080e. Solving Equations Involving Fractions

Learning Intentions

• To understand that to Solve an equation with a fraction, it is helpful to multiply both sides by its denominator
• solve equations involving fractions

Pre-requisite Summary

• Understand that an equation states that two expressions are equal
• Know that equivalent equations have the same solution
• Understand that the same operation can be applied to both sides of an equation
• Know that multiplying by a denominator can remove a fraction
• Recall multiplication facts and simple fraction notation
• Be able to solve one-step and two-step equations with whole numbers
• Be able to Simplify expressions after multiplying brackets where needed
• Understand that solutions should be checked by substitution

Worked Examples

Worked Example 1

a) Explain why multiplying both sides of by is helpful.

b) Solve .

c) Check the solution.

Worked Example 2

Solve each equation involving a fraction:

a)

b)

c)

Worked Example 3

Solve each equation involving a fraction:

a)

b)

c)

Worked Example 4

Solve each equation involving a fraction:

a)

b)

c)

Worked Example 5

Solve each equation involving a fraction:

a)

b)

c)

Worked Example 6

For each equation:

a) multiply both sides by the denominator

b) solve the simpler equation

c) check the solution by substitution

For

Problems

Problem 1

a) Explain why multiplying both sides of is helpful.

b) Solve .

c) Check the solution.

Problem 2

Solve each equation involving a fraction:

a)

b)

c)

Problem 3

Solve each equation involving a fraction:

a)

b)

c)

Problem 4

Solve each equation involving a fraction:

a)

b)

c)

Problem 5

Solve each equation involving a fraction:

a)

b)

c)

Problem 6

For each equation:

a) multiply both sides by the denominator

b) solve the simpler equation

c) check the solution by substitution

For

Exercises

Understanding and Fluency

Exercise 1.

State the denominator in each fraction:

a)

b)

c)

Exercise 2.

State what you would multiply both sides by to remove the fraction:

a)

b)

c)

Exercise 3.

Solve each equation:

a)

b)

c)

Exercise 4.

Solve each equation:

a)

b)

c)

Exercise 5.

Solve each equation:

a)

b)

c)

Exercise 6.

Solve each equation:

a)

b)

c)

Exercise 7.

Solve each equation:

a)

b)

c)

Exercise 8.

Solve each equation and check by substitution:

a)

b)

c)

Reasoning

Exercise 9.

Explain why multiplying both sides of an equation by the denominator does not change the solution.

Exercise 10.

A student solves by adding to both sides. Explain the mistake.

Exercise 11.

Explain why becomes after multiplying both sides by .

Exercise 12.

A student solves by multiplying only the left side by . Explain why this is incorrect.

Exercise 13.

Explain why checking by substitution is useful after solving an equation with a fraction.

Exercise 14.

A student says can be solved by multiplying by first or subtracting first. Explain which approach is clearer and why.

Problem-solving

Exercise 15.

A recipe equation is , where is the number of tablespoons of flour. Solve for .

Exercise 16.

A ticket problem is modelled by , where is the ticket cost in dollars before the fixed charge. Solve for .

Exercise 17.

A container problem is modelled by . Solve for .

Exercise 18.

A phone data problem is modelled by , where is the number of gigabytes used. Solve for .

Exercise 19.

A length problem is modelled by . Solve for .

Exercise 20.

A saving problem is modelled by , where is the number of dollars saved. Solve for .

Potential Misunderstandings

• Students may multiply only one side of the equation by the denominator
• Students may confuse the denominator with the numerator
• Students may try to add or subtract the denominator instead of multiplying by it
• Students may forget that the entire numerator is divided by the denominator
• Students may fail to Use brackets correctly when multiplying both sides
• Students may solve the simplified equation incorrectly after removing the fraction
• Students may not check whether the final value makes the original equation true
• Students may think multiplying by the denominator changes the solution rather than creating an equivalent equation

Next: 081e. Expanding Brackets and Simplifying Expressions