystyle \frac{3}{x}}+{\displaystyle \frac{2}{4x}}$
   b) ${\displaystyle \frac{4}{ab}}-{\displaystyle \frac{1}{a}}$
   c) ${\displaystyle \frac{5x}{6}}+{\displaystyle \frac{x}{3}}$
   d) ${\displaystyle \frac{7a}{8b}}-{\displaystyle \frac{3a}{4b}}$

8. Add or subtract and simplify:
   a) ${\displaystyle \frac{x+1}{5}}+{\displaystyle \frac{2x-1}{10}}$
   b) ${\displaystyle \frac{3m+2}{6n}}-{\displaystyle \frac{m-1}{3n}}$
   c) ${\displaystyle \frac{2p}{q}}+{\displaystyle \frac{p}{2q}}$
   d) ${\displaystyle \frac{5a-1}{4b}}+{\displaystyle \frac{a+3}{4b}}$

Reasoning

9. Explain why ${\displaystyle \frac{3}{a}}$ and ${\displaystyle \frac{5}{ab}}$ do not already have a common denominator.

10. A student says that the lowest common denominator of ${\displaystyle \frac{1}{x}}$ and ${\displaystyle \frac{1}{3x}}$ is $4x$. Explain the mistake.

11. Noah says that to make an equivalent algebraic fraction, you only need to multiply the denominator. Is he correct? Explain.

12. Explain why ${\displaystyle \frac{2}{a}}$ and ${\displaystyle \frac{2b}{ab}}$ are equivalent.

13. A student adds ${\displaystyle \frac{1}{x}}+{\displaystyle \frac{2}{3x}}$ and writes ${\displaystyle \frac{3}{4x}}$. Describe the error.

Problem-solving

14. A formula includes the expression ${\displaystyle \frac{2}{x}}+{\displaystyle \frac{1}{3x}}$. Simplify this expression to a single algebraic fraction.

15. A student writes two parts of a calculation as ${\displaystyle \frac{5}{a}}$ and ${\displaystyle \frac{2}{ab}}$. Rewrite both with a common denominator, then subtract.

16. In a pattern rule, the total change is given by ${\displaystyle \frac{x}{4}}+{\displaystyle \frac{3x}{8}}$. Simplify the result.

17. A science formula is written as ${\displaystyle \frac{m}{6n}}+{\displaystyle \frac{2m}{3n}}$. Simplify it to one algebraic fraction.

18. A quantity changes by ${\displaystyle \frac{7a}{10b}}-{\displaystyle \frac{2a}{5b}}$. Simplify the expression.

19. A designer combines two lengths written as ${\displaystyle \frac{p+1}{2q}}$ and ${\displaystyle \frac{p-2}{4q}}$. Write the total length as a single simplified algebraic fraction.

Potential Misunderstandings

• Students may think an algebraic fraction must have pronumerals in both the numerator and denominator
• Students may confuse the lowest common denominator with simply adding the denominators
• Students may forget that equivalent fractions are made by multiplying both numerator and denominator by the same expression
• Students may multiply only the denominator when making an equivalent algebraic fraction
• Students may try to add numerators and denominators separately
• Students may not recognise when denominators already share a common factor
• Students may simplify too early and lose the common denominator structure
• Students may make arithmetic errors when combining the numerators after rewriting the fractions
• Students may forget to simplify the final result when possible