Mathematics • Year 10 • Unit 2 • Lesson 12
Word Problems — Mixed Challenge
Pull together every idea from Lesson 12: read carefully, define one variable, write an equation, solve, and check against the original sentence. Choose the right approach for each problem, spot another student's mistake, then design your own word problem that hits a target answer.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 12. Decide which technique applies before you start writing. Show your working. 3 marks each
1.1 Three more than four times a number is 31. Define a variable, write the equation, and solve.
1.2 Four consecutive integers sum to 78. State all four integers.
1.3 A swimming pool fills at 250 L/min. After how many minutes does it hold exactly 6000 L? Use d = rt style reasoning.
1.4 A rectangle has perimeter 50 cm and its length is 3 cm more than twice its width. Find both dimensions.
1.5 A bus leaves Newcastle for Sydney at 8:00 am at 80 km/h. A car leaves the same point at 8:30 am at 100 km/h on the same route. At what time does the car catch up?
1.6 A canteen sells 80 drinks in a lunch break. Bottled water costs $2.50 each and juice costs $3.50 each. The total taken is $232. How many of each was sold?
2. Find the mistake
Another Year 10 student has tried to solve a consecutive-numbers problem. The problem is: "The sum of three consecutive odd integers is 81. Find them." Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — three consecutive odd integers, sum = 81:
Line 1: Let n = first odd integer.
Line 2: n + (n + 1) + (n + 2) = 81
Line 3: 3n + 3 = 81
Line 4: 3n = 78 → n = 26
Line 5: Integers: 26, 27, 28.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected three integers.
Stuck? Revisit lesson § "Consecutive Numbers" — consecutive ODD (or EVEN) integers are 2 apart, not 1.3. Open-ended challenge — design your own word problem
This question has many valid answers. Be creative but show every step. 4 marks
3.1 Design your own real-world word problem that uses all four of: a fixed starting cost (flag fall, base fee, sign-up), a variable rate per unit, a total cost given, and the question "how many units?". The problem must have exactly 12 units as its answer.
In your submission, include:
(i) Your word problem written as a complete paragraph.
(ii) Variable definitions and the equation you set up.
(iii) Full solving steps showing the answer equals 12.
(iv) A one-sentence real-world check.
Bonus: Make the fixed cost and the rate use whole-dollar amounts so the arithmetic is clean.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — "Three more than four times a number is 31"
Let n = the number. Equation: 4n + 3 = 31. Subtract 3: 4n = 28. Divide by 4: n = 7. Check: 4(7) + 3 = 31 ✓.
1.2 — Four consecutive integers sum to 78
Let n = first. Equation: n + (n+1) + (n+2) + (n+3) = 78 → 4n + 6 = 78 → 4n = 72 → n = 18. Integers: 18, 19, 20, 21. Check: 18 + 19 + 20 + 21 = 78 ✓.
1.3 — Pool fills at 250 L/min, target 6000 L
Let t = minutes. Equation: 250t = 6000. Divide by 250: t = 24 minutes. Check: 250 × 24 = 6000 ✓.
1.4 — Rectangle perimeter 50, length = 2w + 3
Let w = width. Length = 2w + 3. Perimeter: 2w + 2(2w + 3) = 50 → 2w + 4w + 6 = 50 → 6w = 44 → w ≈ 7.33 cm, length ≈ 17.67 cm. (Non-integer is fine for a real measurement.) Check: 2(7.33) + 2(17.67) = 14.66 + 35.34 = 50 ✓.
1.5 — Bus 8:00 am 80 km/h, car 8:30 am 100 km/h
Bus head start in 0.5 h = 80 × 0.5 = 40 km. Let t = hours after 8:30 am. Equation: 40 + 80t = 100t → 40 = 20t → t = 2 hours. Catch-up time = 8:30 am + 2 h = 10:30 am. Check: both distances = 80(2.5) = 200 km and 100(2) = 200 km ✓.
1.6 — Canteen: 80 drinks, water $2.50, juice $3.50, $232 total
Let w = water, j = juice. w + j = 80 and 2.5w + 3.5j = 232. From the first: j = 80 − w. Substitute: 2.5w + 3.5(80 − w) = 232 → 2.5w + 280 − 3.5w = 232 → −w = −48 → w = 48, j = 32. Check: 48 + 32 = 80 ✓; 2.5(48) + 3.5(32) = 120 + 112 = $232 ✓.
2 — Find the mistake (three consecutive ODD integers, sum 81)
(a) The mistake is on Line 2.
(b) Consecutive ODD integers are 2 apart, not 1 apart. The student used (n + 1) and (n + 2), which would describe three consecutive integers in general. For odd integers, the next two are (n + 2) and (n + 4).
(c) Corrected working:
Let n = first odd integer.
n + (n + 2) + (n + 4) = 81 → 3n + 6 = 81 → 3n = 75 → n = 25.
Integers: 25, 27, 29. Check: 25 + 27 + 29 = 81 ✓ and all three are odd ✓.
Lesson § "Consecutive Numbers" warns: odd/even = +2 each step, not +1.
3 — Open-ended challenge (sample word problem)
Many valid answers — here is one construction with answer n = 12 units.
Sample problem: "Maya joins a yoga studio that charges a $40 once-off sign-up fee plus $15 per session. After several months her total bill is $220. How many sessions did she attend?"
(i) Word problem ✓ above.
(ii) Let n = number of sessions. Equation: 40 + 15n = 220.
(iii) 15n = 180 → n = 12 sessions.
(iv) Real-world check: 12 is a positive whole number — sensible. Total = $40 + $15(12) = $40 + $180 = $220 ✓.
Marking: 1 for a coherent word problem containing all four required elements; 1 for clean variable definition and equation setup; 1 for showing the answer = 12 with full working; 1 for a sensible real-world check. Full marks for any valid construction.