157. Solving Real-World Problems with Equations

Learning Intentions

• To understand that equations can be applied to real-world situations
• solve problems using 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 $b$ be the cost of the book. Write and solve an equation.

Worked Example 2

Three identical pens cost $12. Let $p$ 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 $k$ be the number of kilometres travelled. Write and solve an equation.

Worked Example 4

A rectangle has length $8$ cm and perimeter $26$ cm. Let $w$ be the width. Write and solve an equation using $P=2l+2w$.

Worked Example 5

A number is doubled and then increased by $7$ to give $19$. Let $x$ be the number. Write and solve an equation.

Worked Example 6

A school bus holds $n$ students in each row. There are $6$ rows and $3$ extra seats, making $33$ seats in total. Write and solve an equation.

Problems

Problem 1

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

Problem 2

Four identical notebooks cost $20. Let $n$ 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 $p$ be the number of people. Write and solve an equation.

Problem 4

A rectangle has length $9$ cm and perimeter $30$ cm. Let $w$ be the width. Write and solve an equation using $P=2l+2w$.

Problem 5

A number is tripled and then increased by $4$ to give $25$. Let $x$ be the number. Write and solve an equation.

Problem 6

A theatre has $5$ rows with $m$ seats in each row and $2$ extra seats, making $27$ seats in total. Write and solve an equation.

Exercises

Understanding and Fluency

1. Write an equation for each situation.
   a) A number increased by $5$ is $14$
   b) Twice a number is $18$
   c) A number divided by $4$ is $3$

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 $1$ is added to get $13$

3. Solve each equation.
   a) $x+6=17$
   b) $3x=21$
   c) $2x+5=19$

4. Solve each equation.
   a) $4m=28$
   b) $n+9=20$
   c) $5p-3=17$

5. A gym charges a $4 entry fee and $2 per class. The total cost is $18.
   a) Let $c$ be the number of classes
   b) Write an equation
   c) Solve the equation

6. A ribbon of length $x$ cm is cut into $5$ equal pieces, each $4$ cm long.
   a) Write an equation
   b) Solve the equation

7. A rectangle has length $7$ cm and perimeter $22$ cm.
   a) Let $w$ be the width
   b) Write an equation using $P=2l+2w$
   c) Solve the equation

8. A number is multiplied by $4$ and then decreased by $3$ to give $21$.
   a) Let $x$ be the number
   b) Write an equation
   c) Solve the equation

Reasoning

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

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

11. Noah solves a worded problem and gets $x=-4$, where $x$ is the number of tickets sold. Explain why the answer should be checked against the context.

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

Problem-solving

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

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?

15. A square has perimeter $36$ cm. Let $s$ be the side length. Write and solve an equation to find $s$.

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?

17. A school orders boxes of markers. Each box contains $10$ markers, and there are $6$ extra markers. The total number of markers is $56$. How many boxes were ordered?

18. A number is divided by $3$ and then increased by $2$ to give $10$. 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