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Free 11+ (11 Plus) Maths Papers & Answers (PDF)

• Updated: July 8, 2024

Table of Contents

Introduction.

This is an up-to-date list of only the best free PDF 11+ Maths practice papers, suitable as preparation for the 2024 and 2025 11+ entrance exams.

No sign up or login is required! Everything here is completely free and in PDF format.

Answers are provided for papers where possible.

Quality papers

We keep this page regularly updated and aim to only provide the highest quality free 11 Plus tests available.

For expert practice papers with detailed answers and explanations, take a look at our 11+ Maths practice papers . Check out our reviews  here !

Expert 11+ Maths Practice Papers for 2024 and 2025 exams >>

Independent schools, aldenham school.

Aldenham School – 11 Plus Maths Sample Paper – For 2023 Entry

Aldenham School – 11 Plus Maths Sample Paper – 2022 Aldenham School – 11 Plus Maths Sample Paper – 2019 Aldenham School – 11 Plus Maths Sample Paper – 2017 Aldenham School – 11 Plus Maths Sample Paper – 2015

Alleyn's School

Alleyn’s – 11 Plus Maths Sample Paper 1 – for 2023 Alleyn’s – 11 Plus Maths Sample Paper 2 – for 2023 Alleyn’s – 11 Plus Syllabus 2023

Alleyn’s – 11 Plus Maths Sample Paper 1 – 2022 Alleyn’s – 11 Plus Maths Sample Paper 2 – 2022 Alleyn’s – 11 Plus Syllabus

Bancroft's School

Bancroft’s School – 11 Plus Maths Sample Paper 2023 Bancroft’s School – 11 Plus Maths Sample Paper 2022 Bancroft’s School – 11 Plus Maths Sample Paper 2020 Bancroft’s School – 11 Plus Maths Sample Paper 2017 Bancroft’s School – 11 Plus Maths Sample Paper 2016 Bancroft’s School – 11 Plus Maths Sample Paper 2013

Blackheath High School

EP – Blackheath High School – 11 Plus Maths Practice Paper 2007

Brentwood School

Brentwood School – 11 Plus Maths Sample Paper

Chigwell School

Chigwell School – 11 Plus Maths Sample Paper

Christ's Hospital School

Christ’s Hospital School – 11 Plus Maths Paper 2014

City of London School

City of London School – 11 Plus Maths Sample Paper for 2022 exam with answers City of London School – 11 Plus Maths Checklist

City of London School for Girls

City of London School for Girls – 11 Plus Maths Sample Paper 2021 with answers

City of London School for Girls – 11 Plus Maths Sample Paper

Colfe's School

Colfe’s School – 11 Plus Maths Sample Paper 1

Dulwich College

Dulwich College – Year 7 Maths Specimen Paper B Dulwich College – Year 7 Maths Specimen Paper C Dulwich College – Year 7 Maths Specimen Paper Dulwich College – Year 7 Maths Specimen Paper – Mark Scheme Dulwich College – 11 Plus Maths Specimen Paper A Dulwich College – 11 Plus Maths Specimen Paper A – Mark Scheme Dulwich College – 11 Plus Maths Specimen Paper B Dulwich College – 11 Plus Maths Specimen Paper B – Mark Scheme Dulwich College – 11 Plus Maths Specimen Paper C Dulwich College – 11 Plus Maths Specimen Paper C – Mark Scheme Dulwich College – 11 Plus Maths Specimen Paper D Dulwich College – 11 Plus Maths Specimen Paper D – Mark Scheme Dulwich College – 11 Plus Maths Specimen Paper E Dulwich College – 11 Plus Maths Specimen Paper E – Mark Scheme Dulwich College – 11 Plus Maths Specimen Paper F Dulwich College – 11 Plus Maths Specimen Paper F – Mark Scheme

Eltham College

Eltham College – Sample 11 Plus Maths Paper – 2024 Eltham College – Sample 11 Plus Maths Paper – 2022 Eltham College – Sample 11 Plus Maths Paper – 2020 Eltham College – 11 Plus Maths Sample Paper 2019 Eltham College – 11 Plus Maths Sample Paper 2019 – Mark Scheme

Emanuel School

Emanuel School – Entrance Exam A Mathematics Emanuel School – Entrance Exam B Mathematics Emanuel School – Entrance Examination Paper 2013

Forest School

Forest School – 11 Plus Maths Sample Paper 2016 Forest School – 11 Plus Maths Sample Paper 2020

The Haberdashers' Aske's Boys' School Elstree

The Haberdashers’ Aske’s Boys’ School – 11 Plus Maths Entrance Exam 2017 The Haberdashers’ Aske’s Boys’ School – 11 Plus Maths Entrance Exam 2016 The Haberdashers’ Aske’s Boys’ School – 11 Plus Maths Entrance Exam 2014 The Haberdashers’ Aske’s Boys’ School – 11 Plus Maths Entrance Exam 2011 The Haberdashers’ Aske’s Boys’ School – 11 Plus Maths Entrance Exam 2010 The Haberdashers’ Aske’s Boys’ School – 11 Plus Maths Entrance Exam 2009

Hampton Court House

Hampton Court House – Sample Paper 1 Hampton Court House – Sample Paper 2

Highgate School

Highgate School – 11 Plus Maths Sample Paper A Highgate School – 11 Plus Maths Sample Paper A – Answers Highgate School – 11 Plus Maths Sample Paper B Highgate School – 11 Plus Maths Sample Paper C

James Allen's Girls' School

James Allen’s Girls’ School – 11 Plus Maths Sample Paper 1 James Allen’s Girls’ School – 11 Plus Maths Sample Paper 2

Kent College

Kent College – 11 Plus Maths Sample Paper 2015 Kent College – 11 Plus Maths Sample Paper 2011 Kent College – 11 Plus Maths Sample Paper 1

King's College Junior School - Wimbledon

Kings College School Wimbledon Specimen Paper 2023 – Paper A Kings College School Wimbledon Specimen Paper 2023 – Paper B Kings College School Wimbledon Specimen Paper 2018 – Section A Kings College School Wimbledon Specimen Paper 2018 – Section B Kings College School Wimbledon Specimen Paper 2017 – Section A Kings College School Wimbledon Specimen Paper 2017 – Section B King’s College School Wimbledon – 11 Plus Maths Specimen Paper

The King's School Chester

The King’s School Chester – 11 Plus Maths Specimen Paper 2 The King’s School Chester – 11 Plus Maths Specimen Paper 2 – Mark Scheme The King’s School Chester – 11 Plus Maths Specimen Paper The King’s School Chester – 11 Plus Maths Specimen Paper – Mark Scheme

Latymer Upper School

Latymer Upper School – 11 Plus Maths & English Sample Papers 1 – 2020 Entry Latymer Upper School – 11 Plus Maths & English Sample Papers 2 – 2020 Entry Latymer Upper School – 11 Plus Maths Sample Paper 2014

Magdalen College School

Magdalen College School – 11 Plus Maths Sample Paper

Manchester Grammar School

MGS 2019 – Arithmetic Section A MGS 2019 – Arithmetic Section A – Answers MGS 2019 – Arithmetic Section B MGS 2019 – Arithmetic Section B – Answers

MGS 2018 – Arithmetic Section A MGS 2018 – Arithmetic Section A – Answers MGS 2018 – Arithmetic Section B MGS 2018 – Arithmetic Section B – Answers

MGS 2017 – Arithmetic Section B MGS 2017 – Arithmetic Section A – Answers MGS 2017 – Arithmetic Section B MGS 2017 – Arithmetic Section B – Answers

MGS 2016 – Arithmetic Section A MGS 2016 – Arithmetic Section A – Answers MGS 2016 – Arithmetic Section B MGS 2016 – Arithmetic Section B – Answers

MGS 2014 – Arithmetic Section A MGS 2014 – Arithmetic Section A – Answers MGS 2014 – Arithmetic Section B MGS 2014 – Arithmetic Section B – Answers

MGS 2013 Arithmetic Paper 1 MGS 2013 Arithmetic Paper 1 – Answers MGS 2013 Arithmetic Paper 2 MGS 2013 Arithmetic Paper 2 – Answers

Merchant Taylors' School

Merchant Taylors’ School – 11 Plus Maths Practice Paper 1 Merchant Taylors’ School – 11 Plus Maths Practice Paper 2 Merchant Taylors’ School – 11 Plus Maths Practice Paper 2009

North London Collegiate School

North London Collegiate School – 11 Plus Maths Sample Paper 2023

North London Collegiate School – 11 Plus Maths Sample Paper

Oundle School

Oundle School – Maths – 11 Plus Past Paper 2020 Oundle School – Maths – 11 Plus Past Paper 2019 Oundle School – Maths – 11 Plus Past Paper 2018 Oundle School – Maths – 11 Plus Past Paper 2017 Oundle School – Maths – 11 Plus Past Paper 2016 Oundle School – Maths – 11 Plus Past Paper 2008

The Perse Upper School

The Perse Upper School – 11 Plus Maths Specimen Specimen Paper 1 The Perse Upper School – 11 Plus Maths Specimen Specimen Paper 2 The Perse Upper School – 11 Plus Maths Specimen Specimen Paper 3 The Perse Upper School – 11 Plus Maths Specimen Specimen Paper 4 The Perse Upper School – 11 Plus Maths Specimen Specimen Paper 5

The Queen's School Chester

The Queen’s School Chester – 11 Plus Maths Sample Paper

Reigate Grammar School

Reigate Grammar School – 11 Plus Maths – Entrance Exam Exemplar 2022 entry

Other: Reigate Grammar School – 11 Plus Maths – Maths paper summary 2020

Reigate Grammar School – 11 Plus Maths Sample Paper 2019 Reigate Grammar School – 11 Plus Maths Sample Paper 2018 Reigate Grammar School – 11 Plus Maths Sample Paper 2013 Reigate Grammar School – 11 Plus Maths Sample Paper 2012

Royal Russell School

Royal Russell School – 11 Plus Maths Sample Paper

Sevenoaks School – Maths Sample Paper 2022 Sevenoaks School – Maths Sample Paper 2021 Sevenoaks School – Maths Sample Paper 2020 Sevenoaks School – Maths Sample Paper 2019 Sevenoaks School – Maths Sample Paper 2018 Sevenoaks School – Maths Sample Paper 2017 Sevenoaks School – Maths Sample Paper 2016 Sevenoaks School – Maths Sample Paper 2015 Sevenoaks School – Maths Sample Paper 2014 Sevenoaks School – Maths Sample Paper 2013 Sevenoaks School – Maths Sample Paper 2012 Sevenoaks School – Maths Sample Paper 2011 Sevenoaks School – Maths Sample Paper 2010

Solihull School

Solihull School – 11 Plus Maths Sample Paper

St Albans School - Hertfordshire

St Albans School – 11 Plus Maths Sample Paper 2020 St Albans School – 11 Plus Maths Sample Paper 2019 St Albans School – 11 Plus Maths Sample Paper 2017 St Albans School – 11 Plus Maths Specimen Paper

St Francis' College

St Francis’ College – 11 Plus Maths Sample Paper 2017 St Francis’ College – 11 Plus Maths Sample Paper 2017 Mark Scheme

St George’s College Weybridge

St George’s College Weybridge – 11 Plus Maths Syllabus and Sample Questions 2012 St George’s College Weybridge – 11 Plus Maths Sample Paper

St John's School Leatherhead

St John’s School Leatherhead – 11 Plus Maths Sample Paper

St Paul's Girls' School

St Paul’s Girls’ School – 11 Plus Maths Example Paper 1 2017 St Paul’s Girls’ School – 11 Plus Maths Example Paper 2 2017

Streatham and Clapham High School

Streatham and Clapham High School – 11 Plus Maths Sample Paper Section A 2019 Streatham and Clapham High School – 11 Plus Maths Sample Paper Section B 2019 Streatham and Clapham High School – 11 Plus Maths Specimen Paper

Sydenham High School

Sydenham High School – 11 Plus Maths Sample Paper

Trinity School Croydon

Trinity School – 11 Plus Maths Sample Questions and Answers

Whitgift School

Whitgift School – 11 Plus Maths Entrance Exam Sample Questions and Syllabus

Independent School Boards

The north london independent girls' schools' consortium.

Please note: this board is now uses online tests and is called 'The London 11+ Consortium'.

Group 1 – 11 Plus Maths 2016 Group 1 – 11 Plus Maths 2015 Group 1 – 11 Plus Maths 2014 Group 1 – 11 Plus Maths 2013 Group 1 – 11 Plus Maths 2012 Group 1 – 11 Plus Maths 2011 Group 1 – 11 Plus Maths 2010 Group 1 – 11 Plus Maths 2009 Group 1 – 11 Plus Maths 2008

Group 2 – 11 Plus Maths 2016 Group 2 – 11 Plus Maths 2015 Group 2 – 11 Plus Maths 2014 Group 2 – 11 Plus Maths 2013 Group 2 – 11 Plus Maths 2012 Group 2 – 11 Plus Maths 2011 Group 2 – 11 Plus Maths 2010 Group 2 – 11 Plus Maths 2009  Group 2 – 11 Plus Maths 2008

Grammar Schools

Dame alice owen's school.

Dame Alice Owen’s School – 11 Plus Maths Familiarisation Paper Dame Alice Owen’s School – 11 Plus Maths Familiarisation Paper – Mark Scheme

The Halifax Grammar Schools

The Halifax Grammar Schools – 11 Plus Admissions Test Mathematics Practice Paper

South West Hertfordshire Schools Consortium

South West Hertfordshire Schools Consortium – 11 Plus Maths Specimen Paper

St Anselm's College

St Anselm’s College – 11 Plus Maths Sample Paper – 2022 St Anselm’s College – 11 Plus Maths Sample Paper 1 – 2021 St Anselm’s College – 11 Plus Maths Sample Paper 2 – 2021

St Olave's Grammar School

St Olave’s Grammar School – 11 Plus Maths Sample Paper – 2019 St Olave’s Grammar School – 11 Plus Maths Sample Paper – 2015

Assessment/Exam Providers

GL Assessment – Maths Familiarisation Test 1 GL Assessment – Maths Familiarisation Test 1 – Answer Sheet GL Assessment – Maths Familiarisation Test 2 GL Assessment – Maths Familiarisation Test 2 – Answer Sheet GL Assessment – Maths – Parent’s Guide and Answers

CSSE – 11 Plus Maths Past Paper – 2019 CSSE – 11 Plus Maths Past Paper – 2019 – Answers

CSSE – 11 Plus Maths Past Paper – 2018 CSSE – 11 Plus Maths Past Paper – 2018 – Answers

CSSE – 11 Plus Maths Past Paper – 2017 CSSE – 11 Plus Maths Past Paper – 2017 – Answers

CSSE – 11 Plus Maths Past Paper – 2016 CSSE – 11 Plus Maths Past Paper – 2016 – Answers

CSSE – 11 Plus Maths Past Paper – 2015 CSSE – 11 Plus Maths Past Paper – 2015 – Answers

View more about the CSSE exam >>

Bond 11 plus.

Bond – 11 Plus Practice Maths Test – 2006 Bond – 11 Plus Practice Maths Test – 2006 – Answers Bond – 11 Plus Sample Maths Paper – 2004

CGP – 11 Plus CEM Sample Maths Test 2018 CGP – 11 Plus CEM Sample Maths Test 2018 – Answer Sheet CGP – 11 Plus CEM Sample Maths Test 2018 – Answers

For GL Assessment and other 11+ test providers CGP – 11 Plus GL Sample Maths Test 2018 CGP – 11 Plus GL Sample Maths Test 2018 – Answer Sheet CGP – 11 Plus GL Sample Maths Test 2018 – Answers

Schofield & Sims

Schofield and Sims – 11 Plus Sample Maths Progress Paper with answers

Owl Tutors – 11 Plus Maths Sample Paper 1 with answers Owl Tutors – 11 Plus Maths Sample Paper 2 with answers

IPS – 11 Plus Sample Maths Paper – 2004 with answers

What is in the 11+ maths exam?

The questions in an 11+ maths paper depend on the school, area or exam board that has written the test.

As a general guide, the 11+ maths exam usually covers the topics in the KS2 maths syllabus, including topics taught at the beginning of Year 6.

Some schools also include questions on KS2 maths topics that come up later in the Year 6 academic year. This means that students must learn these topics outside of school classes to prepare for the 11+ exams.

Key topics in the KS2 maths National Curriculum syllabus

• Arithmetic – operations, number properties, decimals, BIDMAS
• Algebra, simple linear equations
• Fractions and Percentages
• Measurements – unit conversions, area and volume
• Graphs – Venn diagrams, line graphs, pie charts, pictograms
• Statistics – mean, median, mode, range
• Geometry – 2D shapes, angles, coordinates
• Word questions

Examberry Papers

Take a look at the expert 11+ Maths practice papers in our shop for the best possible practice for the 2024 and 2025 11+ exams!

We have carefully written each paper to be up-to-date for the current exam requirements.

Answers and explanations are included for all of our papers.

11+ Maths Pack 1 (Download)

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Introduction When it comes to short questions and multiple-choice style answers, your child’s mental maths skills might be their ticket to saving those precious extra

11+ Non-Verbal Reasoning: Understanding Visual Cues

Find out some of the things that your child should be looking out for while solving different categories of questions!

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What is the 11 plus maths paper?

• What maths is covered in 11 plus past paper maths?
• How to prepare your kids for 11 plus maths questions?

11 plus exams test children on their maths , English , verbal reasoning, and non-verbal reasoning skills. Many students find the mathematics questions in the 11 plus past paper maths challenging as they are tricky and require strong reasoning and analytical skills. 

So what exactly is an 11 plus maths exam ? What concepts does the syllabus cover? Let’s find out.

The eleven plus exam is an exam taken by students in the U.K. As the name suggests, it is usually administered to 10-11-year-olds during their final year of primary education. 

Not all primary schools in the UK offer the 11+ but, if your child does take the exam, their result will determine which kind of secondary school they go to, either grammar school or comprehensive. What does this mean? Their scores on these exams are important. 

Many children are known to struggle with the maths eleven plus exam. Maths is a daunting yet important subject to master for primary school children. If your child needs some extra support in maths, why not book a free trial lesson with one of our GoStudent tutors? They can create a personalised learning plan for your child to help them hit top marks.

We've also created some super valuable 11+ maths practice resources. Download your very own GoStudent maths practice paper and answer booklet right here:

What maths is covered in 11 plus maths paper?

11 plus maths papers are primarily used to assess your child’s problem-solving and detail analysis skills. The questions are asked according to the national curriculum, which includes topics like:

Fractions and decimals

Measurements, data handling.

Mean, mode, median and everything in between. 

How to prepare your kids for 11 plus maths questions

Here’s a list of things that you can do to help your child with their 11+ preparation:  

Start early

Develop their skills, using practice tests, use learning apps.

11 plus maths papers can be challenging and tricky. However, with solid preparation, your child can boost their chances of getting good marks.

Get an 11+ tutor

IIf you want your kids to do well in this crucial exam, we recommend hiring an experienced 1:1 tutor .

GoStudent has a pool of experienced tutors who can guide your kids in their studies and prepare them for their 11 plus maths papers. Start with a free trial now and feel the GoStudent difference! 

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11 Plus Maths Papers

The following papers have been written by our 11 Plus Maths tutors, all of whom are qualified teachers with experience of preparing students for entrance exams. They are free for anyone to use for non-commercial use. The papers are designed to reflect the various examination styles used and expected levels at 11+ Maths.

11 Plus Maths Papers - 7 papers available

11 + Maths Non calculator Paper 8

11 + Maths Non calculator Paper 9

11 Plus Maths Paper 1

11 Plus Maths Paper 2

11 Plus Maths Paper 3

11 Plus Maths Paper 4

11 Plus Maths Paper 5

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Sarah is an upper primary teacher who specialises in online 11+ tuition for students in Years 4 , 5 and 6 (8- 11...

Sarah is an upper primary teacher who specialises in online 11+ tuition for students in Years 4 , 5 and 6 (8- 11 year old). Her speciality is preparing students for entry into competitive...

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Raj has a very strong Mathematical background having studied for degrees in Mathematics at Keele University (BA Hons) and Oxford University (MSc), and...

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Are you looking for a tutor with specific school entrance experience?

Browse our school guide, 11 plus maths paper guide.

The 11 plus maths paper is a crucial part of the assessment process. These papers are challenging and test a wide range of mathematical concepts, as well as problem solving skills.

Most maths papers will start with simple one step questions such as 234 x 27 and write 7/8 as a decimal , before moving on to harder word and context based problems. Owl Tutors’ maths papers are designed to support your child’s familiarisation and practice of the mathematical topics in the 11 plus assessment.

Once completed, these papers will give you and/or your tutor a better understanding of the topics your child needs further help with. 

What skills are being tested in 11 Plus Maths Papers?

Firstly, depending on the school you are applying to, there are different assessment processes. For some schools your child will have to sit a multiple choice pretest (usually ISEB) and then sit another written maths paper, other schools just do one or the other. In any case all maths papers cover, in various degrees of difficulty, these six main areas: 

• Properties of numbers, number problems and calculations:  This includes, but is not limited to, the four operations, prime/squared/cubed numbers, HCF and LCM, sequences, divisibility rules and negative numbers. 
• Fractions, decimals and percentages (FDP):  FDP plays a big part in the 11 plus maths paper. In most papers there is a large proportion of questions that have a FDP element to them. Therefore, it is really important that your child is confident in converting between FDP, finding FDP of numbers, as well as applying these to word problems. 
• Ratio: Again, ratio is a big part of most 11 plus maths papers. As well as being able to work out simple ratios, it is really important that your child can identify when to use ratios to solve more complex word problems. 
• Algebra: Depending on the school you are applying to, your child may not get tested on algebra. However, basic equations do feature in some harder exams, and for schools such as St Pauls, Highgate and Latymer Upper there have been problem solving questions where you need to use simultaneous equations. 
• Measurement: A seemingly simple topic. However, conversions can cause confusion and lead to silly mistakes and lost marks. Make sure your child is really confident converting between metric units, as well as knowing the conversion rates between feet/inches and cm/m. 
• Shapes and space : These questions will test your child’s ability to work out area and perimeter of quadrilaterals, triangles, parallelograms and sometimes trapeziums, and the application of this knowledge to word problems. 
• Data handling : For many children data handling is a welcome break from the other more intense topics. These questions involve reading data from charts, drawing bar charts and graphs, as well as using the four averages. From my experience many papers include ‘backwards mean’ questions, so be mindful that your child can find the mean , but can also find the missing number when given the mean . 
• Problem solving : Problem solving can be tricky as there are no specific strategies to teach. Therefore, practice is really important. Often, the problem solving questions get recycled throughout exam papers over the years, so try and expose you child to as many practice papers as possible!  

How to use Owl Tutors’ Maths Papers

Owl Tutors’ maths practice papers will cover all of the above topics through a variety of different questions. These questions are designed to not only test your child’s ability to use a mathematical concept but to also: 

• Develop your child’s confidence in applying these concepts to one and two step word problems.
• Within one question use multiple mathematical concepts such as using ratio, FDP and the four operations to solve more complex word problems. 
• Develop your child’s abilities to problem solve.

If you are in the early stages of your 11 plus practice, I would suggest not to time your child for the first few papers. It is best to give your child space to familiarise themselves with the questions without the impending doom of a ticking clock which can cause rushing and mistakes. In the early stages, if your child needs to complete the paper in two goes, that’s fine. 

Once they have finished their first few papers, then you or a tutor can mark it and find out where the gaps are. This is also a good time to see where your child makes most of their ‘silly mistakes’, so you can support them in checking their work.

Needless to say, it is important for your child to show their workings! Although easier said than done for some, there are allocated marks for the correct use of a mathematical strategy, even if there is a small error in the answer.

And finally, work up to timing your child. Once they have a few papers under their belt, then is the time to start adding a timeframe. Speed and accuracy are key, but it does not come overnight, so keep practising! 

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11 Plus Maths Exam: Key Topics, Tips, and Practice Questions

Navigating the journey to secondary education in the UK often involves the challenging but pivotal 11 Plus exam . A significant component of this assessment is the 11 Plus Maths paper, a critical step in determining whether a child will secure a place in their desired grammar school. As parents or guardians, understanding the nuances of the 11 Plus Maths exam, from the key topics it covers to the types of questions that frequently appear, can greatly enhance your child’s preparation strategy.

In this blog post, we will demystify the 11 Plus Maths exam, shedding light on the breadth of mathematical knowledge required, the format of the exam, and the typical maths questions your child might encounter. We’ll also offer practical tips for preparing for the exam, illustrate how a tutor can make a significant difference, and provide some sample questions for your child to practice.

Whether you’re curious about the maths topics covered in the 11 Plus exam, seeking examples of past maths papers, or looking for ways to boost your child’s performance on hard questions, you’re in the right place. This comprehensive guide is here to support you and your child on the path to 11 Plus success.

Remember, the road to acing the 11 Plus Maths paper begins with understanding what lies ahead. So, let’s dive in!

Understanding the 11 Plus Maths Exam

Embarking on the 11 Plus journey requires a firm understanding of the exam itself. For many students, the Maths section often seems formidable. Yet, with the right insight into the exam’s structure and content, you can guide your child to develop a solid footing.

The 11 Plus Maths exam aims to assess a child’s proficiency in numerical understanding and mathematical reasoning. It delves into a range of mathematical concepts that fall within the Key Stage 2 Maths curriculum. So, if your child has been paying attention in their maths lessons, they’re already on the right path!

The exact format of the maths test can vary from one region to another and from one school to the next. Some schools administer a separate multiple-choice maths test, while others combine English and maths questions into a single paper. The allotted time for the test may also differ, ranging from 30 minutes to an hour. In some cases, schools include numerical reasoning tests to evaluate a student’s ability to use critical thinking and analysis to solve problems. 

It’s worth noting that the 11 Plus Maths exam doesn’t simply test rote memorization of facts. It’s designed to measure a student’s ability to apply their mathematical knowledge to solve problems. The exam includes a diverse set of questions that cover key areas such as numbers, fractions and decimals, geometry, measurements, data handling, and problem-solving.

To ensure your child’s preparation is spot on, it’s vital to visit the school’s website where your child will be taking the exam. This can provide you with detailed information about the specific test format and focus areas.

In the subsequent sections, we’ll delve deeper into the types of questions involved in the 11 Plus Maths exam, as well as proven strategies to help your child prepare effectively. From early preparation to the use of learning apps and the support of experienced tutors, we’ve got you covered. Stay tuned!

Key Topics Covered in the 11 Plus Maths Exam

The 11 Plus maths exam may seem broad, but understanding the key areas of focus can help streamline your child’s preparation. The topics are grounded in the Key Stage 2 National Curriculum, providing an even playground for all students. Here’s an overview of the crucial topics that form the foundation of the 11 Plus Maths paper:

This covers a broad range of topics, from basic arithmetic operations such as addition, subtraction, multiplication, and division to more complex ideas like prime numbers, decimals, and the number line. Mastery of this topic allows students to perform quick calculations, an essential skill for time-based tests.

Fractions and Decimals

This section is all about understanding and working with non-whole numbers. Students should be able to convert fractions to decimals and vice versa, perform operations with fractions and decimals, and solve problems involving these concepts.

This section encompasses geometry. Students are tested on their knowledge of different shapes, their properties, and their interrelationships. They may be asked to calculate areas, perimeters, or identify different types of angles.

Measurements

This is another important aspect of the 11 Plus Maths exam. Students must understand how to calculate distances, use units of measurement appropriately, and apply their knowledge to solve practical problems.

Data Handling

Here, students are expected to interpret and analyse data presented in various formats, such as tables, charts, and graphs. They need to understand concepts such as mean, median, mode, and range, and how to use these statistical measures to make sense of data.

These key topics aim to assess a student’s problem-solving skills and their ability to analyse and interpret information. Understanding these topics and focusing on their practice can provide a solid base for your child’s 11 Plus Maths preparation. However, the understanding of topics alone isn’t enough. Let’s dive into some effective strategies to hone your child’s problem-solving abilities and enhance their exam performance in the next section.

Practical Tips for Excelling in the 11 Plus Maths Exam

As the 11 Plus maths exam approaches, children and parents may find themselves on a roller coaster ride of emotions, stress, and pressure. However, with a strategic approach and consistent effort, success is achievable. Here are some practical tips that can help your child excel in the 11 Plus Maths Exam.

Start Early

The curriculum of the 11 Plus Maths Exam is built on the foundation of Key Stage 2 topics. Hence, an early start can ensure that the child grasps the basic concepts thoroughly, paving the way for a smoother preparation process.

Skill Development

Maths isn’t just about numbers; it’s about logical reasoning, critical thinking, and problem-solving. Encourage your child to engage in activities that stimulate these skills, like puzzles, chess, Sudoku, and other brain games. This will make learning fun and interactive, boosting their enthusiasm for the subject.

Regular Practice

There’s no substitute for regular practice when it comes to maths. Encourage your child to solve 11 Plus maths papers and answer hard 11+ questions regularly. The objective should be to understand different types of questions and improve speed and accuracy.

Effective Time Management

The 11 Plus Maths exam is a timed test, and often, the challenge lies in solving questions accurately within the stipulated time. Regular timed practice can help your child become more comfortable with the pace of the exam.

Use of Learning Apps

The digital world offers a plethora of learning resources. Leverage educational apps and online platforms that offer interactive 11+ maths questions and answers. These can provide a fun and engaging way to prepare for the exam.

Get a Tutor

Sometimes, personalised guidance can make a significant difference. A tutor who is experienced in 11 Plus preparation can offer customised learning plans and address your child’s individual weaknesses, enhancing their understanding and confidence.

Maths | Chemistry | Biology Tutor

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Regular Reviews

Regular reviews of the topics covered can help to ensure that your child retains the information. Use 11 Plus Maths paper examples with answers for this review, to help identify any gaps in knowledge or understanding.

Remember, the goal of the 11 Plus Maths Exam isn’t just to test mathematical knowledge, but also to assess a child’s ability to think critically and solve problems effectively. With careful preparation and the right guidance, your child can approach this exam with confidence and succeed.

Sample 11 Plus Maths Questions and Answers

To provide you with a clearer understanding of the kind of questions your child might encounter in an 11 Plus Maths Exam, here are a few sample questions and their solutions. It’s important to remember that questions in the exam can range from straightforward arithmetic to more complex word problems that require a good understanding of various mathematical concepts.

1. Arithmetic Question:

Question: What is the result when you subtract 1234 from 4321?

Answer: The result is 3087. 

2. Fractions and Decimals Question:

Question: If Sally has a pizza and she eats 0.25 of it, how much pizza does she have left?

Answer: Sally has 0.75 (or 3/4) of the pizza left.

3. Shapes and Geometry Question:

Question: If a rectangle has a length of 6 cm and a width of 4 cm, what is its area?

Answer: The area of a rectangle is calculated by multiplying its length by its width, so the area of this rectangle is 24 square cm. 

4. Data Handling Question:

Question: The following are the test scores of a student: 78, 82, 85, 78, 80. What is the median score?

Answer: Arrange the numbers in order: 78, 78, 80, 82, 85. The median score is the one in the middle, which is 80.

5. Problem-solving Question:

Question: Sarah and Peter are collecting seashells. If Sarah collects 20 seashells and Peter collects three times as many seashells as Sarah, how many seashells do they have in total?

Answer: Peter collects 3 times 20 seashells, which is 60 seashells. Together, they collect 20 (Sarah’s seashells) + 60 (Peter’s seashells) = 80 seashells.

These examples give a glimpse of the kind of 11 Plus maths questions your child will encounter. However, the actual exam will include a wider variety of problems. Regular practice with different types of questions will equip your child with the skills they need to tackle the exam confidently.

The Role of Tutoring in 11 Plus Maths Exam Preparation

When preparing for the 11 Plus Maths Exam, having an experienced, knowledgeable tutor can be a game-changer. A tutor not only guides students through complex mathematical concepts but also provides strategies for problem-solving, builds confidence, and offers personalised attention that can dramatically improve a student’s performance.

Edumentors, a growing online tutoring platform, stands out in this regard. Edumentors boasts a talented pool of tutors who are students from top UK universities such as Cambridge, Oxford, UCL, and Warwick. These tutors have not only been through rigorous academic experiences themselves but also understand the unique demands of the 11 Plus Maths Exam. They have successfully navigated these challenges and are equipped to guide your child on their educational journey.

One significant advantage of choosing an Edumentors tutor is their close age to the students. Being nearer in age, these tutors can relate better to your child’s experiences and challenges, making them not just teachers, but friends and role models. Their relatability creates a comfortable learning environment where students feel free to ask questions and express difficulties, facilitating a more effective learning process.

Remember, the goal of tutoring isn’t simply to learn to answer questions correctly but to develop a deep understanding of the subject matter. The right tutor can inspire a lifelong love for maths in your child, beyond the 11 Plus Maths Exam.

Choosing a tutor from Edumentors can help turn the daunting task of preparing for the 11 Plus Maths Exam into an enjoyable and rewarding learning experience. Start today and see the difference that a dedicated, relatable tutor can make in your child’s 11 Plus preparation journey.

Preparing for the 11 Plus Maths Exam is indeed a significant milestone in your child’s educational journey. It requires consistent practice, strategic studying, and a deep understanding of complex mathematical concepts. However, beyond the exam lies the real reward – the development of critical thinking skills, analytical capabilities, and a solid maths foundation that will serve your child well into their academic future and beyond.

It’s essential to remember that every student’s journey is unique, and so is their learning process. What works for one might not work for another. That’s why it’s so crucial to foster a love of learning and curiosity that will drive your child to explore, question, and grow.

If you find your child struggling or if you simply want to give them an extra push towards achieving their academic goals, remember that help is just a few clicks away. Edumentors, a renowned online tutoring platform, can be the support your child needs. 

The 11+ tutors at Edumentors , all students at top UK universities, have faced similar challenges and emerged victorious. They understand what it takes to excel in exams like the 11 Plus Maths Exam and are ready to share their strategies and insights with your child. More than just tutors, they serve as friends and role models, inspiring students to aim high and boost their confidence.

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Free 11+ Maths Practice Papers

A complete list of 11 plus maths practice papers.

Below is the most complete and up-to-date list of all free 11 Plus Maths practice papers available on the internet.  All of these 11+ Maths practice papers are in pdf format and we have provided the answers where possible.

Click here to access our FREE 11+ Maths Mock Test designed for Year 5 students preparing for 11+ Grammar & Independent School exams.

*Bookmark this page for future reference*

Table of Contents

11 plus maths practice papers - private/independent school.

Aldenham School

• Aldenham School 11 Plus Maths Sample Paper 2015
• Aldenham School 11 Plus Maths Sample Paper 2017
• Aldenham School 11 Plus Maths Sample Paper 2019
• Aldenham School 11 Plus Maths Sample Paper 2022
• Aldenham School 11 Plus Maths Sample Paper 2023

Alleyn’s School

• Alleyn’s 11 Plus Maths Sample Paper 1
• Alleyn’s 11 Plus Maths Sample Paper 2

Bancroft’s School

• Bancroft’s School 11 Plus Maths Sample Paper 2013
• Bancroft’s School 11 Plus Maths Sample Paper 2016
• Bancroft’s School 11 Plus Maths Sample Paper 2017
• Bancroft’s School 11 Plus Maths Sample Paper 2020
• Bancroft’s School 11 Plus Maths Sample Paper 2022

Blackheath High School

• Blackheath High School 11 Plus Maths Sample Paper 2022

Chigwell School

• Chigwell School 11 Plus Maths Specimen Paper 2020
• Chigwell School 11 Plus Maths Specimen Paper 2020 – Mark Scheme

Christ’s Hospital School

• Christ’s Hospital School 11 Plus Maths Paper 2014

City of London School

• City of London School 11 Plus Maths Specimen Paper
• City of London School 11 Plus Maths Paper 2018

City of London School for Girls

• City of London School for Girls 11 Plus Maths Sample Paper
• City of London School for Girls 11 Plus Maths Paper 2021

Colfe’s School

• Colfe’s School 11 Plus Maths Sample Paper

Dulwich College

• Dulwich College 11 Plus Maths Specimen Paper A
• Dulwich College 11 Plus Maths Specimen Paper A – Mark Scheme
• Dulwich College 11 Plus Maths Specimen Paper B
• Dulwich College 11 Plus Maths Specimen Paper B – Mark Scheme
• Dulwich College 11 Plus Maths Specimen Paper C
• Dulwich College 11 Plus Maths Specimen Paper C – Mark Scheme
• Dulwich College 11 Plus Maths Specimen Paper D
• Dulwich College 11 Plus Maths Specimen Paper D – Mark Scheme
• Dulwich College 11 Plus Maths Specimen Paper E
• Dulwich College 11 Plus Maths Specimen Paper E – Mark Scheme
• Dulwich College 11 Plus Maths Specimen Paper F
• Dulwich College 11 Plus Maths Specimen Paper F – Mark Scheme
• Dulwich College 11 Plus Maths Specimen Paper 1
• Dulwich College 11 Plus Maths Specimen Paper 1 – Mark Scheme
• Dulwich College 11 Plus Maths Specimen Paper 2

Eltham College

• Eltham College 11 Plus Maths Sample Paper 2019
• Eltham College 11 Plus Maths Sample Paper 2019 – Mark Scheme
• Eltham College 11 Plus Maths Sample Paper 2020
• Eltham College 11 Plus Maths Sample Paper 2022
• Eltham College 11 Plus Maths Sample Paper

Emmanuel School

• Emmanuel School 11 Plus Maths Sample Paper 2013
• Emmanuel School 11 Plus Maths Entrance Exam A
• Emmanuel School 11 Plus Maths Entrance Exam B

Forest School

• Forest School 11 Plus Maths Sample Paper 2016
• Forest School 11 Plus Maths Sample Paper 2020

Haberdashers’ Boys’ School

• Haberdashers’ Boys’ School 11 Plus Maths Sample Paper 2009
• Haberdashers’ Boys’ School 11 Plus Maths Sample Paper 2010
• Haberdashers’ Boys’ School 11 Plus Maths Sample Paper 2011
• Haberdashers’ Boys’ School 11 Plus Maths Sample Paper 2014
• Haberdashers’ Boys’ School 11 Plus Maths Sample Paper 2016
• Haberdashers’ Boys’ School 11 Plus Maths Sample Paper 2017

Hampton Court House

• Hampton Court House 11 Plus Maths Sample Paper 1
• Hampton Court House 11 Plus Maths Sample Paper 2

Highgate School

• Highgate School 11 Plus Maths Sample Paper 2013
• Highgate School 11 Plus Maths Sample Paper A
• Highgate School 11 Plus Maths Sample Paper A – Mark Scheme
• Highgate School 11 Plus Maths Sample Paper B
• Highgate School 11 Plus Maths Sample Paper C

James Allen’s Girls’ School

• James Allen’s Girls’ School 11 Plus Maths Sample Paper 1
• James Allen’s Girls’ School 11 Plus Maths Sample Paper 2

Kent College

• Kent College 11 Plus Maths Sample Paper
• Kent College 11 Plus Maths Sample Paper 2011
• Kent College 11 Plus Maths Sample Paper 2015

King’s College Wimbledon

• King’s College School Wimbledon 11 Plus Maths Specimen Paper 2014
• King’s College School Wimbledon 11 Plus Maths Specimen Paper A 2017 (2019)
• King’s College School Wimbledon 11 Plus Maths Specimen Paper B 2017 (2019)
• King’s College School Wimbledon 11 Plus Maths Specimen Paper A 2018 (2020)
• King’s College School Wimbledon 11 Plus Maths Specimen Paper B 2018 (2020)
• King’s College School Wimbledon 11 Plus Maths Specimen Paper A 2023 (2025)
• King’s College School Wimbledon 11 Plus Maths Specimen Paper B 2023 (2025)

Latymer Upper School

• Latymer Upper School 11 Plus Maths Sample Paper 2014
• Latymer Upper School 11 Plus Maths Sample Paper 1 2020
• Latymer Upper School 11 Plus Maths Sample Paper 2 2020

Magdalen College School

• Magdalen College School 11 Plus Maths Sample Paper
• Magdalen College School 11 Plus Maths Sample Paper – Mark Scheme

Manchester Grammar School

• Manchester Grammar School 11 Plus Arithmetic 1 2010
• Manchester Grammar School 11 Plus Arithmetic 2 2010
• Manchester Grammar School 11 Plus Arithmetic 1 2011
• Manchester Grammar School 11 Plus Arithmetic 2 2011
• Manchester Grammar School 11 Plus Arithmetic 1 2012
• Manchester Grammar School 11 Plus Arithmetic 2 2012
• Manchester Grammar School 11 Plus Arithmetic 1 2013
• Manchester Grammar School 11 Plus Arithmetic 1 2013 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic 2 2013
• Manchester Grammar School 11 Plus Arithmetic 2 2013 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic A 2014
• Manchester Grammar School 11 Plus Arithmetic A 2014 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic B 2014
• Manchester Grammar School 11 Plus Arithmetic B 2014 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic A 2016
• Manchester Grammar School 11 Plus Arithmetic A 2016 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic B 2016
• Manchester Grammar School 11 Plus Arithmetic B 2016 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic A 2017
• Manchester Grammar School 11 Plus Arithmetic A 2017 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic B 2017
• Manchester Grammar School 11 Plus Arithmetic B 2017 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic A 2018
• Manchester Grammar School 11 Plus Arithmetic A 2018 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic B 2018
• Manchester Grammar School 11 Plus Arithmetic B 2018 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic A 2019
• Manchester Grammar School 11 Plus Arithmetic A 2019 – Mark Scheme
• Manchester Grammar School 11 Plus Arithmetic B 2019
• Manchester Grammar School 11 Plus Arithmetic B 2019 – Mark Scheme

Merchant Taylors’ School

• Merchant Taylors’ School 11 Plus Maths Practice Paper 2008
• Merchant Taylors’ School 11 Plus Maths Practice Paper 2009
• Merchant Taylors’ School 11 Plus Maths Specimen Paper 1
• Merchant Taylors’ School 11 Plus Maths Specimen Paper 2

North London Collegiate School

• North London Collegiate School 11 Plus Maths Sample Paper

Oundle School

• Oundle School 11 Plus Maths Sample paper 2008
• Oundle School 11 Plus Maths Sample paper 2016
• Oundle School 11 Plus Maths Sample paper 2017
• Oundle School 11 Plus Maths Sample paper 2018
• Oundle School 11 Plus Maths Sample paper 2019
• Oundle School 11 Plus Maths Sample paper 2020

Perse Upper School

• Perse Upper School 11 Plus Maths Specimen Paper 1

Perse Upper School 11 Plus Maths Specimen Paper 2

• Perse Upper School 11 Plus Maths Specimen Paper 3
• Perse Upper School 11 Plus Maths Specimen Paper 4
• Perse Upper School 11 Plus Maths Specimen Paper 5
• Perse Upper School 11 Plus Maths Sample Paper 1
• Perse Upper School 11 Plus Maths Sample Paper 2

Reigate Grammar School

• Reigate Grammar School 11 Plus Maths Sample Paper 2012
• Reigate Grammar School 11 Plus Maths Sample Paper 2013
• Reigate Grammar School 11 Plus Maths Sample Paper 2018
• Reigate Grammar School 11 Plus Maths Sample Paper 2019
• Reigate Grammar School 11 Plus Maths Sample Paper 2022

Royal Russel School

• Royal Russel School 11 Plus Maths Sample Paper

Sevenoaks School

• Sevenoaks 11 Plus Maths Sample Paper 2010
• Sevenoaks 11 Plus Maths Sample Paper 2011
• Sevenoaks 11 Plus Maths Sample Paper 2012
• Sevenoaks 11 Plus Maths Sample Paper 2013
• Sevenoaks 11 Plus Maths Sample Paper 2014
• Sevenoaks 11 Plus Maths Sample Paper 2015
• Sevenoaks 11 Plus Maths Sample Paper 2016
• Sevenoaks 11 Plus Maths Sample Paper 2017
• Sevenoaks 11 Plus Maths Sample Paper 2018
• Sevenoaks 11 Plus Maths Sample Paper 2019
• Sevenoaks 11 Plus Maths Sample Paper 2020
• Sevenoaks 11 Plus Maths Sample Paper 2021
• Sevenoaks 11 Plus Maths Sample Paper 2022

Solihull School

• Solihull School 11 Plus Maths Sample Paper

St Alban’s School

• St Alban’s School 11 Plus Maths Specimen Paper
• St Alban’s School 11 Plus Maths Sample Paper 2017
• St Alban’s School 11 Plus Maths Sample Paper 2019
• St Alban’s School 11 Plus Maths Sample Paper 2020

St Francis’ College

• St Francis’ College 11 Plus Maths Sample Paper 2017
• St Francis’ College 11 Plus Maths Sample Paper 2017 – Mark Scheme

St George’s College Weybridge

• St George’s College Weybridge 11 Plus Maths Sample Paper 2012
• St George’s College Weybridge 11 Plus Maths Sample Paper

St John’s School Leatherhead

• St John’s School Leatherhead 11 Plus Maths Sample Paper

St Paul’s Girls’ School

• St Paul’s Girls’ School 11 Plus Maths Sample Paper 1 2017
• St Paul’s Girls’ School 11 Plus Maths Sample Paper 2 2017

Streatham and Clapham High

• Streatham and Clapham High School 11 Plus Maths Sample Paper A 2019
• Streatham and Clapham High School 11 Plus Maths Sample Paper B 2019
• Streatham and Clapham High School 11 Plus Maths Specimen Paper

Sydenham High School

• Sydenham High School 11 Plus Maths Sample Paper

The King’s School Chester

• The King’s School Chester 11 Plus Maths Specimen Paper
• The King’s School Chester 11 Plus Maths Specimen Paper – Mark Scheme
• The King’s School Chester 11 Plus Maths Specimen Paper 2
• The King’s School Chester 11 Plus Maths Specimen Paper 2 – Mark Scheme

The Queen’s School Chester

• The Queen’s School Chester 11 Plus Maths Sample Paper

Trinity School

• Trinity School 11 Plus Maths Sample Paper 1
• Trinity School 11 Plus Maths Sample Paper 2

11 Plus Maths Practice Papers - Grammar School

Crossley Heath & Halifax Grammar Schools

• Crossley Heath & Halifax Grammar Schools 11 Plus Maths Sample Paper

Dame Alice Owens School

• Dame Alice Owens School 11 Plus Maths Familiarisation Paper
• Dame Alice Owens School 11 Plus Maths Familiarisation Paper – Mark Scheme

St Anselm’s College

• St Anselm’s College 11 Plus Maths Sample Paper 1
• St Anselm’s College 11 Plus Maths Sample Paper 2

St Olave’s Grammar School

• St Olave’s 11 Plus Maths Sample Paper 2015
• St Olave’s 11 Plus Maths Sample Paper 2019

SW Hertfordshire Schools Consortium

• SW Hertfordshire Schools Consortium 11 Plus Maths Specimen Paper

11 Plus Maths Practice Papers - Exam Boards

• CSSE 11+ Maths Paper 2015
• CSSE 11+ Maths Paper 2015 – Mark Scheme
• CSSE 11+ Maths Paper 2016
• CSSE 11+ Maths Paper 2016 – Mark Scheme
• CSSE 11+ Maths Paper 2017
• CSSE 11+ Maths Paper 2017 – Mark Scheme
• CSSE 11+ Maths Paper 2018
• CSSE 11+ Maths Paper 2018 – Mark Scheme
• CSSE 11+ Maths Paper 2019
• CSSE 11+ Maths Paper 2019 – Mark Scheme
• GL 11+ Maths Familiarisation Paper 1
• GL 11+ Maths Familiarisation Paper 2
• GL 11+ Maths Familiarisation Papers – Mark Scheme

North London Girls’ Schools Consortium Group 1 & 2 (Now ‘The London 11+ Consortium’)

• Group 1 11 Plus Maths – 2008
• Group 1 11 Plus Maths – 2009
• Group 1 11 Plus Maths – 2010
• Group 1 11 Plus Maths – 2011
• Group 1 11 Plus Maths – 2012
• Group 1 11 Plus Maths – 2013
• Group 1 11 Plus Maths – 2014
• Group 1 11 Plus Maths – 2015
• Group 1 11 Plus Maths – 2016
• Group 2 11 Plus Maths – 2008
• Group 2 11 Plus Maths – 2009
• Group 2 11 Plus Maths – 2010
• Group 2 11 Plus Maths -2011
• Group 2 11 Plus Maths – 2012
• Group 2 11 Plus Maths – 2013
• Group 2 11 Plus Maths – 2014
• Group 2 11 Plus Maths -2015
• Group 2 11 Plus Maths -2016

11 Plus Maths Worksheets PDF

• In stock (11)
• Out of stock (0)

50 Maths Timed Tests - Fractions - Book 1

50 Maths Timed Tests - Fractions - Book 1 Note: This product is a digital download in a PDF format. As this is a digital product, once purchased, r...

Bucks GL 11+ Papers (MathsNVR) - 3 Paper Pack

Bucks GL Maths & Non-Verbal Reasoning Papers - 3 Paper Pack Three exam papers Prepare your child for the Bucks 11 Plus Test with this helpfu...

70 Maths Timed Tests - The 4 Operations - Book 1

70 Maths Timed Tests - The 4 Operations - Book 1 Note: This product is a digital download in a PDF format. As this is a digital product, once purch...

11+ & SATs Maths Workbook

11+ & SATs Maths Workbook The 11 Plus, SATs, and KS3 exams can be stressful experiences, and our children need a helping hand. Look no further ...

70 Maths Timed Tests - The 4 Operations (Advanced) - Book 2

70 Maths Timed Tests - The 4 Operations (Advanced) - Book 2 Note: This product is a digital download in a PDF format. As this is a digital product,...

11 Plus Maths Problem Solving Test 1

Maths Problem Solving - Test 1 - Ages 11 - 12 If you're looking for a maths problem-solving test to help prepare for the 11+ and entrance exams, ...

11 Plus Maths Problem Solving - Test 1

Maths Problem Solving - Test 1 - Ages 10 - 11 This 11 Plus Maths Problem Solving Test is perfect for preparing for the 11+ and entrance exams. Thi...

40 Time Worksheets - Book 1

40 Time Worksheets -  Book 1 Note: This product is a digital download in a PDF format. As this is a digital product, once purchased, refunds are no...

11+ Maths Exam Preparation Bundle

11 Plus Maths Test Preparation Bundle - Fantastic Value! Is your child getting ready to take the 11 Plus exams? If so, you'll want to do everythin...

11 Plus Maths Problem Solving - Test 2

Maths Problem Solving - Test 2 - Ages 11 - 12 The 11 Plus Maths Problem Solving Test is the perfect way to prepare for the entrance exams. It fea...

70 Maths Timed Tests - Fractions - Book 2

70 Maths Timed Tests - Fractions - Book 2 Note: This product is a digital download in a PDF format. As this is a digital product, once purchased, r...

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• Online Assessment Mock Test (11 Plus Problem Solving)

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Time Questions and Answers

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Time and Date for the 11 Plus Exams

Every person’s life revolves around time. You wake up at a particular time every morning. 

The day can be split into two halves. The day has 24 hours in it. 

Midnight to midday: first 12 hours of the day: Called AM

Next 12 hours from noon back to midnight are called PM.

Time can be shown in two formats: 24-hour and 12-hour. Let us learn about these formats in the coming sections.

1.1 What  is 24-hour clock?

It is more often shown on digital clock with the numbers 1-24. The time represented by 24-hour clock is in four digits, with the first two digits denoting the hours while the last two digits denoting the minutes. 

AM or PM is not used in the 24-hour time format.

For example:

Six AM in the 24-hour clock will be 06:00.

Three-thirty PM in the 24-hour clock will be 15:30 (15 hours = 12 hours + 3 hours).

1.2 What is 12-hour clock?

It is more often shown on analogue clock with the numbers 1-12. The  time represented by 12-hour clock is followed by AM or PM.

Examples: 3 am, 4 pm, 5:30 pm

2. Time Conversions

Time can be measured in seconds, minutes, hours, days, weeks, months  and years.

Conversion time guide:

Normal year have 365 days while leap year have 366 days. Leap year happens every four years.

Let us now workout some examples on time conversions:

2.1 Time Conversion: Solved Examples

a) 17 minutes 45 seconds to seconds.

First step is to convert 17 minutes to seconds.

1 minute has 60 seconds.

17 minutes =  17× 60 = 1020 seconds

So 17 minutes 45 seconds = 1020 seconds + 45 seconds

                                            = 1065 seconds

b) 4 weeks 3 days 12 hours to hours

First  step is to convert 4 weeks to days and to hours.

One week has 7 days, so 4 weeks has 

7× 4=28 days

One  day has 24 hours, so 28 days has

24× 28 = 672 hours

Second step is to convert 3 days to hours

One  day has 24 hours, so 3 days has

24× 3 = 72 hours

Now 4 weeks 3 days 12 hours = 672 hours + 72 hours + 12 hours

                                                 = 756 hours

2.2 Time Loss and Gain

When the clock is running fast, time is gained and when is  running slow, the  time is lost.

Let us understand the terms time loss and gain using a simple example.

If a clock shows 7.20 when the correct time is 7.30, the clock is late by 10 minutes and it losses time.

If a clock shows 7.20 when the correct time is 7.10, the clock is fast by 10 minutes and it gains time.

Let us now learn how to workout the correct time given the time gain or loss with the example below.

Example: A clock gains 15 minutes in one hour and was set right at 11 AM. What  time will this clock show at 3 PM on the same day?

The clock gains 15 minutes in an hour.

Number of hours passed from 11 AM to 3 PM = 4

Time gained by the clock in 4 hours =  4× 15 = 60 minutes

So after 4 hours i.e. at 3 PM, the clock will be 60 minutes or one hour ahead.

At 3 PM the clock will show 4PM.

3. Date Problems

Things to remember while solving date problems:

• Number of days in ordinary year = 365
• Number of days in leap year = 366
• Number of days in one month = 30 or 31
• Number of days in one week = 7

Let us now use this information and try to solve an example of date problem.

Example: If the 8th of June was Friday, what day was on 19th of September of the same year?

One week has 7 days.

After every week the days will repeat.

The number of days from 8th June to 19th December = 103 days

And, 103⁄7=14R5

We have 14 full weeks and 5 days extra. 

The 5th day after Friday is Wednesday. 

19th September is a Wednesday. 

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Algebra Questions with Solutions and Answers for Grade 11

algebra questions with answers and detailed solutions , for grade 11, are presented.

• Complete the square in the quadratic function f given by f(x) = 2x 2 - 6x + 4
• Find the point(s) of intersection of the parabola with equation y = x 2 - 5x + 4 and the line with equation y = 2x - 2
• Find the constant k so that : -x 2 - (k + 7)x - 8 = -(x - 2)(x - 4)
• Find the center and radius of the circle with equation x 2 + y 2 -2x + 4y - 11 = 0
• Find the constant k so that the quadratic equation 2x 2 + 5x - k = 0 has two real solutions.
• Find the constant k so that the system of the two equations: 2x + ky = 2 and 5x - 3y = 7 has no solutions.
• Factor the expression 6x 2 - 13x + 5
• Simplify i 231 where i is the imaginary unit and is defined as: i = √(-1).
• What is the remainder when f(x) = (x - 2) 54 is divided by x - 1?
• Find b and c so that the parabola with equation y = 4x 2 - bx - c has a vertex at (2 , 4)?
• Find all zeros of the polynomial P(x) = x 3 - 3x 2 - 10x + 24 knowing that x = 2 is a zero of the polynomial.
• If x is an integer, what is the greatest value of x which satisfies 5 < 2x + 2 < 9?
• Sets A and B are given by: A = {2 , 3 , 6 , 8, 10} , B = {3 , 5 , 7 , 9}. a) Find the intersection of sets A and B. b) Find the union of sets A and B.
• Simplify | - x 2 + 4x - 4 |.
• Find the constant k so that the line with equation y = k x is tangent to the circle with equation (x - 3) 2 + (y - 5) 2 = 4.

Solutions to the Above Questions

• f(x) = 2(x 2 - 3x) + 4 : factor 2 out in the first two terms = 2(x 2 - 3x + (-3/2) 2 - (-3/2) 2 ) + 4 : add and subtract (-3/2) 2 = 2(x - 3/2)) 2 - 1/2 : complete square and group like terms
• 2x - 2 = x 2 - 5x + 4 : substitute y by 2x - 2 x = 1 and x = 6 : solution of quadratic equation (1 , 0) and (6 , 10) : points of intersection
• -x 2 - (k + 7)x - 8 = -(x - 2)(x - 4) : given -x 2 - (k + 7)x - 8 = -x 2 + 6x - 8 -(k + 7) = 6 : two polynomials are equal if their corresponding coefficients are equal. k = -13 : solve the above for k
• x 2 - 2x + y 2 + 4y = 11 : Put terms in x together and terms in y together (x - 1) 2 + (y + 2) 2 - 1 - 4 = 11 (x - 1) 2 + (y + 2) 2 = 4 2 : write equation of circle in standard form center(1 , -2) and radius = 4 : identify center and radius
• 2x 2 + 5x - k = 0 : given discriminant = 25 - 4(2)(-k) = 25 + 8k 25 + 8k > 0 : quadratic equations has 2 real solutions when the discriminant is positive k > -25/8
• Determinant = -6 - 5k -6 - 5k = 0 : when the determinant is equal to zero (and equations independent) the system has no solution k = -6/5 : solve for k
• 6x 2 - 13x + 5 = (3x - 5)(2x - 1)
• Note that i 4 = 1 Note also that 231 = 4 * 57 + 3 Hence i 231 = (i 4 ) 57 * i 3 = 1 57 * -i = -i
• remainder = f(1) = (1 - 2) 54 = 1 : remainder theorem
• h = b / 8 = 2 : formula for x coordinate of vertex b = 16 : solve for b y = 4 for x = 2 : the vertex point is a solution to the equation of the parabola 4(2) 2 - 16(2) - c = 4 c = -20 : solve for c
• divide P(x) by (x - 2) to obtain x 2 - x + 12 P(x) = (x 2 - x + 12)(x - 2) = (x - 4)(x + 3)(x - 2) : factor the quadratic term the zeros are : 4 , -3 and 2
• 5 < 2x + 2 < 9 : given 3/2 < x < 7/2 the greatest integer value of is 3 (the integer less than 7/2)
• A intersection B = {3} : common element to both A and B is 3 A union B = {2 , 3 , 6 , 8, 10 , 5 , 7 , 9} : all elements of A and B are in the union. Elements that are common to both A and B are listed once only since it is a set.
• | - x 2 + 4x - 4 | : given = | -(x 2 + 4x - 4) | = | -(x - 2) 2 | = (x - 2) 2
• (x - 3) 2 + (y - 5) 2 = 4 : given (x - 3) 2 + (kx - 5) 2 = 4 : substitute y by kx x 2 (1 + k 2 ) - x(6 + 10k) + 21 = 0 : expand and write the quadratic equation in standard form. (6 + 10k) 2 - 4(1 + k 2 )(21) = 0 : For the circle and the line y = kx to be tangent, the discriminant of the above quadratic equation must be equal to zero. 16k 2 + 120k - 48 = 0 : expand above equation k = (-15 + √(273)/4 , k = (-15 - √(273)/4 : solve the above quadratic equation.

More References and links

High School Maths (Grades 10, 11 and 12) - Free Questions and Problems With Answers Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers Home Page

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QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
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30 Problem Solving Maths Questions And Answers For GCSE

Sophie Bessemer

Problem solving maths questions can be challenging for GCSE students as there is no ‘one size fits all’ approach. In this article, we’ve compiled tips for problem solving, example questions, solutions and problem solving strategies for GCSE students. 

Since the current GCSE specification began, there have been many maths problem solving exam questions which take elements of different areas of maths and combine them to form new maths problems which haven’t been seen before. 

While learners can be taught to approach simply structured problems by following a process, questions often require students to make sense of lots of new information before they even move on to trying to solve the problem. This is where many learners get stuck.

GCSE MATHS 2024: STAY UP TO DATE Join our email list to stay up to date with the latest news, revision lists and resources for GCSE maths 2024. We’re analysing each paper during the course of the 2024 GCSEs in order to identify the key topic areas to focus on for your revision. Thursday 16th May 2024: GCSE Maths Paper 1 2024 Analysis & Revision Topic List Monday 3rd June 2024: GCSE Maths Paper 2 2024 Analysis & Revision Topic List Monday 10th June 2024: GCSE Maths Paper 3 2024 Analysis GCSE 2024 dates GCSE 2024 results (when published) GCSE results 2023

How to teach problem solving

In the Ofsted maths review , published in May 2021, Ofsted set out their findings from the research literature regarding the sort of curriculum and teaching that best supports all pupils to make good progress in maths throughout their time in school.

Regarding the teaching of problem solving skills, these were their recommendations:

• Teachers could use a curricular approach that better engineers success in problem-solving by teaching the useful combinations of facts and methods, how to recognise the problem types and the deep structures that these strategies pair to.
• Strategies for problem-solving should be topic specific and can therefore be planned into the sequence of lessons as part of the wider curriculum. Pupils who are already confident with the foundational skills may benefit from a more generalised process involving identifying relationships and weighing up features of the problem to process the information. 
• Worked examples, careful questioning and constructing visual representations can help pupils to convert information embedded in a problem into mathematical notation.
• Open-ended problem solving tasks do not necessarily mean that the activity is the ‘ideal means of acquiring proficiency’. While enjoyable, open ended problem-solving activities may not necessarily lead to improved results.  

If you’re a KS2 teacher needing more support and CPD around teaching reasoning, problem solving & planning for depth we have a whole series of word problems and strategies to teach them available for you.

30 Problem Solving Maths Questions, Solutions & Strategies

Help your students prepare for their math GCSE with these free problem solving maths questions, solutions and strategies.

6 tips to tackling problem solving maths questions

There is no ‘one size fits all’ approach to successfully tackling problem solving maths questions however, here are 6 general tips for students facing a problem solving question:

• Read the whole question, underline important mathematical words, phrases or values.
• Annotate any diagrams, graphs or charts with any missing information that is easy to fill in.
• Think of what a sensible answer may look like. E.g. Will the angle be acute or obtuse? Is £30,000 likely to be the price of a coat?
• Tick off information as you use it.
• Draw extra diagrams if needed.
• Look at the final sentence of the question. Make sure you refer back to that at the end to ensure you have answered the question fully.

There are many online sources of mathematical puzzles and questions that can help learners improve their problem-solving skills. Websites such as NRICH and our blog on SSDD problems have some great examples of KS2, KS3 and KS4 mathematical problems.

Read more: KS2 problem solving and KS3 maths problem solving

In this article, we’ve focussed on GCSE questions and compiled 30 problem solving maths questions and solutions suitable for Foundation and Higher tier students. Additionally, we have provided problem solving strategies to support your students for some questions to encourage critical mathematical thinking . For the full set of questions, solutions and strategies in a printable format, please download our 30 Problem Solving Maths Questions, Solutions & Strategies.

Looking for additional support and resources at KS3? You are welcome to download any of the secondary maths resources from Third Space Learning’s resource library for free. There is a section devoted to GCSE maths revision with plenty of maths worksheets and GCSE maths questions . There are also maths tests for KS3, including a Year 7 maths test , a Year 8 maths test and a Year 9 maths test For children who need more support, our maths intervention programmes for KS3 achieve outstanding results through a personalised one to one tuition approach.

10 problem solving maths questions (Foundation tier)

These first 10 questions and solutions are similar to Foundation questions. For the first three, we’ve provided some additional strategies.

In our downloadable resource, you can find strategies for all 10 Foundation questions .

1) L-shape perimeter 

Here is a shape:

Sarah says, “There is not enough information to find the perimeter.”

Is she correct? What about finding the area?

• Try adding more information – giving some missing sides measurements that are valid. 
• Change these measurements to see if the answer changes.
• Imagine walking around the shape if the edges were paths. Could any of those paths be moved to another position but still give the same total distance?

The perimeter of the shape does not depend on the lengths of the unlabelled edges.

Edge A and edge B can be moved to form a rectangle, meaning the perimeter will be 22 cm. Therefore, Sarah is wrong.

The area, however, will depend on those missing side length measurements, so we would need more information to be able to calculate it.

2) Find the missing point

Here is a coordinate grid with three points plotted. A fourth point is to be plotted to form a parallelogram. Find all possible coordinates of the fourth point.

• What are the properties of a parallelogram?
• Can we count squares to see how we can get from one vertex of the parallelogram to another? Can we use this to find the fourth vertex?

There are 3 possible positions.

3) That rating was a bit mean!

The vertical line graph shows the ratings a product received on an online shopping website. The vertical line for 4 stars is missing.

If the mean rating is 2.65, use the information to complete the vertical line graph.

Strategies 

• Can the information be put into a different format, either a list or a table?
• Would it help to give the missing frequency an algebraic label, x ?
• If we had the data in a frequency table, how would we calculate the mean?
• Is there an equation we could form?

Letting the frequency of 4 star ratings be x , we can form the equation \frac{45+4x}{18+x} =2.65

Giving x=2 

4) Changing angles

The diagram shows two angles around a point. The sum of the two angles around a point is 360°.

Peter says “If we increase the small angle by 10% and decrease the reflex angle by 10%, they will still add to 360°.”

Explain why Peter might be wrong.

Are there two angles where he would be correct?

Peter is wrong, for example, if the two angles are 40° and 320°, increasing 40° by 10% gives 44°, decreasing 320° by 10% gives 288°. These sum to 332°.

10% of the larger angle will be more than 10% of the smaller angle so the sum will only ever be 360° if the two original angles are the same, therefore, 180°.

5) Base and power

The integers 1, 2, 3, 4, 5, 6, 7, 8 and 9 can be used to fill in the boxes. 

How many different solutions can be found so that no digit is used more than once?

There are 8 solutions.

6) Just an average problem 

Place six single digit numbers into the boxes to satisfy the rules.

The mean in maths is 5  \frac{1}{3}

The median is 5

The mode is 3.

How many different solutions are possible?

There are 4 solutions.

2, 3, 3, 7, 8, 9

3, 3, 4, 6, 7, 9

3, 3, 3, 7, 7, 9

3, 3, 3, 7, 8, 8

7) Square and rectangle  

The square has an area of 81 cm 2 . The rectangle has the same perimeter as the square.

Its length and width are in the ratio 2:1.

Find the area of the rectangle.

The sides of the square are 9 cm giving a perimeter of 36 cm. 

We can then either form an equation using a length 2x and width x . 

Or, we could use the fact that the length and width add to half of the perimeter and share 18 in the ratio 2:1. 

The length is 12 cm and the width is 6 cm, giving an area of 72 cm 2 .

8) It’s all prime

The sum of three prime numbers is equal to another prime number.

If the sum is less than 30, how many different solutions are possible?

There are 6 solutions. 

2 can never be used as it would force two more odd primes into the sum to make the total even.

9) Unequal share

Bob and Jane have £10 altogether. Jane has £1.60 more than Bob. Bob spends one third of his money. How much money have Bob and Jane now got in total?

Initially Bob has £4.20 and Jane has £5.80. Bob spends £1.40, meaning the total £10 has been reduced by £1.40, leaving £8.60 after the subtraction.

10) Somewhere between

Fred says, “An easy way to find any fraction which is between two other fractions is to just add the numerators and add the denominators.” Is Fred correct?

Solution 

Fred is correct. His method does work and can be shown algebraically which could be a good problem for higher tier learners to try.

If we use these two fractions \frac{3}{8} and \frac{5}{12} , Fred’s method gives us \frac{8}{20} = \frac{2}{5}

\frac{3}{8} = \frac{45}{120} , \frac{2}{5} = \frac{48}{120} , \frac{5}{12} = \frac{50}{120} . So \frac{3}{8} < \frac{2}{5} < \frac{5}{12}

10 problem solving maths questions (Foundation & Higher tier crossover)

The next 10 questions are crossover questions which could appear on both Foundation and Higher tier exam papers. We have provided solutions for each and, for the first three questions, problem solving strategies to support learners.

11) What’s the difference?

An arithmetic sequence has an nth term in the form an+b .

4 is in the sequence.

16 is in the sequence.

8 is not in the sequence.

-2 is the first term of the sequence.

What are the possible values of a and b ?

• We know that the first number in the sequence is -2 and 4 is in the sequence. Can we try making a sequence to fit? Would using a number line help?
• Try looking at the difference between the numbers we know are in the sequence.

If we try forming a sequence from the information, we get this:

We can now try to fill in the missing numbers, making sure 8 is not in the sequence. Going up by 2 would give us 8, so that won’t work.

The only solutions are 6 n -8 and 3 n -5.

12) Equation of the hypotenuse

The diagram shows a straight line passing through the axes at point P and Q .

Q has coordinate (8, 0). M is the midpoint of PQ and MQ has a length of 5 units.

Find the equation of the line PQ .

• We know MQ is 5 units, what is PQ and OQ ?
• What type of triangle is OPQ ?
• Can we find OP if we know PQ and OQ ?
• A line has an equation in the form y=mx+c . How can we find m ? Do we already know c ?

PQ is 10 units. Using Pythagoras’ Theorem OP = 6

The gradient of the line will be \frac{-6}{8} = -\frac{3}{4} and P gives the intercept as 6.

13) What a waste

Harry wants to cut a sector of radius 30 cm from a piece of paper measuring 30 cm by 20 cm. 

What percentage of the paper will be wasted?

• What information do we need to calculate the area of a sector? Do we have it all?
• Would drawing another line on the diagram help find the angle of the sector?

The angle of the sector can be found using right angle triangle trigonometry.

The angle is 41.81°.

This gives us the area of the sector as 328.37 cm 2 .

The area of the paper is 600 cm 2 .

The area of paper wasted would be 600 – 328.37 = 271.62 cm 2 .

The wasted area is 45.27% of the paper.

14) Tri-polygonometry

The diagram shows part of a regular polygon and a right angled triangle. ABC is a straight line. Find the sum of the interior angles of the polygon.

Finding the angle in the triangle at point B gives 30°. This is the exterior angle of the polygon. Dividing 360° by 30° tells us the polygon has 12 sides. Therefore, the sum of the interior angles is 1800°.

15) That’s a lot of Pi

A block of ready made pastry is a cuboid measuring 3 cm by 10 cm by 15 cm. 

Anne is making 12 pies for a charity event. For each pie, she needs to cut a circle of pastry with a diameter of 18 cm from a sheet of pastry 0.5 cm thick.

How many blocks of pastry will Anne need to buy?

The volume of one block of pastry is 450 cm 3 . 

The volume of one cylinder of pastry is 127.23 cm 3 .

12 pies will require 1526.81 cm 3 .

Dividing the volume needed by 450 gives 3.39(…). 

Rounding this up tells us that 4 pastry blocks will be needed.

16) Is it right?

A triangle has sides of (x+4) cm, (2x+6) cm and (3x-2) cm. Its perimeter is 80 cm.

Show that the triangle is right angled and find its area.

Forming an equation gives 6x+8=80

This gives us x=12 and side lengths of 16 cm, 30 cm and 34 cm.

Using Pythagoras’ Theorem

16 2 +30 2 =1156 

Therefore, the triangle is right angled.

The area of the triangle is (16 x 30) ÷ 2 = 240 cm 2 .

17) Pie chart ratio

The pie chart shows sectors for red, blue and green. 

The ratio of the angles of the red sector to the blue sector is 2:7. 

The ratio of the angles of the red sector to the green sector is 1:3. 

Find the angles of each sector of the pie chart.

Multiplying the ratio of red : green by 2, it can be written as 2:6. 

Now the colour each ratio has in common, red, has equal parts in each ratio.

The ratio of red:blue is 2:7, this means red:blue:green = 2:7:6.

Sharing 360° in this ratio gives red:blue:green = 48°:168°:144°.

18) DIY Simultaneously

Mr Jones buys 5 tins of paint and 4 rolls of decorating tape. The total cost was £167.

The next day he returns 1 unused tin of paint and 1 unused roll of tape. The refund amount is exactly the amount needed to buy a fan heater that has been reduced by 10% in a sale. The fan heater normally costs £37.50.

Find the cost of 1 tin of paint.    

The sale price of the fan heater is £33.75. This gives the simultaneous equations

p+t = 33.75 and 5 p +4 t = 167.

We only need the price of a tin of paint so multiplying the first equation by 4 and then subtracting from the second equation gives p =32. Therefore, 1 tin of paint costs £32. 

19) Triathlon pace

Jodie is competing in a Triathlon. 

A triathlon consists of a 5 km swim, a 40 km cycle and a 10 km run. 

Jodie wants to complete the triathlon in 5 hours. 

She knows she can swim at an average speed of 2.5 km/h and cycle at an average speed of 25 km/h. There are also two transition stages, in between events, which normally take 4 minutes each.

What speed must Jodie average on the final run to finish the triathlon in 5 hours?

Dividing the distances by the average speeds for each section gives times of 2 hours for the swim and 1.6 hours for the cycle, 216 minutes in total. Adding 8 minutes for the transition stages gives 224 minutes. To complete the triathlon in 5 hours, that would be 300 minutes. 300 – 224 = 76 minutes. Jodie needs to complete her 10 km run in 76 minutes, or \frac{19}{15} hours. This gives an average speed of 7.89 km/h.

20) Indices

a 2x × a y =a 3

(a 3 ) x ÷ a 4y =a 32

Find x and y .

Forming the simultaneous equations

Solving these gives

10 problem solving maths questions (Higher tier)

This final set of 10 questions would appear on the Higher tier only. Here we have just provided the solutions. Try asking your learners to discuss their strategies for each question.  

21) Angles in a polygon

The diagram shows part of a regular polygon.

A , B and C are vertices of the polygon. 

The size of the reflex angle ABC is 360° minus the interior angle.

Show that the sum of all of these reflex angles of the polygon will be 720° more than the sum of its interior angles.

Each of the reflex angles is 180 degrees more than the exterior angle: 180 + \frac{360}{n}

The sum of all of these angles is n (180 + \frac{360}{n} ). 

This simplifies to 180 n + 360

The sum of the interior angles is 180( n – 2) = 180 n – 360

The difference is 180 n + 360 – (180 n -360) = 720°

22) Prism and force (Non-calculator)

The diagram shows a prism with an equilateral triangle cross-section.

When the prism is placed so that its triangular face touches the surface, the prism applies a force of 12 Newtons resulting in a pressure of \frac{ \sqrt{3} }{4} N/m^{2}

Given that the prism has a volume of 384 m 3 , find the length of the prism.

Pressure = \frac{Force}{Area}

Area = 12÷ \frac{ \sqrt{3} }{4} = 16\sqrt{3} m 2

Therefore, the length of the prism is 384 ÷ 16\sqrt{3} = 8\sqrt{3} m

23) Geometric sequences (Non-calculator)

A geometric sequence has a third term of 6 and a sixth term of 14 \frac{2}{9}

Find the first term of the sequence.

The third term is ar 2 = 6

The sixth term is ar 5 = \frac{128}{9}

Diving these terms gives r 3 = \frac{64}{27}

Giving r = \frac{4}{3}

Dividing the third term twice by \frac{4}{3} gives the first term a = \frac{27}{8}

24) Printing factory

A printing factory is producing exam papers. When all 10 of its printers are working, it can produce all of the exam papers in 12 days.

For the first two days of printing, 3 of the printers are broken.

At the beginning of the third day it is discovered that 2 more printers have broken down, so the factory continues to print with the reduced amount of printers for 3 days. The broken printers are repaired and now all printers are available to print the remaining exams.

How many days in total does it take the factory to produce all of the exam papers?

If we assume one printer prints 1 exam paper per day, 10 printers would print 120 exam papers in 12 days. Listing the number printed each day for the first 5 days gives:

Day 5: 5 

This is a total of 29 exam papers.

91 exam papers are remaining with 10 printers now able to produce a total of 10 exam papers each day. 10 more days would be required to complete the job.

Therefore, 15 days in total are required.

25) Circles

The diagram shows a circle with equation x^{2}+{y}^{2}=13 .

A tangent touches the circle at point P when x=3 and y is negative.

The tangent intercepts the coordinate axes at A and B .

Find the length AB .

Using the equation  x^{2}+y^{2}=13 to find the y value for P gives y=-2 .

The gradient of the radius at this point is - \frac{2}{3} , giving a tangent gradient of \frac{3}{2} .

Using the point (3,-2) in y = \frac {3}{2} x+c gives the equation of the tangent as y = \frac {3}{2} x – \frac{13}{2}

Substituting x=0 and y=0 gives A and B as (0 , -\frac {13}{2}) and ( \frac{13}{3} , 0)

Using Pythagoras’ Theorem gives the length of AB as ( \frac{ 13\sqrt{13} }{6} ) = 7.812.

26) Circle theorems

The diagram shows a circle with centre O . Points A, B, C and D are on the circumference of the circle. 

EF is a tangent to the circle at A . 

Angle EAD = 46°

Angle FAB = 48°

Angle ADC = 78°

Find the area of ABCD to the nearest integer.

The Alternate Segment Theorem gives angle ACD as 46° and angle ACB as 48°.

Opposite angles in a cyclic quadrilateral summing to 180° gives angle ABC as 102°.

Using the sine rule to find AC will give a length of 5.899. Using the sine rule again to find BC will give a length of 3.016cm.

We can now use the area of a triangle formula to find the area of both triangles.

0.5 × 5 × 5.899 × sin (46) + 0.5 × 3.016 × 5.899 × sin (48) = 17 units 2 (to the nearest integer).

27) Quadratic function

The quadratic function f(x) = -2x^{2} + 8x +11 has a turning point at P .

Find the coordinate of the turning point after the transformation -f(x-3) .

There are two methods that could be used. We could apply the transformation to the function and then complete the square, or, we could complete the square and then apply the transformation.

Here we will do the latter.

This gives a turning point for f(x) as (2,19).

Applying -f(x-3) gives the new turning point as (5,-19).

28) Probability with fruit

A fruit bowl contains only 5 grapes and n strawberries.

A fruit is taken, eaten and then another is selected.

The probability of taking two strawberries is \frac{7}{22} .

Find the probability of taking one of each fruit. 

There are n+5 fruits altogether.

P(Strawberry then strawberry)= \frac{n}{n+5} × \frac{n-1}{n+4} = \frac{7}{22}

This gives the quadratic equation 15n^{2} - 85n - 140 = 0

This can be divided through by 5 to give 3n^{2} - 17n- 28 = 0

This factorises to (n-7)(3n + 4) = 0

n must be positive so n = 7.

The probability of taking one of each fruit is therefore, \frac{5}{12} × \frac{7}{11} + \frac {7}{12} × \frac {5}{11} = \frac {70}{132}

29) Ice cream tub volume

An ice cream tub in the shape of a prism with a trapezium cross-section has the dimensions shown. These measurements are accurate to the nearest cm.

An ice cream scoop has a diameter of 4.5 cm to the nearest millimetre and will be used to scoop out spheres of ice cream from the tub.

Using bounds find a suitable approximation to the number of ice cream scoops that can be removed from a tub that is full.

We need to find the upper and lower bounds of the two volumes. 

Upper bound tub volume = 5665.625 cm 3

Lower bound tub volume = 4729.375 cm 3

Upper bound scoop volume = 49.32 cm 3  

Lower bound scoop volume = 46.14 cm 3  

We can divide the upper bound of the ice cream tub by the lower bound of the scoop to get the maximum possible number of scoops. 

Maximum number of scoops = 122.79

Then divide the lower bound of the ice cream tub by the upper bound of the scoop to get the minimum possible number of scoops.

Minimum number of scoops  = 95.89

These both round to 100 to 1 significant figure, Therefore, 100 scoops is a suitable approximation the the number of scoops.

30) Translating graphs

 The diagram shows the graph of y = a+tan(x-b ).

The graph goes through the points (75, 3) and Q (60, q).

Find exact values of a , b and q .

The asymptote has been translated to the right by 30°. 

Therefore, b=30

So the point (45,1) has been translated to the point (75,3). 

Therefore, a=2

We hope these problem solving maths questions will support your GCSE teaching. To get all the solutions and strategies in a printable form, please download the complete resource .

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