083. Solving Real-World Problems with Equations

Learning Intentions

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

Pre-requisite Summary

• Understand that an equation shows two expressions are equal
• Know that a pronumeral can represent an unknown quantity
• Be able to translate simple word statements into algebraic expressions
• Be able to solve one-step and two-step equations
• Be able to Check a solution by substituting it back into the equation
• Understand that the answer in a real-world problem must make sense in the context

Worked Examples

Worked Example 1

A number increased by is . Write an equation and Solve the number.

Worked Example 2

A rectangle has perimeter cm and width cm. Write an equation and find the length.

Worked Example 3

Three equal movie tickets cost $36 in total. Write an equation and find the cost of one ticket.

Worked Example 4

A taxi fare has a $4 flagfall fee plus $3 per kilometre. The total fare is $22. Write an equation and find the number of kilometres travelled.

Worked Example 5

There are students in a class. The number of boys is less than the number of girls. Write an equation and find the number of girls.

Worked Example 6

A school is selling notebooks for $2 each and pens for $3 each. One student buys the same number of notebooks and pens, and spends $25 altogether. Write an equation and find how many of each item were bought.

Problems

Problem 1

A number increased by is . Write an equation and find the number.

Problem 2

A rectangle has perimeter cm and width cm. Write an equation and find the length.

Problem 3

Four equal movie tickets cost $48 in total. Write an equation and find the cost of one ticket.

Problem 4

A taxi fare has a $5 flagfall fee plus $4 per kilometre. The total fare is $29. Write an equation and find the number of kilometres travelled.

Problem 5

There are students in a class. The number of boys is less than the number of girls. Write an equation and find the number of girls.

Problem 6

A student buys the same number of juice boxes at $2 each and sandwiches at $5 each, and spends $28 altogether. Write an equation and find how many of each item were bought.

Exercises

Understanding and Fluency

Exercise 1.

Write an equation for each statement.

a) A number plus is

b) A number multiplied by is

c) A number decreased by is

Exercise 2.

Solve each equation.

a)

b)

c)

Exercise 3.

Write an equation and solve.

a) A number plus is

b) Three equal items cost $27

c) A number divided by is

Exercise 4.

Write an equation and solve.

a) A book costs $7 and a student spends $35. How many books were bought?

b) A ribbon is cut into pieces of cm and the total length is cm. How many pieces are there?

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

Exercise 5.

The cost of hiring skates is a $6 fee plus $4 per hour.

a) Write an equation for a total cost of $18

b) Solve the equation

c) State what the solution means

Exercise 6.

A rectangle has perimeter cm and width cm.

a) Write an equation using

b) Solve for

c) Check the solution

Exercise 7.

A class has students. The number of girls is more than the number of boys.

a) Let the number of boys be and write an equation

b) Solve the equation

c) Find the number of girls

Exercise 8.

A mobile plan charges a $10 base fee and $2 per GB of data.

a) Write an equation for a total cost of $26

b) Solve the equation

c) Interpret the solution

Reasoning

Exercise 9.

Explain why writing an equation can help solve a worded problem.

Exercise 10.

A student writes for a taxi fare with a $4 flagfall fee and $3 per kilometre. Explain what each part of the equation represents.

Exercise 11.

A student solves a real-world problem and gets . Explain why the answer may not make sense in context.

Exercise 12.

Two students model the same problem with different pronumerals. Explain why both equations can still be correct.

Problem-solving

Exercise 13.

A cinema charges $11 per ticket. A family spends $55 on tickets. How many tickets did they buy?

Exercise 14.

A plumber charges a $50 call-out fee and $40 per hour. If the bill was $170, how many hours did the plumber work?

Exercise 15.

The perimeter of a rectangle is cm and the width is cm. Find the length.

Exercise 16.

A sports club buys balls at $6 each and pays a delivery fee of $8. The total cost is $44. Find the number of balls.

Exercise 17.

At a fundraiser, each sausage roll costs $3.50. A student buys some sausage rolls and one drink costing $2. If the total is $16, how many sausage rolls were bought?

Exercise 18.

There are animals at a farm made up of only chickens and cows. If there are cows, Use an equation to find the number of chickens.

Potential Misunderstandings

• A student may Identify the wrong quantity as the unknown
• A student may write an expression instead of an equation
• A student may use the wrong operation when translating words into algebra
• A student may forget to include a fixed starting amount, such as a fee
• A student may solve the equation correctly but not answer the actual question asked
• A student may not check whether the solution is reasonable in the real-world context
• A student may think every worded problem needs a complicated equation
• A student may confuse the meaning of the pronumeral after solving