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hscscience Maths Std · Y11
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Module 1 · L3 of 13 ~45 min ⚡ +90 XP available

Equations from Worded Problems

Turn practical situations into equations by defining the unknown, choosing the correct operations, solving and interpreting the result. Master the four-step translation process and you'll never be stumped by a real-world algebra problem again.

Today's hook, A phone plan costs $\$18$ per month plus $\$0.10$ per text. The bill is $\$32$. How many texts were sent? The four-step process makes this trivial.
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Recall, your gut answer first

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
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Recall, your gut answer first
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A phone plan costs $\$18$ per month plus $\$0.10$ per text. The bill is $\$32$.

Without calculating how could you find the number of texts? What equation would you try first?

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The translation pattern you need to own
+5 XP to read

Most practical algebra problems follow one pattern: a fixed charge plus a rate multiplied by an unknown number equals a known total. Lock in this structure and translation becomes automatic.

Fixed charge is added once regardless of quantity. Rate is charged repeatedly for each unit. The total is the known result. Your job is to find the unknown.

A common practical equation and a worked example.
$T = f + rn$, use when a fixed cost is added to repeated equal charges
Define before everything
Always write what the variable means: "Let $n$ be the number of texts." This makes the equation unambiguous.
Total goes on one side
Place the known total alone on one side. The fixed charge plus rate expression goes on the other side.
Interpret the answer
After solving, write a sentence using the context. "140 texts were sent" is a complete answer; "$n = 140$" alone is not.
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What you'll master
Know

Key facts

  • A variable must be defined before it is used in an equation.
  • Fixed charges and repeated rates play different roles.
  • An equation is needed when the total is known and an unknown must be found.
Understand

Concepts

  • The context decides what the variable represents.
  • The total should be on one side of the equation.
  • The final answer must make sense in the original situation.
Can do

Skills

  • Define an unknown from a worded problem.
  • Write equations from ticket, phone, hire and budget contexts.
  • Solve and interpret the answer with units.
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Key terms
VariableA letter representing an unknown quantity. Must be defined with "Let ... be ..." before use.
Fixed charge ($f$)A one-off cost added regardless of quantity, such as a booking fee or joining fee.
Rate ($r$)A cost per unit charged repeatedly, such as $\$0.10$ per text or $\$15$ per hour.
EquationA statement that two expressions are equal. Used when the total is known and the unknown must be found.
ExpressionAn algebraic rule with no equals sign, e.g. $12 + 4r$. Used to write a cost formula when no total is given.
InterpretWrite the answer as a sentence in context, with correct units, after solving.
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Use a four-step translation process

Good algebra starts before the first calculation.

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Use a four-step translation process
core concept

Good algebra starts before the first calculation. Use four steps every time.

Four ordered steps for solving worded problems, flowing DEFINE to INTERPRET.
Common error: Do not use a variable without defining it. "Let $t$ be the number of tickets" makes the equation clear.

Four-step word-problem method: (1) Define the unknown as a variable with units. (2) Write the equation using the words. (3) Solve the equation. (4) State the answer in a sentence referencing the original question.

Pause, copy the four-step word-problem method: (1) define the unknown with units; (2) write the equation from the words; (3) solve step-by-step; (4) state the answer in a sentence referencing the original question into your book.

Did you get this? True or false: you can use a variable in an equation without writing what it represents, as long as the context makes it obvious.

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Choose an equation, not just an expression

We just saw the four-step translation process: define the unknown, write the equation from the words, solve, and state the...

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Choose an equation, not just an expression
core concept

We just saw the four-step translation process: define the unknown, write the equation from the words, solve, and state the answer in a sentence. That raises a question: after defining the unknown and reading the problem, what turns the situation into something solvable, an expression or an equation? This card answers it → you need an equation (with an equals sign) not just an expression; an equation is created the moment the problem tells you two things are equal or gives a total.

If a question asks for a total cost rule, an expression such as $12 + 4r$ may be enough. If the total is known and you must find the unknown, write an equation such as $12 + 4r = 40$.

Table:
Question type Algebra needed Example
Write the cost rule Expression or formula $C = 12 + 4r$
Find the number of rides if the total is $\$40$ Equation $12 + 4r = 40$

When the total is known, write an equation (not just an expression). Expressions describe a quantity; equations state that two quantities are equal, only an equation can be solved for an unknown.

Pause, copy the equation-vs-expression rule: an expression describes a quantity but can't be solved (3n + 7); an equation claims two things are equal and can be solved (3n + 7 = 28), you write an equation when the problem gives you a total or tells you two quantities are the same into your book.

Quick check: A hall costs $\$80$ to book plus $\$12$ per person. Which correctly sets up the equation to find the number of people when the total is $\$260$?

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Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · TICKET PRICES

Adult tickets cost $\$18$ each. A booking fee of $\$6$ is added. The total cost is $\$78$. How many adult tickets were bought?

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Let $t$ be the number of adult tickets.
Define the unknown. The variable $t$ represents a count, so it must be a whole number.
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PROBLEM 2 · MOBILE PHONE USAGE

A phone plan costs $\$18$ per month plus $\$0.10$ per text. The bill is $\$32$. How many texts were sent?

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Let $n$ be the number of texts.
Define the unknown. The monthly cost is fixed; the text charge varies with $n$.
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PROBLEM 3 · HIRE FEE WITH HOURLY RATE

A kayak hire company charges $\$25$ plus $\$15$ per hour. A customer pays $\$85$. How many hours did they hire the kayak?

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Let $h$ be the number of hours hired.
Define the unknown. Fixed charge is $\$25$; rate is $\$15$ per hour.
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Fill the gap: For the phone plan problem $18 + 0.10n = 32$, subtracting 18 from both sides gives $0.10n = $ , so $n = $ .

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Common errors · the 3 traps that cost marks

Common errors · the 3 traps that cost marks

Common errors · the 3 traps that cost marks

Trap 01
Using a variable without defining it
Writing $6 + 18t = 78$ without first stating "Let $t$ be the number of tickets" is incomplete. Always define the variable in a sentence before writing the equation.
Trap 02
Confusing the fixed charge and the rate
In "$\$25$ plus $\$15$ per hour", $\$25$ is the fixed amount (not multiplied) and $\$15$ is multiplied by the number of hours. Swapping them gives a wrong equation.
Trap 03
Stopping at a number without interpreting
Finding $n = 140$ is not a complete answer. Write "140 texts were sent" to answer the original question. HSC questions award a mark for the interpretation.
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Revisit your thinking

Quick-fire practice · 4 translations

Quick-fire practice · 4 translations

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A hall costs $\$80$ to book plus $\$12$ per person for catering. The total is $\$260$. Find the number of people.

2

A student has $\$125$ and spends $\$9$ each week. After some weeks, $\$53$ remains. Find the number of weeks.

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A gym charges a $\$20$ joining fee plus $\$15$ per class. The total paid is $\$95$. Find the number of classes.

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Write down the key formula $T = f + rn$ and explain what each variable represents in the context of a phone bill.

Match it: Which step in the four-step process matches each action?

  • "Let $h$ be the number of hours hired."
  • "$25 + 15h = 85$"
  • "$h = 4$"
  • "The kayak was hired for 4 hours."
  • 4. Interpret
  • 3. Solve
  • 2. Translate
  • 1. Define

Think it through: A gym charges a $\$20$ joining fee plus $\$15$ per class. The total is $\$95$. What is the correct equation to find the number of classes $c$?

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Revisit your thinking

Earlier you predicted an equation for the phone plan. Let's check: the fixed monthly cost is $\$18$, the text rate is $\$0.10$, and the total bill is $\$32$. The correct equation is $18 + 0.10n = 32$.

Solving: $0.10n = 14$, so $n = 140$. The number 32 is the total (placed alone on one side), not the rate. The fixed $\$18$ and the variable text charge are together on the other side.

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Show what you have learned

Multiple choice, then short answer under exam conditions.

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

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Short answer
ApplyBand 44 marks

Q1. Movie tickets cost $\$14$ each plus a $\$5$ booking fee. The total is $\$61$. How many tickets were bought? (4 marks)

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ApplyBand 44 marks

Q2. A van hire costs $\$45$ plus $\$20$ per hour. The total cost is $\$145$. Find the hire time. (4 marks)

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AnalyseBand 53 marks

Q3. A student writes $30 + 5 = x$ for "a $\$30$ starting amount plus $\$5$ each week becomes $80". Explain the error and write the correct equation. (3 marks)

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📖 Comprehensive answers (click to reveal)

Drill 1: Let $p$ = number of people. $80 + 12p = 260$. $12p = 180$, $p = 15$. Answer: 15 people.

Drill 2: Let $w$ = number of weeks. $125 - 9w = 53$. $9w = 72$, $w = 8$. Answer: 8 weeks.

Drill 3: Let $c$ = number of classes. $20 + 15c = 95$. $15c = 75$, $c = 5$. Answer: 5 classes.

Q1 (4 marks): Let $t$ = number of tickets [1]. $5 + 14t = 61$ [1]. $14t = 56$, $t = 4$ [1]. Answer: 4 tickets were bought [1].

Q2 (4 marks): Let $h$ = hours hired [1]. $45 + 20h = 145$ [1]. $20h = 100$, $h = 5$ [1]. Answer: the van was hired for 5 hours [1].

Q3 (3 marks): Error: the right-hand side should be the known total ($\$80$), not an unknown $x$; the number of weeks is the unknown, not the total [1]. Correct equation: Let $w$ = number of weeks. $\$30 + 5w = 80$ [2].