"Our son had always been good at maths in school, but the numerical reasoning questions in his practice tests were a completely different challenge. Learning to read and interpret the problem — not just calculate — was the breakthrough." — David P., Parent of a 2025 NSW Selective Offer Recipient
What This Guide Covers
This guide explains every aspect of the numerical reasoning test as it appears in Australian selective school exams — including NSW Selective, OC, HAST, and ASET. You will find a complete breakdown of question types, 15+ worked practice questions with answers, and evidence-based preparation strategies tailored for Years 4, 5, and 6 students.
What Is Numerical Reasoning?
Numerical reasoning — also called mathematical reasoning or quantitative reasoning — is the ability to analyse, interpret, and draw conclusions from numerical information. It goes well beyond arithmetic: while strong number fluency is essential, the emphasis is on applying mathematical thinking to solve problems in real-world and abstract contexts.
In a numerical reasoning test, students encounter questions where the challenge is not simply "can you calculate?" but rather "can you work out what calculation is needed, and why?" This distinction is crucial. A student who has memorised procedures for division and percentages may still struggle if they cannot identify which procedure applies to an unfamiliar problem.
Numerical reasoning is a strong predictor of success in academically demanding environments — which is precisely why it features prominently in every major Australian selective school entrance exam.
What Makes Numerical Reasoning Different from School Maths
Standard school maths tests assess whether students have learned the content taught in class. Questions are typically set in familiar formats, and students know in advance which topics they are being tested on.
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 — students must identify the approach themselves
- No calculator — mental arithmetic speed and accuracy matter
- Time pressure — approximately 70 seconds per question
A student scoring 90% on school maths assessments may score 60% on their first numerical reasoning practice test. This is normal — and it is exactly why targeted preparation makes such a significant difference.
Numerical Reasoning at a Glance
Key facts for Australian selective school exams
Of Total Score
Mathematical Reasoning weight in NSW Selective
Questions
Mathematical Reasoning questions in 40 minutes
Per Question
Average time available per question
Calculators
No calculators permitted — mental maths essential
Everything you need to understand and practise numerical reasoning for selective school exams
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How Numerical Reasoning Appears in Australian Exams
Numerical reasoning is a core component across every major Australian selective school and gifted-education entrance test. Understanding the specific demands of each exam helps you prepare efficiently.
NSW Selective High School Placement Test
The Mathematical Reasoning component of the NSW Selective test consists of 35 multiple-choice questions completed in 40 minutes. No calculator is permitted. Questions draw primarily from Years 5–6 curriculum content but present it in challenging contexts that require genuine understanding.
The test is fully computer-based from 2026, which means students must be comfortable reading and interpreting problem scenarios on screen, without the ability to easily annotate or underline key information the way they might on paper.
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, sat by Year 4 students, includes a Mathematical Reasoning component at an age-appropriate difficulty level. Content focuses heavily on number operations, basic fractions and decimals, early proportional reasoning, and simple data interpretation. The underlying reasoning demands are similar to the Selective test, but the mathematical content is calibrated to the Year 3–4 curriculum.
HAST (Higher Ability Selection Test)
HAST includes a Quantitative Reasoning section that explicitly tests numerical and mathematical reasoning skills. HAST questions often involve data-rich contexts — interpreting tables, charts, and graphs — alongside traditional arithmetic reasoning and pattern-based number problems.
ASET (Academic Selective Entrance Test)
ASET, used for South Australian selective school entry, includes mathematical reasoning questions with a strong emphasis on problem-solving in unfamiliar contexts. Students who have practised numerical reasoning for NSW Selective exams are well-prepared for ASET.
Year-Level Note
The numerical reasoning content in selective school tests draws primarily from the Year 5–6 curriculum for the NSW Selective test, and Year 3–4 curriculum for the OC test. However, the difficulty lies in the reasoning demands, not the complexity of the mathematics. A Year 4 student who truly understands multiplication — not just the procedure — will outperform a Year 6 student who relies on memorised methods without understanding.
Types of Numerical Reasoning Questions
There are seven main categories of numerical reasoning questions in Australian selective school exams. 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
This is not a separate content area but rather a reasoning layer applied to any of the above. Multi-step problems require students 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.
The Calculation Trap
Many students lose marks by rushing into calculations before fully reading the question. In numerical reasoning tests, understanding what is being asked is just as important as knowing how to calculate it. Train students to read the question twice and identify the final goal before writing anything down.
Numerical Reasoning Practice Questions with Answers
Work through each question independently before reading the answer and explanation. Questions progress from straightforward to challenging.
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.
Preparation Strategies for Numerical Reasoning
Effective numerical reasoning preparation combines strong mathematical foundations with deliberate problem-solving practice. Here is a proven framework built on fifteen years of selective school coaching experience.
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:
- Study the concept — ensure the underlying mathematics is understood, not just memorised
- Work through examples — solve problems at low difficulty with no time pressure
- Increase difficulty — move to harder questions in the same category
- 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.
Solving Multi-Step Numerical Reasoning Problems
Read Carefully
Read the entire problem before writing anything. Identify the final question being asked — it is often at the end of a long paragraph.
Extract the Information
Write down the key numbers and what each one represents. Cross out irrelevant information if working on paper.
Plan the Steps
Identify the operations needed and the order to perform them. Say the plan aloud or write it in words before calculating.
Calculate Carefully
Perform each calculation step by step. For mental arithmetic, use number-friendly approaches (e.g., 37 × 8 = 40 × 8 − 3 × 8 = 320 − 24 = 296).
Check Reasonableness
Does the answer make sense in the context of the question? An answer of 0.03 hours for a journey is implausible; 3 hours is reasonable.
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 approximately 69 seconds per question. This 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 — there is no penalty for incorrect answers, so always guess if unsure
- 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. At BraintreeCoaching, our programmes include weekly mock tests under authentic exam conditions.
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
The Accuracy-First Principle
Many students training for numerical reasoning make the mistake of practising speed before accuracy. This embeds errors into their habits. Always establish accurate, systematic approaches first — speed develops naturally as strategies become automatic. A student who solves 30 questions accurately in 40 minutes will outscore a student who attempts all 35 but makes careless errors on 10.
Mental Maths Strategies Worth Knowing
Strong mental arithmetic is a genuine competitive advantage in numerical reasoning tests. 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%
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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 — including students two or three years ahead academically. The emphasis shifts from "can you apply the procedure?" to "can you figure out which procedure to use, and why?"
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. Number patterns are also common in Thinking Skills questions that overlap with mathematical reasoning. 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; 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. With targeted practice, school-strong maths students typically see rapid improvement.
How early should we start preparing for numerical reasoning?
For the NSW Selective test (Year 6 entry), beginning Year 5 systematic preparation allows 12–18 months of progressive skill building. For OC test preparation (Year 4 entry), starting in Year 3 is ideal. However, even 3–4 months of focused preparation can make a meaningful difference — the key is the quality of practice, not just the duration.
Are there specific numerical reasoning resources for Years 4, 5, and 6?
Yes. BraintreeCoaching provides practice resources and coaching programmes specifically calibrated to each year level. Year 4 (OC preparation) resources focus on OC-appropriate content; Year 5–6 (Selective preparation) resources extend to the full range of content and difficulty found in the Selective test. See our Free Practice Tests section for year-level-appropriate resources.
Does speed matter in numerical reasoning tests?
Yes, but accuracy matters more. The time pressure in numerical reasoning tests — around 69 seconds per question in the NSW Selective — is real, but the bigger risk for most students is careless errors on questions they could solve correctly, not running out of time. Focus first on developing systematic, accurate approaches. Speed develops naturally 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. Specifically: 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 preparation routine is valuable, particularly in the months leading up to the exam.
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Understand every section of the NSW Selective test — Reading, Maths, Thinking Skills, and Writing
Access ResourceAbstract Reasoning Test Guide
Master the abstract and non-verbal reasoning questions in the Thinking Skills component
Access ResourceSelective School Preparation Programme
Comprehensive coaching programme covering all four test components with expert support
Access ResourceHave questions about numerical reasoning preparation? Contact our team or explore our comprehensive Selective School Preparation Programme.
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