8.2 Multi-Step Operational Word Problems
Key Takeaways
- A multi-step stem is not finished when the first subtraction or multiplication is complete — apply every later clause to the current remainder, not only to the opening quantity.
- Name start, out, in, and left as spoken labels so you can track remainder on OnVUE without paper.
- Order of operations still applies when the brackets are hidden in words such as each, then, twice, remaining, and more than.
- Stop-early numbers (the result of clause one) are usually listed as traps; seeing your first result among the choices is a warning, not a confirmation.
- Fractions and rates from Chapter 7 appear here only as one step inside a longer chain — the extra skill is continuing the chain.
Why multi-step items are problem solving, not a second numeracy chapter
Chapter 7 numeracy asks whether you can compute a rate, a percentage, or a fraction of a quantity as a single move. Multi-step operational word problems in Core Abilities problem solving ask three further things: whether you keep going after the first calculation, whether you track what is left, and whether you apply order of operations when several numbers sit in one sentence.
The flavour is Queensland Fire and Rescue work: tankers, hose lengths, BA sets, portable tanks, crews, and kilometres between a rural property and a station. The numbers in this section are invented practice numbers. They are not copied from QFD's unpublished live bank, and QFD does not publish how many multi-step items you will see or what score advances you.
OnVUE still bans scratch paper and calculators. You cannot draw a column of working. You must park intermediate results as labelled mental objects: "used," "left," "each crew," "total hose." If you cannot name the object, you will accidentally reuse the opening figure.
Do not stop after the first operation
The classic trap is a two-action stem where the first action is easy and the second is a short clause at the end of the sentence.
Worked example. A rural tanker starts the shift with 3600 L of water. The first job uses 850 L. The second job uses twice as much as the first job. How much water remains?
- Step 1: first job 850 L.
- Step 2: second job 2 × 850 = 1700 L.
- Step 3: total used 850 + 1700 = 2550 L.
- Step 4: remaining 3600 − 2550 = 1050 L.
If you stop after the first operation you answer 3600 − 850 = 2750 L, which will almost certainly be a listed choice. If you treat "twice as much" as "200 extra litres" you invent a number the stem did not give. The word twice multiplies the first job's quantity, not a convenient round number you prefer.
Second example. 24 lengths of hose on the rack. Three crews take 5 lengths each. Then 4 lengths are tagged damaged and removed. How many serviceable lengths remain? Three crews × 5 = 15 taken. 24 − 15 = 9 after the issue. 9 − 4 = 5 after the damage tag. Stopping after "three crews" without multiplying by 5 yields a nonsense remainder. Stopping after 24 − 15 = 9 forgets the damaged lengths. Subtracting 4 from the original 24 and ignoring the crews is the same family of error: you applied one clause to the opening stock and abandoned the rest.
Track the remainder as its own quantity
Remainder is whatever is left after a removal (and after any return). Name it. Do not leave it as "the tank" or "the cage," because those nouns still sound like the opening quantity.
Worked example. A 2000 L tank fills four portable tanks of 350 L each. The remainder is then split equally across two dampening runs. How many litres per run?
- Filled: 4 × 350 = 1400 L.
- Remainder: 2000 − 1400 = 600 L.
- Each run: 600 ÷ 2 = 300 L.
Traps: 2000 ÷ 4 = 500 (ignored portable size); 2000 − 350 = 1650 (one portable only); 1400 ÷ 2 = 700 (split the used water, not the remainder); 2000 ÷ 2 = 1000 (split the original tank).
Second example. 18 BA sets in the cage. Morning: two crews take 4 sets each. Afternoon: one crew returns 3 sets, then another crew takes 5. How many sets are in the cage at stand-down?
- Start 18.
- After morning: 18 − 8 = 10.
- After the return: 10 + 3 = 13.
- After the last issue: 13 − 5 = 8.
The remainder changes three times. If you only subtract 8 and 5 you get 5 and miss the return. If you stop after the return you answer 13. If you subtract 4 once instead of 4 per crew you never left the first clause.
A remainder ledger you can say aloud
On OnVUE, replace a paper ledger with four spoken labels:
- Start — the opening quantity.
- Out — everything that left.
- In — everything that came back.
- Left — start minus out plus in.
Update Left after every clause, not at the end of the paragraph. "Then" in the stem is your cue to refresh Left before you read the next number.
Order of operations in operational English
Written maths uses a convention often taught as BODMAS or PEMDAS: brackets first, then orders (powers), then division and multiplication (left to right), then addition and subtraction (left to right). Word problems hide the brackets in English. Your job is to restore them.
Worked example. A station has 2 urban appliances and 1 rural tanker. Each urban appliance carries 4 lengths of 30 m hose. The tanker carries 6 lengths of 30 m. Total hose metres?
- Urban: 2 × 4 × 30 = 240 m.
- Tanker: 6 × 30 = 180 m.
- Total: 240 + 180 = 420 m.
Wrong grouping: (2 + 1) × 4 × 30 = 360 m treats the tanker as a third urban appliance. Wrong grouping: 2 + 4 × 30 + 6 × 30 = 2 + 120 + 180 = 302 m adds the appliance count as if it were metres. Multiplication binds the "each … carries" clause; addition joins two different carrying rules.
Second example. "8 firefighters plus 4 recruits each carry 3 lengths." If each of the 12 people carries 3, restore brackets as (8 + 4) × 3 = 36 lengths. If only the 4 recruits carry 3 and the 8 are a separate headcount, the grouping is 8 + (4 × 3) = 20. The word each usually multiplies across the whole named group. Plus without each is more likely addition of two different groups. Read each, per, then, remaining, twice, and more than as operation words, not flavour.
Fractions appear only as a step, not as a lesson. If a first crew takes one quarter of 48 road cones, that step is 12 cones, remainder 36. If those 36 are then shared equally among 3 stations, each station gets 12. The skill being tested is "do not treat 48 ÷ 3 = 16 as the answer" — that divided the opening stock and skipped the remainder. Chapter 7 already taught what a quarter is; here the quarter is just clause one.
| Stem fragment | Operation | What not to do |
|---|---|---|
| twice as much as the first job | multiply the first job's quantity by 2 | subtract the first job and stop |
| three crews take 5 each | 3 × 5, then subtract from stock | subtract 3, or subtract 5 once |
| remaining water is split equally | divide the remainder, not the start | divide the original tank |
| each of 2 appliances carries 4 × 30 m | 2 × 4 × 30 | add 2 + 4 + 30 |
| then 4 lengths are damaged | subtract 4 from the current remainder | subtract 4 from the opening stock only |
| 200 L more than A used | A's litres plus 200 | treat 200 as B's whole usage |
Putting a full stem together
Full worked problem. A QFR urban pump starts with 3200 L. Pump operator A uses 750 L on a car fire. Pump operator B then uses 200 L more than A used. How many litres remain for the next job?
A uses 750. B uses 750 + 200 = 950. Used together 750 + 950 = 1700. Remain 3200 − 1700 = 1500 L.
Checks that match listed traps:
- 3200 − 750 = 2450 stops after the first operation.
- 3200 − 750 − 200 = 2250 treats "200 more" as "200 litres," not "200 more than A."
- 750 + 200 = 950 offered as the remainder confuses B's usage with what is left.
Second full problem. 30 road cones in a bay. A first crew takes 6 to a crash scene. The remaining cones are shared equally among 3 stations. How many cones does each station receive?
Remainder after the first crew: 30 − 6 = 24. Each station: 24 ÷ 3 = 8. 30 ÷ 3 = 10 ignored the first removal. 24 left sitting as the answer stopped before the share-out. 30 − 6 − 3 = 21 subtracted the number of stations as if it were cones.
Third full problem. A locker holds 40 sets of structural gloves. Morning issue: 3 crews take 4 pairs each. Midday: 2 pairs come back from the workshop. Last issue: 5 pairs go to a relief crew. Pairs left?
Out in the morning: 12. Left: 28. In: 2. Left: 30. Out: 5. Left: 25. The return sits in the middle on purpose. Skipping it yields 40 − 12 − 5 = 23, which looks tidy and is still wrong.
Time pressure without paper
Use this five-move method on every multi-step stem:
- Mentally mark every operation word: used, remaining, each, twice, then, returned, split, more than.
- Compute one clause; store the result under a name ("after job 1").
- Apply the next clause to that named result, not to the opening figure, unless the stem says so.
- If choices include the result of step 1, treat that as a stop-early trap, not as confirmation.
- Skip a six-clause monster if the timer is red; the Candidate Information Pack does not require a perfect paper, and QFD does not publish a percentage you must hit.
Traps in one list
- Stop-early answers that match clause one only.
- Using the opening quantity for clause two as well as clause one.
- Ignoring each and subtracting a crew count instead of a quantity per crew.
- Treating "200 more than A" as 200.
- Dividing the start instead of the remainder.
- Restoring brackets so that addition happens before a stated multiplication.
- Sliding into shape or matrix items; those are Chapter 9.
The flowchart below is the same method drawn as a single pass.
A QFR urban pump starts with 3200 L. Operator A uses 750 L. Operator B then uses 200 L more than A used. How many litres remain?
A rack holds 24 hose lengths. Three crews take 5 lengths each. Four lengths are then tagged damaged and removed. How many serviceable lengths remain?
A cage holds 18 BA sets. Two crews take 4 sets each in the morning. In the afternoon one crew returns 3 sets and another crew then takes 5. How many sets are in the cage at stand-down?