Guides · Mathematical Reasoning
Every Mathematical Reasoning question type on the NSW Selective Test, with examples.
Mathematical Reasoning is 35 multiple-choice questions, five options (A–E) each, in 40 minutes, no calculator. That sounds like 35 unrelated puzzles, but it isn't — our own question-by-question analysis of the official past papers shows the same nine families recurring, in roughly the same proportions each time. This page goes through all nine, in plain language, with a fully worked example — stem, options, answer, and why — for four of them.
The format, confirmed
| Questions | 35, multiple choice |
| Options per question | 5 (A–E) |
| Time | 40 minutes |
| Calculator | Not permitted |
| Wrong-answer penalty | None — only the correct option scores |
Source: NSW Department of Education, placement test and practice test pages (accessed 9 July 2026). See the complete parent's guide for how this section fits alongside Reading, Thinking Skills and Writing.
The nine families, and how the paper divides between them
We built this breakdown ourselves, working through four official past papers question by question — sorting all 140 questions into repeating types, then checking the proportions held. They do, closely enough that we build every one of our practice papers to the same family mix. Roughly 60% of questions carry a diagram; the rest are text-only.
| Family | Share of paper | Topics |
|---|---|---|
| A Number sense & operations | 6 of 35 (≈17%) | Divisibility & remainders, number sequences, place value, decimal ordering |
| B Fractions, ratios & probability | 5 of 35 (≈14%) | Fraction chains, true/false fraction statements, ratio chains, probability |
| C Algebra & logical reasoning | 5 of 35 (≈14%) | Two-unknown word problems, working backwards, symbol equations, inequalities, digit puzzles |
| D Time & date | 2 of 35 (≈6%) | Elapsed time & 12/24-hour conversion, time zones, clock angles |
| E Measurement & scales | 3 of 35 (≈9%) | Dial & gauge reading, balance-scale weights, unit conversion |
| F 2D geometry | 6 of 35 (≈17%) | Shaded area, perimeter, protractor angles, symmetry, rotation & reflection, compass turns |
| G 3D geometry & spatial reasoning | 3 of 35 (≈9%) | Cube counting, nets of solids, face/edge/vertex counts |
| H Data & graphs | 2 of 35 (≈6%) | Column graphs, line graphs |
| I Combinatorics, logic & grid | 3 of 35 (≈9%) | Optimisation (largest/smallest value), grid coordinates & paths |
A handful of rarer question shapes from the source papers — map-scale questions, Venn diagrams, magic squares — turn up too rarely across the sample to pin down a reliable share of their own; they're a small share of any real paper too.
Number sense & operations
6 of 35 questions (≈17%) · text-only
The most literal "maths" questions on the paper — no word-problem dressing, just numbers. Four repeating shapes: spot the number that doesn't share a remainder with the rest (or find a common multiple), work out the rule behind a sequence and extend or backsolve it, reason about digit positions and place value, and order a mixed set of decimals or find the one closest to a target — often with a trailing-zero trap (58.2 vs 58.208) built in.
Worked example — divisibility & remainders
"Four of the numbers below have the same remainder when divided by 4. Which one has a DIFFERENT remainder?"
Why: Divide each option by 4. 66, 82, 42 and 54 all leave remainder 2. Only 16 divides exactly, leaving remainder 0 — the odd one out. The four wrong options are exactly the numbers that fit the shared pattern; picking one of them means you found the matching group, not the outlier.
Fractions, ratios & probability
5 of 35 questions (≈14%) · mostly text, one spinner diagram
Chains of fractional or ratio reasoning laid over a small story — someone gives away thirds of a sticker collection, three friends split a prize in a stated ratio, three statements about fraction size need a true/false call, or a bag of marbles or spinner sets up a probability question, sometimes after an item has already been removed.
Worked example — fraction-of-quantity chain
"Leo has a collection of 105 stickers. He gives 3/5 of them to Uma. He then gives 2/3 of the remaining stickers to Sam. How many stickers does Leo have left?"
Why: Work through the chain in order, not all at once. 3/5 of 105 is 63, leaving 42. 2/3 of that 42 is 28, leaving 14. Two of the wrong options are checkpoints along the way — 42 is what's left after just the first give-away, 63 is how many were given away first — rather than the final amount. Stopping one step early is the classic slip on multi-stage fraction questions.
Algebra & logical reasoning
5 of 35 questions (≈14%) · mostly text, some balance/box diagrams
Algebra without algebra notation. Two unknown prices or weights are encoded across two scenarios that need combining (a burger and a drink, chips and a drink, then both together), a number is put through a chain of operations that must be undone in reverse, shapes stand in for numbers across two or three equations, or a small set of candidate digits has to be searched for the one placement that satisfies every constraint.
Worked example — symbol equations
"Each shape stands for a whole number. If □ + ○ = 13, △ × ○ = 14, and □ + △ = 8, what is the value of □?"
Why: The three equations pin down one whole-number solution: □ = 6, △ = 2, ○ = 7. Two of the wrong options, 2 and 7, are the values of the other two shapes, not the one asked for — the most common mistake here is solving the system correctly and then reading off the wrong symbol.
Time & date
2 of 35 questions (≈6%) · mostly text, one clock-face diagram
Elapsed time and 12/24-hour conversions, an international flight itinerary crossing one or more time zones (with a layover to add in), or the angle between the hour and minute hands of a clock at a stated time.
Worked example — clock arithmetic
"A train is scheduled to leave at 11:25 am but is delayed by 4 hours 25 minutes. What is the new departure time in 24-hour format?"
Why: Add the delay in two clean steps: 4 hours takes 11:25 am to 3:25 pm, then 25 minutes takes it to 3:50 pm — convert to 24-hour format last, giving 15:50. The wrong options isolate the two most common slips: adding the hours but forgetting the minutes (15:25), and getting the clock face right but the am/pm half of the day wrong (03:50 the next day).
Measurement & scales
3 of 35 questions (≈9%) · mostly diagram-based
Reading a value off a drawn dial, circular gauge or measuring cylinder — sometimes with the pointer not sitting at zero when empty, a deliberate trap — working out an unknown weight from a balance-scale diagram (so many of shape A equals so many of shape B, plus one known weight), or converting and combining measurements in one problem (millimetres to centimetres, millilitres to litres).
2D geometry
6 of 35 questions (≈17%) · diagram-based
The single largest family and the most diagram-heavy: the area of an oddly-shaped shaded region built from rectangles and triangles, the perimeter of a composite shape (often with a shortcut once you notice cut notches don't change the outer boundary), an angle read off a protractor with a dual inner/outer scale, how many lines of symmetry a partly-shaded grid pattern has, what a shape looks like after a rotation or reflection, and which compass direction you're facing after a sequence of turns.
3D geometry & spatial reasoning
3 of 35 questions (≈9%) · mostly diagram-based
Counting cubes hidden inside a larger stack, or touched when every outer edge is painted; working out which flat net folds up into a given solid (or where a painted corner ends up once it's folded); or computing a solid's face-plus-edge-plus-vertex count, or the total length of all its edges, from its name and a couple of dimensions.
Data & graphs
2 of 35 questions (≈6%) · diagram-based
Reading, comparing or reconstructing a column (bar) or line graph — the value of a bar left off the chart, the difference between two categories, a weighted total across several bars, or which of several statements about the graph is actually true.
Combinatorics, logic & grid
3 of 35 questions (≈9%) · mostly text, one grid diagram
Finding the largest or smallest value that fits a set of constraints — the most items buyable for a fixed budget across a few pricing tiers, the fewest coins that make an amount — or working with coordinates on a grid: tracing a stepped path to its final cell, or finding the point equidistant from two markers. A few overlapping visual and logic question shapes from the source papers (growing patterns, number pyramids) sit closer to number work and 2D geometry, so we count them under families A and F — which is also how a real paper actually distributes them.
Where these examples come from
The worked examples above are questions from our own practice papers — written by us, not copied from an official test, and not reworded copies of anything published. The taxonomy itself is our proprietary analysis: we sorted 140 questions across four official NSW Selective Test past papers into repeating types, checked the proportions held from paper to paper, and built our practice papers to that same map — the same mix of families, the five-option multiple-choice format, and wrong answers that encode a specific, plausible mistake rather than random numbers. Every question type goes through a review pass before it appears in a paper, and we re-check our papers against the official ones as new material is released. This page isn't a claim about how any of it affects results — it's a map of what the paper actually contains, so you know what you're looking at before test day.
See the full mix in one sitting
The clearest way to check how these nine families feel in practice is to sit one paper under real timing — 35 questions, 40 minutes, no calculator.