157. Solving Real-World Problems with Equations

Learning Intentions

• To understand that equations can be applied to real-world situations
• Solve problems Use equations

Pre-requisite Summary

• An equation states that two expressions are equal
• A pronumeral can be used to represent an unknown quantity
• Solving an equation means finding the value that makes the equation true
• The same operation can be applied to both sides of an equation to form an equivalent equation
• A real-world situation can often be translated into words, then into an equation
• A solution should be checked to see whether it makes sense in the context of the problem

Worked Examples

Worked Example 1

A book and a $4 bookmark cost $15 altogether. Let be the cost of the book. Write and solve an equation.

Worked Example 2

Three identical pens cost $12. Let be the cost of one pen. Write and solve an equation.

Worked Example 3

A taxi fare is a $5 flagfall plus $3 per kilometre. The total fare is $20. Let be the number of kilometres travelled. Write and solve an equation.

Worked Example 4

A rectangle has length cm and perimeter cm. Let be the width. Write and solve an equation using .

Worked Example 5

A number is doubled and then increased by to give . Let be the number. Write and solve an equation.

Worked Example 6

A school bus holds students in each row. There are rows and extra seats, making seats in total. Write and solve an equation.

Problems

Problem 1

A sandwich and a $3 drink cost $11 altogether. Let be the cost of the sandwich. Write and solve an equation.

Problem 2

Four identical notebooks cost $20. Let be the cost of one notebook. Write and solve an equation.

Problem 3

A movie ticket costs $2 booking fee plus $6 per person. The total cost is $26. Let be the number of people. Write and solve an equation.

Problem 4

A rectangle has length cm and perimeter cm. Let be the width. Write and solve an equation using .

Problem 5

A number is tripled and then increased by to give . Let be the number. Write and solve an equation.

Problem 6

A theatre has rows with seats in each row and extra seats, making seats in total. Write and solve an equation.

Exercises

Understanding and Fluency

Exercise 1.

Write an equation for each situation.

a) A number increased by is

b) Twice a number is

c) A number divided by is

Exercise 2.

Write an equation for each situation.

a) A $7 item and a $2 item cost $15 altogether

b) Three identical pencils cost $9

c) A number is doubled and then is added to get

Exercise 3.

Solve each equation.

a)

b)

c)

Exercise 4.

Solve each equation.

a)

b)

c)

Exercise 5.

A gym charges a $4 entry fee and $2 per class. The total cost is $18.

a) Let be the number of classes

b) Write an equation

c) Solve the equation

Exercise 6.

A ribbon of length cm is cut into equal pieces, each cm long.

a) Write an equation

b) Solve the equation

Exercise 7.

A rectangle has length cm and perimeter cm.

a) Let be the width

b) Write an equation using

c) Solve the equation

Exercise 8.

A number is multiplied by and then decreased by to give .

a) Let be the number

b) Write an equation

c) Solve the equation

Reasoning

Exercise 9.

Explain why a variable should be defined clearly before writing an equation for a real-world situation.

Exercise 10.

A student writes the equation for “a $5 flagfall plus $3 per kilometre totals $20”. Explain what the variable must represent and why the equation is suitable.

Exercise 11.

Noah solves a worded problem and gets , where is the number of tickets sold. Explain why the answer should be checked against the context.

Exercise 12.

Explain why solving a real-world problem with an equation involves both forming the equation correctly and interpreting the solution correctly.

Problem-solving

Exercise 13.

A concert charges a $6 booking fee plus $8 per ticket. The total cost is $46. How many tickets were bought?

Exercise 14.

A plumber charges a $15 call-out fee and $25 per hour. The total bill is $90. How many hours did the plumber work?

Exercise 15.

A square has perimeter cm. Let be the side length. Write and solve an equation to Solve .

Exercise 16.

A phone plan costs $12 per month plus a one-off setup fee of $8. The total paid is $56. How many months were paid for?

Exercise 17.

A school orders boxes of markers. Each box contains markers, and there are extra markers. The total number of markers is . How many boxes were ordered?

Exercise 18.

A number is divided by and then increased by to give . Find the number.

Potential Misunderstandings

• Thinking equations are only abstract and cannot represent real situations
• Choosing a variable but not defining what it stands for
• Writing an expression when an equation is needed
• Translating words into operations incorrectly, especially phrases such as “altogether”, “per”, “more than” and “less than”
• Forgetting to include fixed amounts as well as variable amounts in the equation
• Solving the equation correctly but giving the wrong unit or interpretation in the final answer
• Accepting a numerical solution that does not make sense in the context, such as a negative number of people
• Using the wrong formula when the problem involves perimeter, area, cost or rate
• Forgetting to check whether the solution satisfies the original situation

Next: 158. Understanding and Representing Inequalities