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.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
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?
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.
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.
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.
Skills
- Define an unknown from a worded problem.
- Write equations from ticket, phone, hire and budget contexts.
- Solve and interpret the answer with units.
Good algebra starts before the first calculation. Use four steps every time.
What to write in your book
- Four steps every time: Define the unknown → Translate to an equation → Solve using inverse operations → Interpret the answer in words with units.
- Common practical equation: $T = f + rn$ where $T$ = total, $f$ = fixed charge, $r$ = rate, $n$ = number of units.
- Always define the variable before using it: "Let $n$ be the number of texts."
- Check the answer makes sense in the original context.
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.
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 $12 + 4r = 40$.
| 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 | $12 + 4r = 40$ |
What to write in your book
- Expression (no equals sign): used when writing a cost rule, e.g. $C = 12 + 4r$.
- Equation (has equals sign): used when the total is known and you must find the unknown.
- Key test: is the total given in the problem? If yes, write an equation.
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?
Worked examples · 3 in a row, reveal as you go
Adult tickets cost $18 each. A booking fee of $6 is added. The total cost is $78. How many adult tickets were bought?
A phone plan costs $18 per month plus $0.10 per text. The bill is $32. How many texts were sent?
A kayak hire company charges $25 plus $15 per hour. A customer pays $85. How many hours did they hire the kayak?
What to write in your book
- Ticket problem: Let $t$ = number of tickets. $6 + 18t = 78 \Rightarrow t = 4$. Answer: 4 tickets.
- Phone problem: Let $n$ = number of texts. $18 + 0.10n = 32 \Rightarrow n = 140$. Answer: 140 texts.
- Hire problem: Let $h$ = hours. $25 + 15h = 85 \Rightarrow h = 4$. Answer: 4 hours.
- Reasonableness check: substitute your answer back into the original equation to verify.
Fill the gap: For the phone plan problem $18 + 0.10n = 32$, subtracting 18 from both sides gives $0.10n = $ , so $n = $ .
Common errors · the 3 traps that cost marks
Quick-fire practice · 4 translations
A hall costs $80 to book plus $12 per person for catering. The total is $260. Find the number of people.
A student has $125 and spends $9 each week. After some weeks, $53 remains. Find the number of weeks.
A gym charges a $20 joining fee plus $15 per class. The total paid is $95. Find the number of classes.
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?
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 $c$?
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.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. Movie tickets cost $14 each plus a $5 booking fee. The total is $61. How many tickets were bought? (4 marks)
Q2. A van hire costs $45 plus $20 per hour. The total cost is $145. Find the hire time. (4 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)
📖 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].
Five timed questions on translating worded problems to equations and solving. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering worded-problem equations. Pool: lesson 3.
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