Equations from Worded Problems

Turn practical situations into equations by defining the unknown, choosing the correct operations, solving and interpreting the result.

45 min Algebra Formulas and equations Lesson 3 of 13
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Use the printable version for translating scenarios, writing equations and interpreting solutions.

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Think First

A phone plan costs $18 per month plus $0.10 per text. The bill is $32. How could you find the number of texts?

Type the equation you would try first.

Write the equation you would try first in your book.

Write your equation in your book
Saved

Know

  • 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

  • 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

  • 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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Common Practical Equation

total = fixed charge + rate x number
Use when a starting cost is added to repeated equal charges.
$T = f + rn$
$T$ = total, $f$ = fixed charge, $r$ = rate, $n$ = number of units

Core Content

1. Use a Four-Step Translation Process

Good algebra starts before the first calculation.

1. Define

Choose the unknown and write what it means.

2. Translate

Turn the words into an equation.

3. Solve

Use inverse operations to find the unknown.

4. Interpret

Write a sentence answering the question.

Common error: Do not use a variable without defining it. "Let $t$ be the number of tickets" makes the equation clear.
Worked Example 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?

Define: Let $t$ be the number of adult tickets.

Translate: $6 + 18t = 78$

Solve: $18t = 72$, so $t = 4$.

Interpret: 4 adult tickets were bought.

Worked Example 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?

Let $n$ be the number of texts.

$18 + 0.10n = 32$

$0.10n = 14$

$n = 140$

Answer: 140 texts were sent.

Reasonableness check: $140 \times 0.10 = 14$, and $18 + 14 = 32$, so the answer matches the bill.
Worked Example 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?

Let $h$ be the number of hours hired.

$25 + 15h = 85$

$15h = 60$

$h = 4$

Answer: The kayak was hired for 4 hours.

2. Choose an Equation, Not Just an Expression

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 typeAlgebra neededExample
Write the cost ruleExpression or formula$C = 12 + 4r$
Find the number of rides if the total is $40Equation$12 + 4r = 40$
Activity

Translate, Solve, Interpret

  1. 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.
  3. A gym charges a $20 joining fee plus $15 per class. The total paid is $95. Find the number of classes.
Complete the translation practice in your book.

Revisit the Phone Plan

The phone problem is $18 + 0.10n = 32$. The fixed monthly cost is $18, the text rate is $0.10, and the unknown is the number of texts.

Explain why 32 is the total in your book.

Assessment

MC

Multiple Choice

Random questions from the lesson bank - feedback appears immediately.

SA

Short Answer

Define the variable, write the equation, solve and interpret.

1. Movie tickets cost $14 each plus a $5 booking fee. The total is $61. How many tickets were bought? 4 MARKS

Answer in your book.

2. A van hire costs $45 plus $20 per hour. The total cost is $145. Find the hire time. 4 MARKS

Answer in your book.

3. 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

Answer in your book.

Translation Sprint

For each scenario, identify the fixed amount, the repeated rate and the unknown before solving.

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