Mathematics • Year 8 • Unit 2 • Lesson 16
Equations in Everyday Life
Turn five real-life scenarios — phone bills, taxi rides, gym memberships, consecutive numbers and shared bills — into linear equations and solve. Then explain in words how an equation models a situation.
1. Word problems
For each: (i) define your variable, (ii) write an equation, (iii) solve it showing every step. A final number alone earns half marks.
1.1 — Taxi fare. A taxi charges a $5 flagfall plus $2 per kilometre. Mia paid $19 for her ride.
(a) Letting k = kilometres travelled, write an equation for the total cost.
(b) Solve the equation to find how many kilometres Mia travelled. 3 marks
1.2 — Gym membership. Jay's gym charges a one-off $40 sign-up fee plus $15 per month. His total spend so far is $145.
(a) Let m = number of months. Write and solve an equation for m.
(b) Check your answer by substituting back. 3 marks
1.3 — Splitting a bill. Three friends share a $32 pizza bill, but Sam also paid an extra $5 for a drink. Together they paid $32. Each share was the same number of dollars, x.
(a) Write an equation in x for "three equal shares plus $5".
(b) Solve for x. How much did each friend put in for the pizza? 3 marks
1.4 — Consecutive numbers. Three consecutive whole numbers add to 72. (Consecutive means each is one bigger than the last.) Let the middle number be n.
(a) Write the three numbers in terms of n.
(b) Form an equation in n and solve it.
(c) State the three numbers. 3 marks
1.5 — Phone plans match-up. Plan A charges $30 plus $0.50 per call. Plan B charges $20 plus $0.70 per call. Let c = number of calls per month.
(a) Write an equation that says "Plan A cost = Plan B cost".
(b) Solve for c — the number of calls at which the two plans cost the same.
(c) Below that c-value, which plan is cheaper? Explain in one sentence. 3 marks
2. Explain your thinking
This question is about communication. Full sentences. 4 marks
2.1 A classmate solves 3x + 4 = 16 like this:
3x + 4 = 16
3x = 16 ÷ 4
3x = 4
x = 4/3
In your own words explain: (i) what they did wrong on the second line, (ii) which rule from this lesson they broke, (iii) what the correct first step should have been, and (iv) the correct value of x. Use the phrase "do the same thing to both sides" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Taxi fare
(a) Cost = 5 + 2k, so 5 + 2k = 19.
(b) Subtract 5: 2k = 14. Divide by 2: k = 7 km. Check: 5 + 2(7) = 19 ✓.
1.2 — Gym membership
(a) 15m + 40 = 145. Subtract 40: 15m = 105. Divide by 15: m = 7 months.
(b) Check: 15(7) + 40 = 105 + 40 = 145 ✓.
1.3 — Splitting a bill
(a) 3x + 5 = 32.
(b) Subtract 5: 3x = 27. Divide by 3: x = $9 each (for the pizza). Check: 3(9) + 5 = 32 ✓.
1.4 — Consecutive numbers
(a) (n − 1), n, (n + 1).
(b) (n − 1) + n + (n + 1) = 72 → 3n = 72 → n = 24.
(c) The three numbers are 23, 24, 25. Check: 23 + 24 + 25 = 72 ✓.
1.5 — Phone plans match-up
(a) 30 + 0.5c = 20 + 0.7c.
(b) Subtract 0.5c: 30 = 20 + 0.2c. Subtract 20: 10 = 0.2c. Divide by 0.2: c = 50 calls.
(c) Below 50 calls, the per-call rate matters less, so the plan with the SMALLER fixed fee wins — that's Plan B (cheaper for fewer than 50 calls).
2.1 — Explain your thinking (sample response)
On line 2 the classmate divided ONLY the right-hand side by 4 — they treated "3x + 4 = 16" as if dividing one side by 4 was the same as undoing the +4. That breaks the rule "do the same thing to both sides", which is the Golden Rule of solving equations. The +4 should be undone by subtracting 4 from both sides, giving 3x = 12. Then divide both sides by 3 to get x = 4. Check: 3(4) + 4 = 12 + 4 = 16 ✓. The classmate's answer x = 4/3 is wrong because they used the wrong inverse operation on the wrong side.
Marking: 1 mark identifying the error on line 2; 1 mark naming the rule broken (same op both sides / balance); 1 mark for the correct first step (subtract 4 from both sides); 1 mark for the correct final answer x = 4 with check.