083. Solving Real-World Problems with Equations

Learning Intentions

• To know that equations can be applied to real-world situations
• solve problems using 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 $7$ is $19$. Write an equation and find the number.

Worked Example 2

A rectangle has perimeter $26$ cm and width $5$ 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 $28$ students in a class. The number of boys is $4$ 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 $9$ is $23$. Write an equation and find the number.

Problem 2

A rectangle has perimeter $34$ cm and width $6$ 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 $32$ students in a class. The number of boys is $6$ 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

1. Write an equation for each statement.
   a) A number plus $6$ is $17$
   b) A number multiplied by $4$ is $28$
   c) A number decreased by $3$ is $11$

2. Solve each equation.
   a) $x+8=21$
   b) $5x=35$
   c) $x-9=13$

3. Write an equation and solve.
   a) A number plus $12$ is $30$
   b) Three equal items cost $27
   c) A number divided by $4$ is $6$

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 $5$ cm and the total length is $40$ cm. How many pieces are there?
   c) A number is doubled and then $3$ is added to make $19$

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

6. A rectangle has perimeter $30$ cm and width $4$ cm.
   a) Write an equation using $P=2l+2w$
   b) Solve for $l$
   c) Check the solution

7. A class has $24$ students. The number of girls is $2$ more than the number of boys.
   a) Let the number of boys be $b$ and write an equation
   b) Solve the equation
   c) Find the number of girls

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

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

10. A student writes $4+3k=19$ for a taxi fare with a $4 flagfall fee and $3 per kilometre. Explain what each part of the equation represents.

11. A student solves a real-world problem and gets $x=-3$. Explain why the answer may not make sense in context.

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

Problem-solving

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

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?

15. The perimeter of a rectangle is $44$ cm and the width is $8$ cm. Find the length.

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

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?

18. There are $36$ animals at a farm made up of only chickens and cows. If there are $10$ 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