Numerical Reasoning Test Practice: Complete Guide for Selective School (2026)

What Is Numerical Reasoning?

Numerical reasoning is the ability to analyse, interpret, and draw conclusions from numerical information. It is also called mathematical reasoning or quantitative reasoning. It goes well beyond arithmetic: strong number fluency is essential, but the emphasis is on applying mathematical thinking to solve problems in real-world and abstract contexts.

Children who score well in school maths frequently underperform on their first numerical reasoning practice test. Across our timed mock reviews, tutors record the same pattern: the shift required is from applying a familiar procedure to identifying which procedure applies in an unfamiliar context, and that shift comes through deliberate practice rather than extra arithmetic drill.

In a numerical reasoning test, your child meets questions where the challenge is not simply "can you calculate?" but rather "can you work out what calculation is needed, and why?" A child who has memorised procedures for division and percentages may still struggle if they cannot identify which procedure applies to an unfamiliar problem.

Numerical reasoning carries real weight in the selection decision. It is a scored component of every major Australian selective school entrance exam, and in the NSW Selective test it accounts for a quarter of the total score.

What Makes Numerical Reasoning Different from School Maths

Standard school maths tests assess whether children have learned the content taught in class. Questions are typically set in familiar formats, and your child knows in advance which topics are being tested.

Numerical reasoning tests are deliberately different:

• Unfamiliar contexts: familiar mathematical ideas are dressed in novel scenarios
• Multi-step problems: no single operation solves the question
• Minimal scaffolding: your child must identify the approach themselves
• No calculator: mental arithmetic speed and accuracy matter
• Time pressure: about 68 seconds per question (40 minutes ÷ 35 questions)

A large drop between school maths marks and a first numerical reasoning practice test is the norm rather than the exception. It is the pattern our tutors see most often in first mock reviews, and it is a comment on the format, not on your child.

Which Australian Exams Test Numerical Reasoning?

Numerical reasoning is a scored component of every major Australian selective school and gifted-education entrance test. The four your child is most likely to sit are the NSW Selective test, the NSW Opportunity Class test, ACER's HAST, and Western Australia's ASET.

NSW Selective High School Placement Test

The Mathematical Reasoning component of the NSW Selective test is 35 multiple-choice questions in 40 minutes, weighted at 25 per cent of the total score, with no calculator permitted. Those figures are published on the NSW Department of Education practice-tests page (opens in a new tab).

The test is delivered by computer, so your child needs to be comfortable reading and interpreting problem scenarios on screen, without easily annotating or underlining key information the way they might on paper. One practical note from the same DoE page: the downloadable PDF sample tests are older paper versions and do not reflect the current on-screen format. The Department's online practice tests simulate the actual software, so use those for format familiarity and keep the PDFs for question practice.

Key content areas tested:

• Number and operations: fractions, decimals, percentages, ratios, proportions
• Number patterns and algebra: sequences, rules, simple equations, algebraic thinking
• Measurement: area, perimeter, volume, time, speed, distance
• Geometry: angles, properties of shapes, transformations
• Statistics and probability: data interpretation, averages, basic probability
• Multi-step word problems: real-world contexts requiring multiple operations

NSW Opportunity Class (OC) Placement Test

The OC test is sat in Year 4 for Year 5 entry, and its Mathematical Reasoning component is also 35 multiple-choice questions in 40 minutes, with five options per question and no calculator. It is one of three components (with Reading and Thinking Skills), each weighted at 33.3 per cent, and there is no writing paper. These figures come from the Department's OC practice-tests page (opens in a new tab).

Content focuses on number operations, basic fractions and decimals, early proportional reasoning, and simple data interpretation. The reasoning demands resemble the Selective test at a younger level. The Department does not publish a year-by-year curriculum mapping for the component, so treat any content breakdown, including ours, as a preparation guide rather than an official syllabus.

HAST (Higher Ability Selection Test)

HAST is ACER's selection test, and its numerical component is called Mathematical Reasoning at the Primary level and Mathematical & Scientific Reasoning at the Secondary level, where science content is folded into the same paper. The full component list, and the fact that HAST is a paper-only assessment with no computer-based option, are set out on the ACER HAST programme page (opens in a new tab).

That paper format is worth planning around: a child preparing for both HAST and a NSW test needs practice in two different modes, on paper and on screen. ACER does not publish per-component question counts or time limits, so be sceptical of any site, ours included, that quotes them as fact.

ASET (Academic Selective Entrance Test)

ASET is the Western Australian academic selective entrance test, run by the WA Department of Education as the entry pathway to its GATE academic select-entry programmes. It is not a South Australian test. Sitting dates and the application timeline are published on the WA Department of Education key-dates page (opens in a new tab); the Year 7 sitting for 2027 entry was Saturday 7 March 2026, with Years 9 to 11 the following day.

ASET includes mathematical reasoning questions with a strong emphasis on problem-solving in unfamiliar contexts, so the practice below transfers. The exam families differ in format and timing, though, so treat NSW practice as skill-building for ASET rather than a rehearsal of it.

What Types of Numerical Reasoning Questions Are Asked?

Australian selective school exams draw numerical reasoning questions from seven main categories. Each category requires specific knowledge and problem-solving strategies.

1. Number Operations and Properties

These questions test fluency with the four operations (addition, subtraction, multiplication, division) and understanding of number properties such as factors, multiples, primes, and divisibility rules.

The distinguishing feature is that questions often involve relationships between numbers rather than straightforward calculation. Examples include:

• Finding the largest factor of two numbers
• Identifying which number in a set is divisible by both 6 and 9
• Determining how many prime numbers fall between two given values
• Working with odd and even number properties in multi-step contexts

Key skill: Number sense — the ability to reason about numbers without performing lengthy written calculations.

2. Fractions, Decimals, and Percentages

This category is consistently one of the most heavily tested areas. Questions assess:

• Converting between fractions, decimals, and percentages
• Comparing and ordering rational numbers
• Calculating percentage increases and decreases
• Finding fractions of quantities
• Solving problems involving GST, discounts, and profits (applied percentage contexts)
• Mixed number and improper fraction operations

Key skill: Proportional thinking — understanding that fractions, decimals, and percentages are all different expressions of the same ratio relationship.

3. Ratios and Proportional Reasoning

Ratio and proportion questions are among the most challenging for students to master, because they require a qualitative understanding of multiplicative relationships — not just procedural skill.

Common question formats include:

• Sharing a quantity in a given ratio
• Scaling recipes or mixtures up or down
• Equivalent ratios and simplification
• Rate problems (e.g., speed, cost per item, workers per task)
• Comparing quantities using ratios

Key skill: Understanding that ratio describes a multiplicative (not additive) relationship. Many students make the error of adding to both parts of a ratio when scaling, rather than multiplying.

4. Number Patterns and Algebraic Thinking

These questions present sequences or relationships and ask students to identify rules, extend patterns, or find missing values.

Types include:

• Arithmetic sequences: Constant difference between terms (e.g., 3, 7, 11, 15, ...)
• Geometric sequences: Constant ratio between terms (e.g., 2, 6, 18, 54, ...)
• Quadratic sequences: Second differences are constant (e.g., 1, 4, 9, 16, ...)
• Mixed-rule sequences: Two alternating rules (e.g., ×2, +3, ×2, +3, ...)
• Algebraic word problems: "A number is multiplied by 4 and then 7 is subtracted. The result is 21. What is the number?"
• Function machines: Input → operation(s) → output; find the rule or missing value

Key skill: Describing rules precisely and working both forwards and backwards through a sequence.

5. Measurement and Geometry

These questions apply numerical reasoning to spatial and measurement contexts:

• Calculating area and perimeter of composite shapes
• Finding volumes of rectangular prisms and other solids
• Converting between units (metres to centimetres, kilograms to grams, litres to millilitres)
• Time calculations (elapsed time, time zones, scheduled events)
• Speed, distance, and time relationships (D = S × T)
• Angle properties (angles in a triangle sum to 180°, on a straight line sum to 180°, in a revolution sum to 360°)

Key skill: Setting up the correct formula and working through multi-step calculations without a calculator.

6. Statistics and Data Interpretation

Data interpretation questions present information in graphs, tables, or charts, and ask students to read, interpret, and calculate.

Common question formats:

• Reading values from bar graphs, line graphs, pie charts, and tables
• Calculating mean, median, mode, and range from a data set
• Comparing two data sets using statistics
• Identifying trends or drawing conclusions from data
• Basic probability calculations (e.g., probability = favourable outcomes ÷ total outcomes)
• Evaluating whether a conclusion is supported by the data

Key skill: Reading carefully and resisting the temptation to use information not shown in the graph or table.

7. Multi-Step Word Problems

Multi-step word problems apply a reasoning layer over any of the six categories above. They require your child to:

• Identify what the question is actually asking (often different from what it appears to ask at first)
• Extract relevant numerical information from a wordy context
• Plan and execute two or more operations in the correct sequence
• Keep track of intermediate results
• Check whether the final answer is reasonable

These questions differentiate top performers from average candidates, because many students can perform each step individually but struggle to chain them together under time pressure.

20 Numerical Reasoning Practice Questions With Answers

These 20 questions were written by Braintree Coaching Australia tutors and are laddered from Easy to Challenging across all seven question types. Have your child work through each one independently before reading the answer and explanation.

Question 1 — Number Operations (Easy)

What is the largest two-digit number that is divisible by both 6 and 9?

Answer: 90

Explanation: Numbers divisible by both 6 and 9 must be divisible by LCM(6, 9) = 18. Multiples of 18 up to 99: 18, 36, 54, 72, 90. The largest two-digit multiple is 90.

Question 2 — Fractions (Easy)

If three-quarters of a class of 28 students play sport, how many students do NOT play sport?

Answer: 7 students

Explanation: 3/4 of 28 = 21 students play sport. 28 − 21 = 7 students do not play sport.

Question 3 — Percentages (Easy)

A jacket originally costs $80. It is on sale for 25% off. What is the sale price?

Answer: $60

Explanation: 25% of $80 = $20. Sale price = $80 − $20 = $60.

Question 4 — Number Patterns (Easy)

What is the next number in the sequence? 5, 9, 13, 17, 21, ...

Answer: 25

Explanation: The difference between consecutive terms is 4 (an arithmetic sequence). 21 + 4 = 25.

Question 5 — Ratios (Easy)

Blue and red paint are mixed in a ratio of 3:2. If 15 litres of blue paint are used, how many litres of red paint are needed?

Answer: 10 litres

Explanation: The ratio is 3:2. For every 3 parts blue, 2 parts red are used. 15 litres of blue = 5 × 3. So red paint = 5 × 2 = 10 litres.

Question 6 — Measurement (Easy)

A rectangular room is 6 metres long and 4.5 metres wide. What is the area of the room in square metres?

Answer: 27 square metres

Explanation: Area = length × width = 6 × 4.5 = 27 m².

Question 7 — Data Interpretation (Easy)

A bag contains 4 red marbles, 3 blue marbles, and 5 green marbles. What is the probability of randomly drawing a blue marble?

Answer: 3/12 = 1/4 (or 25%)

Explanation: Total marbles = 4 + 3 + 5 = 12. Probability of blue = 3/12 = 1/4.

Question 8 — Fractions and Percentages (Moderate)

A school survey found that 45% of students prefer science, 30% prefer maths, and the rest prefer English. If 360 students were surveyed, how many prefer English?

Answer: 90 students

Explanation: English = 100% − 45% − 30% = 25%. 25% of 360 = 0.25 × 360 = 90 students.

Question 9 — Number Patterns (Moderate)

What is the missing number? 2, 6, 18, 54, __, 486

Answer: 162

Explanation: Each term is multiplied by 3 (a geometric sequence). 54 × 3 = 162.

Question 10 — Multi-Step Word Problem (Moderate)

Amira earns $15 per hour working at a supermarket. She worked 6 hours on Saturday and 4 hours on Sunday. She spent $35 on transport over the weekend. How much money did she have left from her weekend earnings?

Answer: $115

Explanation: Total hours = 6 + 4 = 10. Total earnings = 10 × $15 = $150. Money left = $150 − $35 = $115.

Question 11 — Ratios (Moderate)

Three friends share a prize of $480 in the ratio 1:3:4. What is the largest share?

Answer: $240

Explanation: Total parts = 1 + 3 + 4 = 8. Each part = $480 ÷ 8 = $60. Largest share (4 parts) = 4 × $60 = $240.

Question 12 — Measurement (Moderate)

A train travels at 80 km/h. How long does it take to travel 300 km? Give your answer in hours and minutes.

Answer: 3 hours 45 minutes

Explanation: Time = Distance ÷ Speed = 300 ÷ 80 = 3.75 hours. 0.75 hours = 0.75 × 60 = 45 minutes. Total: 3 hours 45 minutes.

Question 13 — Data Interpretation (Moderate)

Five students scored the following marks on a test: 72, 85, 68, 91, 74. What is the mean score?

Answer: 78

Explanation: Mean = (72 + 85 + 68 + 91 + 74) ÷ 5 = 390 ÷ 5 = 78.

Question 14 — Algebraic Thinking (Moderate)

A number is doubled, then 9 is added, giving a result of 31. What is the original number?

Answer: 11

Explanation: Working backwards: 31 − 9 = 22. 22 ÷ 2 = 11. Check: (11 × 2) + 9 = 22 + 9 = 31.

Question 15 — Multi-Step Word Problem (Challenging)

A shop buys a pair of shoes for $60 and sells them at a 40% profit. During a sale, the marked price is discounted by 20%. What is the final sale price?

Answer: $67.20

Explanation: Selling price (before discount) = $60 × 1.40 = $84. Sale discount = 20% of $84 = $16.80. Final price = $84 − $16.80 = $67.20.

Question 16 — Proportional Reasoning (Challenging)

A 600 mL bottle of juice contains 15% real fruit. A 400 mL bottle contains 25% real fruit. If both bottles are combined into one container, what percentage of the mixture is real fruit?

Answer: 19%

Explanation: Fruit in 600 mL bottle = 15% of 600 = 90 mL. Fruit in 400 mL bottle = 25% of 400 = 100 mL. Total fruit = 90 + 100 = 190 mL. Total mixture = 600 + 400 = 1000 mL. Percentage = 190/1000 × 100 = 19%.

Question 17 — Number Patterns (Challenging)

The first four terms of a sequence are: 1, 5, 13, 29. What is the fifth term?

Answer: 61

Explanation: Differences between terms: 4, 8, 16 — each difference doubles (geometric progression of differences). The next difference is 32. Fifth term = 29 + 32 = 61.

Question 18 — Multi-Step Problem (Challenging)

A tank is 3/5 full. After 120 litres are removed, the tank is 2/5 full. What is the total capacity of the tank?

Answer: 600 litres

Explanation: The difference in fill levels is 3/5 − 2/5 = 1/5 of the tank. This equals 120 litres. So 1/5 of the tank = 120 litres. Total capacity = 120 × 5 = 600 litres.

Question 19 — Data and Statistics (Challenging)

The mean of four numbers is 18. When a fifth number is added, the mean drops to 15. What is the fifth number?

Answer: 3

Explanation: Sum of four numbers = 4 × 18 = 72. New mean of five numbers = 15, so new sum = 5 × 15 = 75. Fifth number = 75 − 72 = 3.

Question 20 — Multi-Step Word Problem (Challenging)

A school camp costs $420 per student. Students can pay the full amount upfront, or pay a deposit of 30% and the remainder in 6 equal monthly instalments. How much is each monthly instalment?

Answer: $49

Explanation: Deposit = 30% of $420 = $126. Remainder = $420 − $126 = $294. Monthly instalment = $294 ÷ 6 = $49.

How Should My Child Prepare for Numerical Reasoning?

Effective numerical reasoning preparation combines strong mathematical foundations with deliberate problem-solving practice. The five phases below follow the order our tutors use when working through timed mock reviews with a student, starting with fluency gaps because they distort everything measured after them.

Phase 1: Build Strong Foundations

Before tackling complex numerical reasoning problems, students must have complete fluency in foundational skills. Gaps in basic number knowledge cause errors on even straightforward questions.

Mental arithmetic targets (no calculator):

• Multiplication tables to 12 × 12, instant recall
• Division facts derived from multiplication tables
• Common percentage calculations: 10%, 25%, 50%, 75% of round numbers, instant
• Fraction-decimal-percentage equivalents: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/5 = 0.2 = 20%, 1/8 = 0.125 = 12.5%
• Squaring numbers to 15²: 1², 2², ..., 15²
• Doubling and halving multi-digit numbers quickly

If any of these are slow or uncertain, address them first through daily drill practice. Fluency in these foundations frees cognitive resources for the actual reasoning required by the questions.

Phase 2: Master Each Question Type

Work systematically through each of the seven question categories described above. For each category:

1. Study the concept — ensure the underlying mathematics is understood, not just memorised
2. Work through examples — solve problems at low difficulty with no time pressure
3. Increase difficulty — move to harder questions in the same category
4. Identify weak spots — find which sub-types cause consistent errors

Use a question journal to track error patterns. When you make an error, record:

• What the question asked
• What mistake you made
• The correct approach

Over time, this journal reveals systematic misunderstandings that need targeted attention.

Phase 3: Develop Problem-Solving Skills

The transition from knowing mathematics to solving unfamiliar problems requires specific practice. Use these strategies:

Read before calculating. Read the entire question before writing anything. Identify: What does the question ask? What information is given? What operation(s) connect them?

Draw a diagram. Even for purely numerical problems, a quick sketch often reveals relationships that are hidden in the text. A tape diagram (bar model) is particularly effective for ratio and proportion problems.

Work backwards. When a question gives you the result and asks for the starting value, reverse the operations. This strategy solves a surprising number of multi-step problems efficiently.

Estimate first. Before calculating, estimate the likely magnitude of the answer. This prevents wild errors from going unnoticed (e.g., getting an answer of 3,000 when the estimate suggested the answer should be around 30).

Check units. Many errors in measurement questions come from mixing units — metres with centimetres, minutes with hours. Always convert to consistent units before calculating.

Phase 4: Practise Under Exam Conditions

Once foundational skills and problem-solving strategies are established, shift to timed practice under exam conditions.

The NSW Selective Mathematical Reasoning component allows about 68 seconds per question, which is simply 40 minutes divided by 35 questions rather than a figure the Department publishes. It feels tight for complex multi-step problems but is manageable with practice.

Time management strategies:

• Don't get stuck: if a question is taking more than 90 seconds, mark it and move on
• Answer every question: the Department's OC practice materials state that no marks are lost for an incorrect, blank, or double answer, so on the OC test a considered guess costs nothing. No equivalent statement is published for the Selective test, so treat the instructions given on the day as authoritative
• Prioritise accuracy: careless errors on easy questions are more costly than skipping a hard question
• Review remaining time: in the final 5 minutes, revisit skipped questions

Regular full-length timed practice tests build the stamina and time-awareness needed on exam day. In our 2025 NSW Selective coaching cohort, fortnightly full computer-based mocks in the final term were the preparation step tutors most often credited in parent check-ins before outcomes day.

Phase 5: Targeted Weakness Remediation

After each practice test, analyse results by question category. Students typically have specific weak areas — perhaps proportional reasoning, or multi-step problems involving rates — rather than uniform weakness across all categories.

Address weaknesses through:

• Focused concept review (understanding the mathematics, not just procedures)
• Targeted practice sets in that specific category
• Reworking missed questions without time pressure to understand the correct approach
• Revisiting the same category in the next week to confirm improvement

Mental Maths Strategies Worth Knowing

Mental arithmetic speed buys thinking time: every second not spent on a calculation is a second available for working out what the question is asking. Here are high-value techniques:

Multiplication shortcuts:

• × 5: multiply by 10 then halve (36 × 5 = 360 ÷ 2 = 180)
• × 9: multiply by 10 then subtract the number (27 × 9 = 270 − 27 = 243)
• × 11 (two-digit numbers): add the digits and insert between them (35 × 11 = 3 _ 5 where _ = 3+5 = 8, so 385)
• × 25: multiply by 100 then divide by 4 (16 × 25 = 1600 ÷ 4 = 400)

Percentage shortcuts:

• 10% = divide by 10
• 5% = half of 10%
• 15% = 10% + 5%
• 20% = divide by 5
• 1% = divide by 100
• Any % = build from 1% and 10%

Fraction-decimal-percentage conversion: Fluency with the most common equivalents eliminates the need to calculate on the spot:

• 1/3 ≈ 33.3%, 2/3 ≈ 66.7%
• 1/6 ≈ 16.7%, 5/6 ≈ 83.3%
• 1/8 = 12.5%, 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5%

What to do in the first week

Five phases is a lot to hold in your head on a Sunday night. If you want one concrete week to start with, this is the sequence our tutors use in a first fortnight:

1. Sit the 20 questions above, untimed. No help, no calculator, one sitting. Mark them together and write down which of the seven categories each wrong answer came from.
2. Test the two fluency gates. Multiplication tables to 12 × 12 and the fraction-decimal-percentage equivalents listed above. If either is slow, that is the week's priority, ahead of any reasoning work.
3. Pick the single weakest category from step 1 and spend three short sessions on it, concepts first, then questions, with no clock running.
4. Start a question journal. For every error: what the question asked, what went wrong, the correct approach. This is the artefact that makes week four useful.
5. End the week with one timed section. Thirty-five questions, 40 minutes, no calculator. Compare the untimed and timed accuracy from steps 1 and 5. The gap between them is what timed practice is for.

On workload, our academic panel's guidance for Year 5 and 6 selective and OC preparation is a maximum of around five to six hours per week including class time, tapering in the final fortnight. That is a ceiling we recommend, not a target to hit, and it should come down if sleep, mood, or family time suffer.

Frequently Asked Questions

How is numerical reasoning different from the maths my child does at school?

School maths typically assesses whether students have learned the content taught in that term, in familiar formats. Numerical reasoning tests present mathematics in unfamiliar, multi-step contexts and assess students against a wider ability range. The emphasis shifts from applying a procedure to figuring out which procedure to use, and why. In our tutors' mock reviews, this is the single most common reason a school-strong child stumbles on a first practice test.

Which areas of numerical reasoning are most important to focus on?

For NSW Selective and OC tests, fractions, percentages, and ratios are consistently the most heavily tested areas. Multi-step word problems are where the highest-difficulty questions appear, and number patterns are also common. Prioritise these areas while ensuring core arithmetic foundations are solid.

My child is strong at school maths but struggling with practice tests. Why?

This is very common and has a simple explanation. School maths tests assess content knowledge in familiar formats, while numerical reasoning tests assess reasoning ability in unfamiliar formats. Improving requires explicit practice with the problem-solving approach: reading carefully, planning before calculating, and working methodically.

How early should we start preparing for numerical reasoning?

For the NSW Selective test, which your child sits in Year 6 for Year 7 entry, beginning systematic preparation in Year 5 allows 12 to 18 months of progressive skill building. For the OC test, sat in Year 4 for Year 5 entry, starting in Year 3 is ideal. Even 3 to 4 months of focused preparation can make a meaningful difference, because the key is the quality of practice rather than the duration alone.

Are there numerical reasoning resources for Years 4, 5, and 6?

Yes. Braintree Coaching Australia publishes practice resources calibrated to each year level. Year 4 resources focus on OC-appropriate content, while Year 5 and Year 6 resources extend to the full range of content and difficulty found in the Selective test. The free mock tests library is the starting point for both.

Does speed matter in numerical reasoning tests?

Yes, but accuracy matters more. The time pressure is real: 35 questions in 40 minutes works out to about 68 seconds each in the NSW Selective Mathematical Reasoning component. The bigger risk for most children is careless errors on questions they could solve correctly. Develop systematic, accurate approaches first, and speed will follow as those approaches become automatic.

How does the computer-based format affect numerical reasoning preparation?

The question content is unchanged, but students must be comfortable working on screen. They cannot easily underline or annotate the question text, and they must interpret graphs and tables on a monitor rather than paper. Including some computer-based practice in your routine is valuable, particularly in the months before the exam.

For a structured pathway, explore the Selective School Preparation Programme hub and the in-depth NSW Selective test format guide. To turn this practice into a routine, the NSW Selective test preparation strategies guide and our NSW Selective practice tests and resources library extend the practice covered above. Families wanting full coverage often choose the Selective Ultimate Pack course. If your child is also sitting scholarship exams, the worked EduTest Year 5 sample paper is a useful contrast in format.

Related Reading:

• NSW Selective School Test Components: Complete 2026 Guide
• Abstract Reasoning Test Practice: Complete Guide
• Selective School Success Strategies for 2026
• Victoria SEHS preparation hub — Victorian selective entry for Melbourne High, Mac.Robertson, Nossal and Suzanne Cory