GM Lesson 060 Linear Equations with Fractions and Decimals

Learning Intentions

By the end of this lesson, students will be able to:

• Solve linear equations involving decimal coefficients.
• Solve linear equations involving fractions.
• Use efficient strategies to remove fractions where appropriate.

Prerequisites

Students should already be able to:

• Solve one-step and two-step linear equations.
• Solve equations with variables on both sides.
• Collect like terms.
• Use inverse operations to isolate an unknown.
• Find common denominators for simple fractions.

Key Idea Summary

Linear equations with fractions and decimals can be solved using the same equivalent-equation principles as other linear equations.

For decimal coefficients, it is often helpful to multiply every term by , or to remove decimals.

For fractional coefficients, it is often helpful to multiply every term by the lowest common denominator.

The important rule is:

Whatever operation is applied to one side of the equation must also be applied to the other side.

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Decimal Equations

When an equation contains decimals, multiply every term by a power of to create an equivalent equation with whole-number coefficients.

Worked Example 1: Solving an Equation with Decimal Coefficients

Solve:

Multiply every term by :

Therefore, .

Check:

The solution is correct.

Worked Example 2: Decimals on Both Sides

Solve:

Multiply every term by :

Therefore, .

Fraction Equations

When an equation contains fractions, multiply every term by the lowest common denominator to remove the fractions.

Worked Example 3: Solving an Equation with One Fraction

Solve:

Multiply every term by :

Therefore, .

Worked Example 4: Solving an Equation with Several Fractions

Solve:

The lowest common denominator of and is .

Multiply every term by :

Therefore, .

Worked Example 5: Fractions on Both Sides

Solve:

The lowest common denominator of and is .

Multiply both sides by :

Therefore, .

Worked Example 6: Practical Decimal Equation

A ride-share company charges a booking fee of $ plus $ per kilometre. A trip costs $ .

Let be the number of kilometres travelled.

Multiply every term by or solve directly:

The trip was km.

Understanding Checks

Check 1

Why is it useful to multiply by ?

Expected answer:

Multiplying by changes the equation into , which is easier to solve.

Check 2

What number should be used to remove the fractions from this equation?

Expected answer:

The lowest common denominator of and is , so multiply every term by .

Check 3

Solve mentally after removing the decimal:

Expected answer:

.

Check 4

Identify the error:

Expected answer:

Only was multiplied by . Every term must be multiplied by , so the correct next line is:

.

Exercises

Simple Familiar Exercises

Exercise 1

Solve:

Exercise 2

Solve:

Exercise 3

Solve:

Exercise 4

Solve:

Exercise 5

Solve:

Exercise 6

Solve:

Exercise 7

Solve:

Exercise 8

Solve:

Complex Familiar Exercises

Exercise 9

Solve:

Exercise 10

Solve:

Exercise 11

Solve:

Exercise 12

Solve:

Exercise 13

Solve:

Exercise 14

Solve:

Exercise 15

A gym charges a joining fee of $ plus $ per class. A student pays $ in total.

Let be the number of classes attended.

Write and solve a linear equation to find .

Exercise 16

A plumber charges $ call-out fee plus $ per hour. The total bill is $ .

Let be the number of hours worked.

Write and solve a linear equation to find .

Homework Problems

Problem 1

Solve:

Problem 2

Solve:

Problem 3

Solve:

Problem 4

Solve:

Problem 5

Solve:

Problem 6

Solve:

Problem 7

Solve:

Problem 8

A taxi fare is calculated using a fixed charge of $ plus $ per kilometre. A trip costs $ .

Let be the distance travelled in kilometres.

Write and solve a linear equation to find .

Problem 9

A student shares the cost of a group gift equally with friends. After adding $ for a card, each person pays $ .

Let be the total cost of the gift before the card is added.

Write and solve a linear equation to find .

Problem 10

Explain why multiplying every term by the lowest common denominator is a valid strategy when solving equations involving fractions.

Next: GM Lesson 061 Equations from Words