6.3 Multi-Step Quantitative & Logical Problems
Key Takeaways
- Multi-step problems follow a fixed rhythm: extract the numbers, apply a relationship, then interpret what the result means
- Know the workhorse relationships cold: speed = distance ÷ time, density = mass ÷ volume, plus percentages and simple ratios
- Estimation and order-of-magnitude checks catch most arithmetic slips before they cost a mark
- Logic puzzles wear science costumes: classification keys, ordering rock layers or life-cycle stages, and if-then chains in food webs
- Under time pressure the classic slips are dividing the wrong way round, misreading the scale, and dropping a zero — recompute with round numbers to check
6.3 Multi-Step Quantitative & Logical Problems
The trickiest Reasoning & Problem Solving questions make you do two or three things in a row: pull numbers out of a graph or table, apply a relationship, then decide what the answer means. None of the steps is hard by itself — the marks are lost in the handover between steps. This section gives you the rhythm for the quantitative problems and the patterns for the logic puzzles, with worked examples at Paper D (around Year 6) and Paper F (around Year 8) level.
The Three-Step Rhythm for Quantitative Problems
- Extract. Find the exact numbers the question needs. Check the units on axes and headings, and check the scale before reading a bar height or line position. Write the numbers down if you can.
- Apply. Use the relationship the question calls for. The workhorses are speed = distance ÷ time, density = mass ÷ volume, plus percentages and simple ratios. Choose the version that matches the question: asked for time? Rearrange to time = distance ÷ speed.
- Interpret. The final answer to an ICAS question is often not the number itself but what the number tells you — which car was faster, which material would float, whether the claim in the stimulus is supported. Never stop at step 2 without checking what was actually asked.
Worked Example (Paper D level): Speed from a Graph
A distance–time graph shows a cyclist reaching 600 metres after 4 minutes. The question asks for the cyclist's speed in metres per minute, then which of four walkers could keep up with her.
- Extract: 600 metres, 4 minutes.
- Apply: speed = distance ÷ time = 600 ÷ 4 = 150 metres per minute.
- Interpret: Now compare — a walker doing 100 m/min falls behind, one doing 160 m/min keeps up easily. The calculation was one line; the interpretation was the real question.
Worked Example (Paper F level): Density and Floating
A table gives a mystery block: mass 54 g, volume 20 cm³. Will it float on water (density 1.0 g/cm³)?
- Extract: 54 g and 20 cm³.
- Apply: density = mass ÷ volume = 54 ÷ 20 = 2.7 g/cm³.
- Interpret: anything denser than water sinks, and 2.7 > 1.0, so the block sinks. Note how the final answer is a prediction, not a number — that is typical of the senior papers.
Estimation and Order-of-Magnitude Checks
Before and after calculating, ask whether the answer is roughly the right size. 54 ÷ 20 must be a bit more than 2, because 20 × 2 = 40 and 20 × 3 = 60 — so if your working produced 0.27 or 27, something slipped. A useful habit: redo the calculation with round numbers in your head. If the exact answer and the estimate disagree wildly, find the slip before you answer.
Logic Puzzles in Science Dress
Many ICAS items contain no arithmetic at all but still demand step-by-step reasoning. Three patterns recur:
- Classification keys. A branching key sorts creatures or objects by yes/no features ("has six legs?" → yes/no). Work from question 1, answer only what the specimen in front of you actually shows, and never skip a branch. One wrong branch early guarantees a wrong name at the end.
- Ordering events. Rock layers, life cycles and the water cycle all have a natural sequence. For undisturbed rock layers, the oldest layer is at the bottom (the law of superposition), so read the stack from bottom to top to tell the story in time order. For a life cycle — egg, larva, pupa, adult for a butterfly — anchor on the one stage you are sure of and place the rest around it.
- If-then chains. Food-web questions ask you to push a change through a system: if the rabbits are removed, then the grass increases (nothing eating it) and the foxes decrease (less food). Trace the arrows one hop at a time, and beware options that jump straight to a far-away effect without the middle steps.
Arithmetic Slips Under Time Pressure — and How to Catch Them
The four classic slips, with their antidotes:
| Slip | Example | Antidote |
|---|---|---|
| Dividing the wrong way round | 20 ÷ 54 instead of 54 ÷ 20 | Ask which unit the answer should have — g/cm³ needs grams on top |
| Misreading the scale | Reading a bar as 30 when each gap is worth 5 | Compute the gap value before reading any bar |
| Dropping or adding a zero | 600 ÷ 4 = 15 instead of 150 | Order-of-magnitude check: 600 ÷ 4 must be in the hundreds |
| Answering step 2 instead of step 3 | Writing "150" when the question asked which walker keeps up | Re-read the final sentence of the question after calculating |
The deep pattern is the same one from Sections 6.1 and 6.2: slow, exact reading at the start, a small amount of careful working in the middle, and a deliberate check at the end. Multi-step problems reward the student who treats each step as its own mini-question — and they are some of the most learnable marks on the whole paper.
A toy car travels 240 centimetres down a ramp in 6 seconds. What is its average speed?
A rock sample has a mass of 90 grams and a volume of 30 cm³. Water has a density of 1.0 g/cm³. What is the rock's density, and what will it do in water?
In an undisturbed cliff face, the rock layers from bottom to top are: sandstone, then shale, then limestone. Fossil shells appear only in the limestone. Which statement follows logically?