Mathematics Standard • Year 11 • Module 1 • Lesson 3

Equations from Worded Problems

Build fluency in the four-step translation process — define the unknown, translate, solve, interpret — across ticket, phone, hire and budget contexts.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 List the four steps of translating a worded problem (lesson § Four-Step Translation Process).

1. ____________ 2. ____________ 3. ____________ 4. ____________

Q1.2 In the general equation T = f + rn, label each letter.

T = ____________ f = ____________ r = ____________ n = ____________

Q1.3 Which one of these is an EQUATION (not an expression)? Circle one: (a) 12 + 4r (b) 12 + 4r = 40 (c) 80 + 12n.

Stuck? Revisit lesson § Choose an Equation, Not Just an Expression — equations contain "=".

2. Worked example — translate, solve, interpret (phone plan)

Follow each step. The variable is defined before any algebra is written.

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

Step 1 — Define.

Let n = the number of texts sent.

Reason: name the unknown before using it in an equation.

Step 2 — Translate.

18 + 0.10n = 32

Reason: total = fixed monthly fee + rate × number of texts.

Step 3 — Solve (subtract 18, then divide by 0.10).

0.10n = 14 ⇒ n = 14 / 0.10 = 140

Reason: inverse operations — addition first, then division.

Step 4 — Interpret with a sentence.

140 texts were sent that month. Reasonableness check: 140 × $0.10 = $14, plus $18 fixed = $32 ✓.

3. Faded example — fill in the missing steps

A kayak hire company charges $25 plus $15 per hour. A customer pays $85. How many hours did they hire the kayak? Fill in each blank. 4 marks

Step 1 — Define:

Let h = ____________________________________________

Step 2 — Translate:

____________ + ____________ h = ____________

Step 3 — Solve:

Subtract: ____________ h = ____________ ⇒ h = ____________

Step 4 — Interpret:

The customer hired the kayak for ____________ hours.

Stuck? Revisit lesson § Worked Example 3 — Hire fee with hourly rate.

4. Graduated practice — translate, solve, interpret

For each, write a definition line, an equation, the solution steps and a one-sentence interpretation.

Foundation — match the situation to its equation (4 questions)

Q	Situation	Equation
4.1 1	A booking fee of $5 plus $12 per ticket totals $41 (t = tickets).	____________________
4.2 1	A $30 starting savings plus $5 each week makes $80 (w = weeks).	____________________
4.3 1	A taxi costs $7 plus $3 per km, total $31 (k = km).	____________________
4.4 1	A hire is $25 plus $15 per hour, total $85 (h = hours).	____________________

Standard — full translation + solve (6 questions)

Write a definition line, the equation, the solving steps and a sentence answer.

4.5 Movie tickets cost $14 each plus a $5 booking fee. The total is $61. How many tickets were bought? 3 marks

4.6 A hall costs $80 to book plus $12 per person for catering. The total is $260. Find the number of people. 3 marks

4.7 A gym charges a $20 joining fee plus $15 per class. The total paid is $95. Find the number of classes. 3 marks

4.8 A van hire costs $45 plus $20 per hour. The total cost is $145. Find the hire time. 3 marks

4.9 A pre-paid card has $50 on it. Each call costs $1.20. After some calls the balance is $20. How many calls were made? (Use $50 − 1.20c = 20.) 3 marks

4.10 A photographer charges $120 plus $35 per edited photo. The invoice is $400. How many photos were edited? 3 marks

Extension — subtraction pattern and unit conversion (2 questions)

4.11 A student has $125 and spends $9 each week. After some weeks, $53 remains. Find the number of weeks. (Hint: 125 − 9w = 53.) 3 marks

4.12 A phone plan costs $25 per month plus $0.15 per minute of calls. One month's bill was $52. (a) Find the number of minutes used. (b) Convert that to hours and minutes. 3 marks

Stuck on 4.11? "Spends" gives subtraction. Subtract 125 from both sides first to get −9w = −72.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 (tickets) I wrote 5 + 12t = 41, not 5 + 12 + t = 41.

For 4.2 (savings) I wrote 30 + 5w = 80, with w as the unknown multiplied by the weekly rate.

For every answer I wrote a full sentence at the end (e.g. "3 tickets were bought.").

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Four-step process

1. Define the unknown. 2. Translate words into an equation. 3. Solve using inverse operations. 4. Interpret the answer in words.

Q1.2 — Labelling T = f + rn

T = total. f = fixed charge. r = rate per unit. n = number of units.

Q1.3 — Equation vs expression

(b) 12 + 4r = 40 is the equation (it has an equals sign). (a) and (c) are expressions.

Q3 — Faded kayak example

Step 1: Let h = the number of hours hired.
Step 2: 25 + 15h = 85.
Step 3: 15h = 60 ⇒ h = 4.
Step 4: The customer hired the kayak for 4 hours.

Q4.1-4.4 — Matching equations

4.1: 5 + 12t = 41. 4.2: 30 + 5w = 80. 4.3: 7 + 3k = 31. 4.4: 25 + 15h = 85.

Q4.5 — Movie tickets

Let t = tickets. 5 + 14t = 61 ⇒ 14t = 56 ⇒ t = 4. 4 tickets were bought.

Q4.6 — Hall catering

Let p = people. 80 + 12p = 260 ⇒ 12p = 180 ⇒ p = 15. 15 people attended.

Q4.7 — Gym classes

Let c = classes. 20 + 15c = 95 ⇒ 15c = 75 ⇒ c = 5. 5 classes were paid for.

Q4.8 — Van hire

Let h = hours. 45 + 20h = 145 ⇒ 20h = 100 ⇒ h = 5. The van was hired for 5 hours.

Q4.9 — Pre-paid card balance

Let c = calls. 50 − 1.20c = 20 ⇒ −1.20c = −30 ⇒ c = 25. 25 calls were made.

Q4.10 — Photographer invoice

Let p = photos. 120 + 35p = 400 ⇒ 35p = 280 ⇒ p = 8. 8 photos were edited.

Q4.11 — Student savings (subtraction)

Let w = weeks. 125 − 9w = 53 ⇒ −9w = −72 ⇒ w = 8. After 8 weeks the balance is $53.

Q4.12 — Phone-call minutes

Let m = minutes. 25 + 0.15m = 52 ⇒ 0.15m = 27 ⇒ m = 180 minutes.
(b) 180 ÷ 60 = 3 hours. 180 minutes = 3 hours exactly.