152. Solving Equations with Algebraic Fractions

Learning Intentions

• To understand that fractions are used in algebra to indicate division
• solve equations involving algebraic fractions

Pre-requisite Summary

• A fraction bar represents division, so ${\displaystyle \frac{x}{3}}$ means $x÷3$
• An equation states that two expressions are equal
• Solving an equation means finding a value that makes the equation true
• Equivalent equations have the same solution
• Multiplying both sides of an equation by the same non-zero number gives an equivalent equation
• To remove a fraction, it is often helpful to multiply both sides by the denominator
• Solutions should be checked by substitution into the original equation

Worked Examples

Worked Example 1

Solve ${\displaystyle \frac{x}{4}}=3$.

Worked Example 2

Solve ${\displaystyle \frac{m}{7}}=5$.

Worked Example 3

Solve ${\displaystyle \frac{a}{3}}+2=7$.

Worked Example 4

Solve ${\displaystyle \frac{p}{5}}-4=2$.

Worked Example 5

Solve ${\displaystyle \frac{2x}{3}}=8$.

Worked Example 6

Solve ${\displaystyle \frac{3y}{4}}=6$.

Worked Example 7

Solve ${\displaystyle \frac{x+1}{2}}=5$.

Worked Example 8

Solve ${\displaystyle \frac{n-3}{6}}=4$.

Problems

Problem 1

Solve ${\displaystyle \frac{x}{6}}=4$.

Problem 2

Solve ${\displaystyle \frac{m}{8}}=3$.

Problem 3

Solve ${\displaystyle \frac{a}{5}}+1=6$.

Problem 4

Solve ${\displaystyle \frac{p}{4}}-3=5$.

Problem 5

Solve ${\displaystyle \frac{2x}{5}}=6$.

Problem 6

Solve ${\displaystyle \frac{3y}{2}}=12$.

Problem 7

Solve ${\displaystyle \frac{x+2}{3}}=4$.

Problem 8

Solve ${\displaystyle \frac{n-5}{7}}=2$.

Exercises

Understanding and Fluency

1. Write each fraction as a division statement.
   a) ${\displaystyle \frac{x}{5}}$
   b) ${\displaystyle \frac{a}{2}}$
   c) ${\displaystyle \frac{3m}{4}}$

2. Solve each equation.
   a) ${\displaystyle \frac{x}{3}}=7$
   b) ${\displaystyle \frac{y}{9}}=2$
   c) ${\displaystyle \frac{a}{4}}=6$

3. Solve each equation.
   a) ${\displaystyle \frac{m}{5}}+3=9$
   b) ${\displaystyle \frac{p}{6}}+1=4$
   c) ${\displaystyle \frac{t}{8}}+2=7$

4. Solve each equation.
   a) ${\displaystyle \frac{k}{7}}-2=3$
   b) ${\displaystyle \frac{n}{4}}-5=1$
   c) ${\displaystyle \frac{b}{9}}-1=6$

5. Solve each equation.
   a) ${\displaystyle \frac{2x}{3}}=10$
   b) ${\displaystyle \frac{4m}{5}}=12$
   c) ${\displaystyle \frac{3p}{7}}=9$

6. Solve each equation.
   a) ${\displaystyle \frac{x+4}{2}}=6$
   b) ${\displaystyle \frac{y-1}{3}}=5$
   c) ${\displaystyle \frac{a+2}{4}}=3$

7. Solve each equation.
   a) ${\displaystyle \frac{n-6}{5}}=2$
   b) ${\displaystyle \frac{p+3}{7}}=4$
   c) ${\displaystyle \frac{m-2}{8}}=1$

8. Solve and check each equation.
   a) ${\displaystyle \frac{x}{6}}+4=9$
   b) ${\displaystyle \frac{y}{3}}-2=5$
   c) ${\displaystyle \frac{2a}{5}}=14$

Reasoning

9. Explain why ${\displaystyle \frac{x}{4}}=3$ can be solved by multiplying both sides by $4$.

10. A student solves ${\displaystyle \frac{x}{5}}=7$ by dividing both sides by $5$ and gets $x={\displaystyle \frac{7}{5}}$. Explain the error.

11. Explain why the fraction bar in ${\displaystyle \frac{a+2}{3}}$ means the whole expression $a+2$ is being divided by $3$.

12. Noah says that ${\displaystyle \frac{x+1}{2}}=5$ means $x+{\displaystyle \frac{1}{2}}=5$. Explain why Noah is incorrect.

Problem-solving

13. A ribbon of length $x$ cm is cut into $4$ equal pieces. Each piece is $6$ cm long. Write and solve an equation to find $x$.

14. A plumber divides a pipe of length $m$ metres into $5$ equal sections. Each section is $3$ metres long. Write and solve an equation to find $m$.

15. The total cost of $x$ dollars is shared equally among $3$ students, and each student pays $8. Write and solve an equation to find $x$.

16. A number is increased by $1$ and then divided by $2$. The result is $9$. Write and solve an equation to find the number.

Potential Misunderstandings

• Thinking a fraction in algebra means only “part of a whole” rather than division
• Forgetting that the fraction bar means divide by the denominator
• Treating only part of the numerator as being divided when the whole numerator should be divided
• Dividing both sides by the denominator when the denominator should be cleared by multiplication
• Multiplying one side of the equation by the denominator but forgetting to do the same to the other side
• Solving ${\displaystyle \frac{x+2}{3}}=5$ as $x+2=5÷3$ instead of first removing the denominator
• Making errors with inverse operations after the fraction has been removed
• Forgetting to check the solution in the original equation
• Confusing ${\displaystyle \frac{2x}{3}}$ with ${\displaystyle \frac{2+x}{3}}$
• Thinking that solving equations with algebraic fractions uses different algebra rules from ordinary equations, rather than the same idea of forming equivalent equations