Mathematical Reasoning is one of the most important components of the NSW Opportunity Class (OC) Placement Test. Unlike traditional school mathematics, this section focuses on how students apply mathematical knowledge to solve unfamiliar problems rather than simply recalling formulas or procedures.
Success in Mathematical Reasoning depends on logical thinking, recognising patterns, selecting appropriate strategies and working accurately under timed conditions.
Mathematical Reasoning assesses a student's ability to think logically using mathematical concepts. Questions often require several steps, encouraging students to analyse information rather than rely on memorisation.
Students may encounter topics such as:
The emphasis is on applying knowledge to new situations. Each practice question below has five answer options, matching the Mathematical Reasoning format.
Strong mathematical reasoning skills help students:
These skills are valuable not only for the OC Placement Test but also for future learning in mathematics and other subjects.
Prepare your child for the NSW Opportunity Class (OC) Test with comprehensive OC practice tests, sample questions and realistic online mock exams designed for Year 4 students. The OC Test is a competitive assessment for students seeking entry to Year 5 Opportunity Classes in NSW public schools, assessing key skills including Reading, Mathematical Reasoning and Thinking Skills.
Our OC Test preparation resources are developed using the NSW curriculum, previous test patterns and NSW Department of Education sample materials to help students become familiar with the question types, difficulty level, test format and time-management requirements. With thousands of real-style OC Test questions, full-length practice tests, detailed answers and performance analysis, students can identify their strengths and weaknesses and build a more effective preparation strategy.
Regular practice helps Year 4 students improve reading comprehension, mathematical problem-solving, logical thinking, reasoning and exam confidence before the Opportunity Class Test. Whether your child is looking for OC practice tests, OC sample papers, Opportunity Class Test questions or online OC Test preparation, practising with realistic questions can provide valuable experience and help them approach the assessment with greater confidence.
This page focuses on improving the ability to:
Find the next number.
4, 9, 8, 17, 16, 33, ?
Answer: B
Explanation:
The rule alternates: multiply by 2 and add 1, then subtract 1.
16 x 2 + 1 = 33, so the next step is 33 - 1 = 32.
Find the missing number.
7, 10, 16, 25, 37, ?
Answer: D
Explanation:
The increases are 3, 6, 9 and 12.
The next increase is 15, so 37 + 15 = 52.
Find the next number.
2, 6, 5, 15, 14, 42, 41, ?
Answer: E
Explanation:
The operations alternate between multiplying by 3 and subtracting 1.
41 x 3 = 123.
A school buys 18 boxes of pencils. Each box contains 24 pencils.
The pencils are shared equally among 9 classes. Each class then uses 7 pencils.
How many pencils remain altogether?
Answer: B
Explanation:
There are 18 x 24 = 432 pencils, so each class receives 432 / 9 = 48.
Each class has 48 - 7 = 41 left. Altogether, 9 x 41 = 369 pencils remain.
A bus begins a trip with 72 passengers.
At the first stop, one-third get off and 14 get on. At the second stop, 18 get off and half as many get on.
How many passengers are then on the bus?
Answer: D
Explanation:
One-third of 72 is 24, so after the first stop there are 72 - 24 + 14 = 62 passengers.
Half of 18 is 9, so after the second stop there are 62 - 18 + 9 = 53.
Four identical notebooks and two identical pens cost $26 altogether.
Each pen costs $3.50. What is the cost of seven notebooks?
Answer: B
Explanation:
The two pens cost 2 x $3.50 = $7, so four notebooks cost $26 - $7 = $19.
One notebook costs $19 / 4 = $4.75. Seven cost 7 x $4.75 = $33.25.
Which fraction is the greatest?
Answer: E
Explanation:
Compare the closest values: 5/8 = 15/24, while 2/3 = 16/24, so 2/3 is greater.
Also, 11/18 is less than 12/18 = 2/3, and the other fractions are smaller than 5/8.
A library display contains 96 books. Three-eighths are removed in the morning.
In the afternoon, one-quarter of the books that remain are removed.
How many books are left?
Answer: B
Explanation:
Three-eighths of 96 is 36, leaving 60 books.
One-quarter of 60 is 15, so 60 - 15 = 45 books are left.
A tank is three-fifths full. After 18 litres are added, it is three-quarters full.
What is the tank's total capacity?
Answer: D
Explanation:
The increase is 3/4 - 3/5 = 15/20 - 12/20 = 3/20 of the tank.
If 3/20 is 18 litres, then 1/20 is 6 litres and the whole tank holds 20 x 6 = 120 litres.
Many students immediately begin calculating after reading a question.
A better approach is to:
This simple process reduces careless mistakes.
Students often lose marks because they:
Accuracy is usually more valuable than speed during practice sessions.
To improve Mathematical Reasoning:
Regular exposure to varied question styles helps students become more confident and flexible problem solvers.
After completing this section, continue with:
Students looking for a more structured preparation experience can also explore the Opportunity Class OC Test 2027 course, which includes full-length mock exams, detailed explanations and progress tracking.
Decimals are frequently used in Mathematical Reasoning questions to assess place value, calculation skills and logical thinking.
Which number is closest to 3?
Answer: D
Explanation:
The distances from 3 are 0.16, 0.015, 0.08, 0.005 and 0.091.
The smallest distance is 0.005, so 2.995 is closest.
A rope is 8.4 metres long. Pieces of 1.75 metres and 2.6 metres are cut off.
The remaining rope is cut into 3 equal pieces. How long is each piece?
Answer: A
Explanation:
The remaining length is 8.40 - 1.75 - 2.60 = 4.05 metres.
Each of the three equal pieces is 4.05 / 3 = 1.35 metres long.
A container holds 3.5 litres when full. Nine cups, each holding 275 millilitres, are filled from it.
How much liquid remains?
Answer: E
Explanation:
3.5 litres = 3,500 millilitres. The cups use 9 x 275 = 2,475 millilitres.
The amount left is 3,500 - 2,475 = 1,025 millilitres.
Percentages often appear in real-life contexts such as discounts, surveys and data interpretation.
A jar is 60% full of beads. After 24 beads are added, it is 75% full.
How many beads fit in the full jar?
Answer: B
Explanation:
The added 24 beads fill 75% - 60% = 15% of the jar.
If 15% is 24, then 5% is 8 and 100% is 20 x 8 = 160 beads.
A game costs $80. It is reduced by 25%, and then a $6 voucher is used.
What is the final price?
Answer: E
Explanation:
A 25% reduction is one-quarter of $80, which is $20. The sale price is $60.
Using the $6 voucher gives $60 - $6 = $54.
A class has 40 students. Twenty-five per cent bring lunch from home.
Of the remaining students, one-third buy a hot meal and the rest buy sandwiches.
What percentage of the whole class buy sandwiches?
Answer: A
Explanation:
Twenty-five per cent of 40 is 10, leaving 30 students.
One-third of 30 is 10, so 20 buy sandwiches. Since 20 is half of 40, the answer is 50%.
Measurement questions test practical mathematical reasoning using length, perimeter, area, time and capacity.
A 3 cm by 3 cm square is attached outside a 12 cm by 8 cm rectangle.
One full 3 cm side of the square is joined to part of a 12 cm side of the rectangle.
What is the perimeter of the combined shape?
Answer: C
Explanation:
The separate perimeters total 2 x (12 + 8) + 4 x 3 = 52 cm.
The shared 3 cm edge was counted twice but is inside the combined shape, so 52 - 6 = 46 cm.
A ribbon is 3.6 metres long. Eight pieces, each 35 centimetres long, are cut from it.
How much ribbon remains?
Answer: D
Explanation:
3.6 metres = 360 centimetres. The eight pieces use 8 x 35 = 280 centimetres.
The length left is 360 - 280 = 80 centimetres.
A journey begins at 11:48 am. The first part takes 1 hour 37 minutes, followed by a 28-minute stop.
The final part takes 2 hours 46 minutes. When does the journey end?
Answer: B
Explanation:
11:48 am + 1 hour 37 minutes = 1:25 pm, and the stop ends at 1:53 pm.
1:53 pm + 2 hours 46 minutes = 4:39 pm.
The square base of a square-based pyramid is glued exactly onto one face of a cube.
How many faces are visible on the combined solid?
Answer: C
Explanation:
A cube has 6 faces and a square-based pyramid has 5 faces, making 11 before joining.
The two glued faces become internal, so 11 - 2 = 9 faces are visible.
Three adjacent angles lie on a straight line. Their sizes are 38 degrees, x degrees and (x + 24) degrees.
What is x?
Answer: B
Explanation:
Angles on a straight line total 180 degrees.
38 + x + x + 24 = 180, so 2x = 118 and x = 59 degrees.
An arrow points north-east. It is reflected in a vertical mirror and then turned through a half-turn.
Which direction does it finally point?
Answer: D
Explanation:
Reflection in a vertical mirror changes north-east to north-west.
A half-turn changes north-west to south-east.
Strong mathematical reasoning is not about solving questions quicklyâit is about solving them accurately and efficiently.
Try this approach for multi-step problems:
Following a consistent method helps reduce careless mistakes and builds confidence.
Many students spend too much time on difficult questions.
A better strategy is to:
Remember that every question is worth marks, so managing your time wisely can improve your overall score.
Students often lose marks because they:
Practising carefully and reviewing mistakes are two of the best ways to improve.
A balanced routine can help students build confidence over time.
| Day | Activity | Suggested Time |
|---|---|---|
| Monday | Number Patterns | 20 minutes |
| Tuesday | Fractions & Decimals | 25 minutes |
| Wednesday | Percentages | 20 minutes |
| Thursday | Measurement & Geometry | 25 minutes |
| Friday | Mixed Mathematical Reasoning | 30 minutes |
| Saturday | Timed Practice Test | 45â60 minutes |
| Sunday | Review Incorrect Answers | 20 minutes |
Consistent practice is more effective than occasional long study sessions.
These questions require students to identify the correct sequence of mathematical operations before calculating the answer.
A school orders 18 packs of 12 exercise books. One-third of all the books are used.
The remaining books are shared equally among 8 classes. How many books does each class receive?
Answer: B
Explanation:
There are 18 x 12 = 216 books. One-third, or 72 books, are used, leaving 144.
Each class receives 144 / 8 = 18 books.
A family buys four child tickets at $13 each and three adult tickets at $21 each.
They use a $15 voucher and pay with $150. How much change do they receive?
Answer: D
Explanation:
The tickets cost 4 x $13 + 3 x $21 = $52 + $63 = $115.
After the voucher, the cost is $100, so the change from $150 is $50.
A runner completes 5 laps in the morning and 3 laps in the afternoon on a 375-metre track.
How much farther must the runner travel to reach a total of 4 kilometres?
Answer: C
Explanation:
The runner completes 8 laps, covering 8 x 375 = 3,000 metres.
Four kilometres is 4,000 metres, so another 1,000 metres are needed.
Students should be able to read tables and interpret information accurately.
Books Read During a Two-Week Challenge
| Student | Week 1 | Week 2 |
|---|---|---|
| Ava | 8 | 12 |
| Noah | 11 | 9 |
| Mia | 7 | 16 |
| Liam | 13 | 11 |
For every 6 books read, the team earns one badge. Using the table, how many complete badges does the team earn, and how many books count towards the next badge?
Answer: D
Explanation:
The team reads 8 + 12 + 11 + 9 + 7 + 16 + 13 + 11 = 87 books.
87 / 6 = 14 remainder 3, so the team earns 14 badges with 3 books towards the next badge.
Each Week 1 book is worth 2 points and each Week 2 book is worth 3 points.
Who earns the greatest number of points?
Answer: B
Explanation:
The point totals are Ava 52, Noah 49, Mia 62 and Liam 59.
Mia has the greatest total, with 62 points.
A fifth student reads the same number of books in each week.
After this student's books are added, the Week 2 total is six-fifths of the Week 1 total.
How many books did the fifth student read in each week?
Answer: D
Explanation:
The original totals are 39 in Week 1 and 48 in Week 2, a difference of 9.
If Week 2 is six-fifths of Week 1, that difference is one-fifth of Week 1. The new Week 1 total is 45, so the student added 45 - 39 = 6 books.
A number is doubled, then 9 is added. The result is multiplied by 3 to give 105.
What was the original number?
Answer: A
Explanation:
Work backwards: 105 / 3 = 35, then 35 - 9 = 26.
The original number is 26 / 2 = 13.
Four of these numbers are even and have digits that add to 9.
Which number does not belong?
Answer: E
Explanation:
Every option has digits that add to 9.
However, 18, 36, 54 and 72 are even, while 81 is odd.
A clock shows 3:30. What is the smaller angle between the hour hand and the minute hand?
Answer: B
Explanation:
At 3:30, the minute hand is at 180 degrees from 12. The hour hand is halfway between 3 and 4, at 105 degrees from 12.
The smaller angle is 180 - 105 = 75 degrees.
Estimation helps students quickly identify unreasonable answers.
Which is the best estimate for 398 x 21?
Answer: D
Explanation:
398 is close to 400, so 398 x 21 is close to 400 x 21 = 8,400.
This is also consistent with 400 x 20 = 8,000 plus about 400.
Estimate 1,987 + 3,024 - 496 by rounding each number to the nearest hundred.
Answer: B
Explanation:
The rounded numbers are 2,000, 3,000 and 500.
2,000 + 3,000 - 500 = 4,500.
High-performing students rarely rely on one method for every question. Instead, they choose the most efficient strategy based on the information provided.
Some useful approaches include:
If the final result is given, reverse each step to find the original value.
Simple sketches can help with geometry, measurement and word problems.
Number sequences often follow predictable rules involving addition, subtraction, multiplication or alternating operations.
If you're unsure, remove options that are clearly impossible before making your final choice.
The OC Placement Test assesses mental and written mathematical reasoning.
To improve without a calculator:
Developing strong number sense makes complex questions easier to solve.
Students often struggle with:
Regular exposure to a variety of question types helps students become more confident and adaptable.
Parents can encourage mathematical reasoning by incorporating maths into everyday activities.
Examples include:
These practical experiences reinforce mathematical thinking in meaningful ways.
Improvement in Mathematical Reasoning comes from consistent practice and thoughtful review.
Students should:
This approach develops both confidence and long-term problem-solving ability.
Complete the following eight challenging questions in about 10 minutes without using a calculator.
A bakery makes 18 trays of 16 muffins. One-quarter of the muffins are reserved for an order, and 59 are sold.
How many unreserved muffins remain unsold?
Answer: E
Explanation:
There are 18 x 16 = 288 muffins. One-quarter, or 72, are reserved, leaving 216 unreserved.
After 59 are sold, 216 - 59 = 157 remain unsold.
Forty per cent of a number is 72. What is 25% of the same number?
Answer: A
Explanation:
If 40% is 72, then 10% is 18, so the whole number is 180.
Twenty-five per cent is one-quarter, and 180 / 4 = 45.
Which sum is greater than 1 but less than 1 1/10?
Answer: B
Explanation:
2/3 + 5/12 = 8/12 + 5/12 = 13/12 = 1 1/12.
This is greater than 1 but less than 1 1/10. The other sums are 1 1/10, 1 3/20, exactly 1 and 1 1/5.
A concert starts at 6:47 pm. The first part lasts 1 hour 18 minutes, followed by a 25-minute interval.
The second part lasts 1 hour 36 minutes. When does the concert finish?
Answer: B
Explanation:
The first part ends at 8:05 pm, and the interval ends at 8:30 pm.
Adding 1 hour 36 minutes gives a finishing time of 10:06 pm.
A square has a perimeter of 64 cm. A rectangle has the same perimeter, and its length is three times its width.
What is the area of the rectangle?
Answer: D
Explanation:
For the rectangle, length + width = 64 / 2 = 32 cm.
If the length is three widths, four widths total 32 cm. The width is 8 cm and the length is 24 cm, so the area is 8 x 24 = 192 square cm.
Find the next number.
1, 2, 6, 15, 31, 56, ?
Answer: D
Explanation:
The increases are the square numbers: 1, 4, 9, 16 and 25.
The next increase is 36, so 56 + 36 = 92.
A 1,200-litre tank is three-quarters full. Another 180 litres are added, then two-fifths of the water is used.
How much water remains?
Answer: B
Explanation:
Three-quarters of 1,200 litres is 900 litres. After 180 litres are added, there are 1,080 litres.
If two-fifths is used, three-fifths remains: 1,080 / 5 x 3 = 648 litres.
A backpack and a hat cost $104 altogether. The backpack costs $8 more than twice the price of the hat.
The backpack is reduced by 25%, while the hat stays at full price. What is the new total cost?
Answer: D
Explanation:
If the hat costs $32, then the backpack costs 2 x $32 + $8 = $72, and together they cost $104.
A 25% reduction on $72 is $18. The new total is $72 - $18 + $32 = $86.
Use this checklist whenever you complete a practice session.
Following a consistent review process helps build long-term mathematical confidence.
On the day of the test:
Students who manage their time well often perform better than those who rush.
Mathematical Reasoning assesses how students apply mathematical knowledge to solve unfamiliar problems. It focuses on logical thinking, problem-solving and numerical reasoning rather than memorising formulas.
Students may encounter:
Short, regular practice sessions several times each week are generally more effective than occasional long sessions.
Yes. Strong mental maths skills help students solve problems efficiently and reduce calculation errors.
Students improve by:
Well-designed practice questions mirror the style and reasoning required in the official assessment, although they do not reproduce actual exam questions.
Continue your preparation by exploring:
Working through these resources helps students develop a balanced preparation strategy across all tested skill areas.
Mathematical Reasoning is a skill that develops with regular exposure to different problem types.
Students who practise consistently are more likely to:
Rather than trying to memorise solutions, focus on understanding the reasoning behind each question. This approach builds flexible problem-solving skills that can be applied in a wide range of situations.
Success in the NSW Opportunity Class Placement Test depends on developing strong reasoning skills through steady, purposeful practice. By working through a variety of mathematical problems, reviewing mistakes and applying effective strategies, students can steadily improve both confidence and performance.
Use this page alongside your Reading and Thinking Skills practice to create a balanced study routine. As your confidence grows, progress to full-length mock tests to experience realistic exam conditions and identify areas for further improvement.
When you're ready for a more structured preparation program, explore our Opportunity Class OC Test 2027 course, where you'll find comprehensive lessons, exam-style mock tests, detailed answer explanations and tools to monitor your progress.
