All 40 Maths 11 Plus syllabus points are listed with the corresponding 40 sample 11+ practice paper questions, fully explained answers and helpful tips for the independent school 11 plus exams.
Sample 11 Plus Question: Find the option that best completes the sequence. Fill in the missing blank by selecting one of the following 5 options.
5, 10, 15, __?
Answer and Explanation: 20, option A. This is because between 5 and 10, there is a difference of 5. Between 10 and 15 there is also a difference of 5. Therefore, it logically follows that there should be a difference of 5 between 15 and the next number. 15 + 5 = 20, and therefore our answer.
Tips: Sequences act to test your child’s ability to apply logic and identify patterns. Always think to yourself, what is the pattern between these numbers?
Use your four pillars of mathematics to help break down this 11+ exam problem if you are stuck.
Do the sequences follow any of these patterns? If not, you could consider using square or cube numbers, or a combination of all of the above.
2) Manipulation of Bracketed Numbers
Sample 11 Plus Maths Question: Inside the brackets there is a number that has been formed through a function to the two numbers outside the brackets. The function to the two outside numbers stays the same for the question. What is the missing number?
90(9)10, 3(3)?, 50(10)5
Answer and Explanation: 1, option E. Look at the first set of 3 numbers. 90 (9) 10. The two outside numbers, 90 and 10 have, through some sort of sum, have got to the number in the brackets, 9. This is also seen with 50 and 5, which have reached 10.
Both 90 and 10 have somehow got to 9, and this is division, as 90 divided by 10 is 9. 50 divided by 5 is 10, which is the number in the brackets also. Therefore 3 divided by something will get us to 3. The only possible option is 1, as 3 divided by 3 is 1, and therefore the answer is E.
Tips: This style of 11+ questions assesses your child’s ability to spot a pattern between three numbers, and then if this can be reciprocated with an unknown. If your child is stuck and cannot find the pattern use your four pillars of mathematics to help break down this problem.
For harder questions, try to build up this by using a combination of two or maybe even three of the 4 pillars. In the 11+ exam there is always going to be a challenging question. If you are still stuck, look at the 5 options, and try to see which one will help give you a clue.
Sample 11 Plus Maths Question: The first number has been manipulated in some way to get to the next number. What is the function that leads the first number to become the second number? NB: some options take the first number as y. Choose your answer from the 5 options:
2 --> 6
Answer and Explanation: x 2 + 2, A. This is because there are many ways to get from 2 to 6. However, from the 5 options you are presented with, only multiplying 2 by 2, to get 4, followed by adding 2 gets you to 6. You can check the other options, and they present you with different numbers. Option B gives an answer of 8, Option C gives an answer of 7, Option D gives an option of 8 and E gives 4. Therefore, the only possible answer could be A.
Tips: With this style of 11+ question, it may be quickest to look at the answers, and solve the function this way. TOP 11+ TIP: There could be endless ways of reaching the answer otherwise, and out of the 5 options only one will be correct.
4) Sets of Numbers
Sample 11 Plus Maths Question: The set of numbers are all related in some way. Pick the number that would best fit into the rules of the set from the 5 options:
1, 27, 8, 64, 125
Answer and Explanation: 216, A. From the group of numbers, they all are cubed numbers. 1 = 13 , 33 = 27, 23 = 8, 43 = 64 and 53 = 125. Therefore, which of 216, 315, 144, 169 and 555 is a cubed number. The answer is 216 as 63 = 216.
Tips: This style of question could test a large variety of mathematical abilities and is frequent in the 11+ exam. The 5 numbers may be related in being odd or even, squared or cubed numbers, or perhaps factors of numbers.
Sample 11 Plus Maths Question: Solve the following problem below. Choose your answer from the 5 options given: In a block of flats there are 1266 rooms. In 212 rooms there are single beds, in 611 rooms there are double beds. How many rooms have neither single nor double beds?
Answer and Explanation: 443, D. 212 rooms + 611 rooms = 823 rooms. If there are 1266 rooms in total, therefore 1266 – 823 = 443. These are rooms that are neither have single or double beds and therefore the answer to the question.
Tips: These are also frequent in our 11+ practice paper questions, generally testing subtraction and the subsequent remainders from this. Most times by removing the sometimes confusing words, you may be able to decipher what the question is asking by looking at the numbers and how they are related, and not be baffled by the confusing terminology.
6) Change Received
Sample 11 Plus Maths Question: Solve the following problem below involving monetary change. Choose your answer from the 5 options given:
Sean and Ravi go to a café and get a coffee that was £2.30 each. They then split a slice of cake that is £1.20 in total. They ask to leave a tip of 10%, and pay with a £10 note. How much change do they get?
Answer and Explanation: £3.62, B. Both Sean and Ravi get a coffee, that is £2.30 EACH. This is a very important word. They then get a cake that was £1.20 in TOTAL. Therefore, they have together bought 2 coffees totalling £4.60 and one cake at £1.20. £4.60 + £1.20 = £5.80. They leave a tip of 10%, which is 58p, (convert £5.80 into pennies to get 580p, and divide by 10 to get 58p). £5.80 + 58p = £6.38. £10.00 - £6.38 = £3.62 and therefore answer B.
Tips: Percentages can be tricky, but always try to think in terms of 10s. For example, 10% of £5.80 is 58p as you simply take off one 0 from 580p. 20% would simply be double 58, which is 116 (break up 58 into 50 and 8, and double each and add them together). These questions require multiple steps, so be sure to draw out your working and double check your answer works before moving onto the next question!
Sample 11 Plus Maths Question: What is the missing number in the grid? Choose your answer from the following 5 options:
The Library 645 has books. The table shows the number of fiction and non-fiction books with different colour covers. How many green fiction books are there?
Answer and Explanation: D, 24. This is really just one big addition sum. 46 + 43 + 32 + 53… = 621. 645 – 621 = 24 and therefore D..
Tips: Short cuts are the key to being quick and accurate in this style of question. Look for two numbers that will add up to factors of ten. For example, 54 and 46 = 100. This makes your final addition much easier and quicker, and you can even do these questions in your head!
8) Shapes - Perimeter and Area
Sample 11 Plus Maths Question: Calculate the area of the shape. Choose your answer from the following 5 options:
Answer and Explanation: B, 17 cm2. Area questions can be easy! Break up this shape into two separate rectangles. Shape A can be 5cm by 3cm and Shape B can be 2 cm by 1 cm. To work out the area of shape A, multiply 5cm by 3 cm. 5 x 3 = 15 cm2. Shape B area can be calculated using 2 x 1 = 2 cm2. 15 + 2 = 17 cm2 and therefore the answer is B.
Tips: Splitting irregular shapes into regular shapes such as squares, rectangles and right angled triangles is the quickest way to solve these sometimes mean area and perimeter questions. Remember that if you don’t know the length of one side, there are hints that you can use to help you. For examples, For the shape above, knowing that the left side is 5 cm means that the right side must also be 5 cm. You know two lengths of the right side are 2 cm and 1 cm, totalling 3 cm. Therefore 5 – 3 = 2. You now know shape B has a length of 2 cm and a width of 1 cm. Easy!
9) Graphs with Interpretation
Sample 11 Plus Maths Question: Solve the following problem using the pictogram. Choose your answer from the 5 options given:
The pictogram shows the number of different colours of lipstick that were sold this week. How many pink lipsticks were sold this week?
Answer and Explanation: D, 17. A black box represents 5 lipsticks. Therefore 5 + 5 + 5 = 15 lipsticks. A white box represents less than 5 lipsticks. Therefore the number of pink lipsticks must be more than 15, but less than 20. The only answer that is greater than 15, and less than 20, is 17, Therefore the answer is D, 17.
Tips: Graphs will come in all shapes and sizes within our papers. They include bar graphs, pie charts, line graphs and pictograms, but they ultimately test your ability to interpret data accurately. Beware of traps in the increments on the x and y axis! Make sure you write on the graph in you have large data to interpret, and always double check your answer!
10) Navigation along a Path
Sample 11 Plus Maths Question: Solve the following problem involving navigation. Choose your answer from the 5 options given:
Sidhika is programming her lawnmower with a simple algorithm that will allow it to avoid the trees in the back yard. The algorithm only allows the directions FORWARD, TURN RIGHT 90°, and TURN LEFT 90° to be programmed in. What instructions will take it from the point on the map marked S and facing east to the point marked F, without hitting any trees?
Answer and Explanation: B. Below is the route required to reach the finish line. If you follow the route the lawnmower must first turn RIGHT 90° as it is facing East and needs to face SOUTH. Then it moves Forward 1, then turns left to face EAST, then moves Forward 2 etc. The only option that follows this rule is B. All the others have an error at some point in their algorithm that does not allow for the lawnmower to reach the final destination.
Tips: Draw out the route with the instructions first, and then find the option that matches your answer. Trying to solve each option one by one is too lengthy and confusing, particularly when the options are very similar, as they try to trick you.
11) Division (sometimes included with Verbal Reasoning)
Sample 11 Plus Maths Question: Solve the following problem involving division. Choose your answer from the 5 options given:Work out:
Answer and Explanation: A, 39. This is a relatively easy question as it simply requires you to do division. 5 goes into 195 39 times, and therefore the answer is A..
Tips: Whilst this question does not use verbal description, many of the questions you will encounter will do! Try to bring it back to numbers, and translate all the words you see into a sum of some sort. A lot of wordy questions are actually very easy once you understand what the question is asking.
12) Symmetry and Rotational Symmetry
Sample 11 Plus Maths Question: Solve the following problem involving rotation and reflection. Choose your answer from the 5 options given:
How has shape X been transformed?
Answer and Explanation: D, reflected and rotated 90 degrees counter-clockwise. From the image below it is easier to see how the shape has initially been reflected, before then being turned 90 degrees COUNTER CLOCKWISE (anticlockwise).
Tips: For questions where shapes are being rotated, it may be easier to draw out the shape on a spare piece of paper, to visualise the changes you are using. Know the difference between counter-clockwise and clockwise, as well as rotations and reflections.
13) Coordinates and Maps
Sample 11 Plus Maths Question: Solve the following problem involving the graph and its coordinates. Choose your answer from the 5 options given:
The grid shows the coordinates of players in a paintball fight. The camouflage tent is at the coordinates (4,6). Who is closest to the camouflage tent?
Answer and Explanation: Ursula, B. The Red X in the image below shows where the camouflage tent is. From this it is easy to see that Ursula is the closest and therefore the right answer. However, check Ursula’s coordinates too! She is at (5,6), which is only one unit away from (4,6). Gizela is the next closest at (2,5) which is 2 units to the left and one unit below from the tent. Therefore further away from the tent than Ursula.
Tips: Make sure you know that the x-axis units always come first, and then the y-axis! Students always mix up their x and y axis! X comes before Y in the alphabet, and this is your tip for remembering how to interpret coordinates.
Sample 11 Plus Maths Question: Solve the following problem involving fractions. Choose your answer from the 5 options given:
The teacher marks 48 questions in a spelling test. 16 questions are wrong. What fraction of the questions were right?
Answer and Explanation: B, 2/3. There are 48 questions in total. This will form the bottom number of your fraction (denominator). If 16 are wrong, then the number of right questions would be 48 - 16 = 32. Therefore, the number of right questions is 32, and the total number of questions is 48. Therefore 32 out of 48 can be written as a fraction 32/48. Simply this fraction by dividing both by a common factor, such as 8. This gives 4/6. This could be simplified further to give 2/3 by dividing by 2. (If you were particularly astute, you may notice that the largest common factor you could divide each number by is 16, and therefore do the simplification in one step).
Tips: With fractions, it is most important that you become quick at spotting common factors to simplify your fractions. This can be done by either breaking down the fraction into small chunks, by perhaps dividing both your numerator and denominators by small numbers like 2 or 3. However, a quicker and more effective way is to divide by the largest factor of both numbers, and that way cut the number of steps your calculation takes. In this case it is 16.
15) Shading of Shapes and Percentages
Sample 11 Plus Maths Question: How many squares must be filled in for 25% of the grid to be shaded?
Answer and Explanation: D, 5. The square has 4 squares across and 5 down. This gives a total of 20 squares. If you know what 25% of 20 from memory it can be solved instantly. However, 25% is also 1/4, and working out a quarter of 20 requires simply dividing by 4, to get 5. Therefore the answer is 5, D.
Tips: Whilst learning some maths calculations is helpful, understanding the origin of percentages and how to work out a question by converting fractions to percentages and vice versa will mean you can solve the more challenging questions, that cannot simply be learn.
16) Pie Charts
Sample 11 Plus Maths Question: Solve the following problem involving pie charts. Choose your answer from the 5 options given:
The pie chart shows how a class of 30 children travel to school. How many travel by bus?
Answer and Explanation: A, 18. A total of 30 students represents the whole circle. The number of degrees in a circle is 360. A bus represents 216 degrees out of the full 360 degrees. We can now form a fraction, 216/360 to represent this. Simplifying this, either by dividing by small numbers, or attempting to look for a larger common factor, such as 18, would give you 12/20 which can now simplify to 3/5. 3/5 of 30 is 18 and therefore the answer, A.
Tips: All circles contain 360 degrees. Knowing how to simplify fractions within a circle, and the factors of 360, such as 18, 60 and 180 will help to break down the scary fractions.
Sample 11 Plus Maths Question: Solve the following algebraic problem. Choose your answer from the 5 options given:
Given the following equation, if b = 8, what is y?
12b + 6 x 7 = y
Answer and Explanation: A, 138. If b = 8, then the equation reads as such: 12(b) + 6 x 7 = y. Don’t forget BODMAS, and do brackets first, then multiplication. So your equation now reads 96 + 42. This total 138 and therefore the answer is A.
Tips: These questions will continually test your understanding of BODMAS or BIDMAS, (Brackets, Orders or Indices, Division, Multiplication, Addition, Subtraction). Furthermore, remember that letters can easily represent numbers, but that the laws of mathematics remain the same, so do not be phased by the presence of letters, and treat just as you would numbers.
Sample 11 Plus Maths Question: Solve the following multiplication problem. Choose your answer from the 5 options given:
This year, my age is a multiple of 5. Next year it will be a multiple of 6. I am older than 10 but younger than 60. How old am I?
Answer and Explanation: 35, D. Whilst it may be intuitive to solve this question by trail and error, you have 5 options in front of you, and therefore should be able to solve this simply by looking at your options. This person has an age that is currently a multiple of 5, therefore the answer cannot be C. Rule this option out. Next year, they will be a multiple of 6. Therefore go through your options and choose which ones will be multiples of 6, when 1 is added to them. A: 30 + 1 = 31. This is not a multiple of 6. B: 25 + 1 = 26. This is also not a multiple of 6. C: 35 + 1 = 36. This is a multiple of 6 and therefore out answer. Just to be sure, E: 40 + 1 = 41. This is not a multiple of 6 either, therefore our answer of 35, D is confirmed.
Tips: Complicated questions do have shortcuts! You have options in front of you, and therefore save time and valuable effort by trailing your options, and logically working through your options!
Sample 11 Plus Maths Question: Solve the following problem. Choose your answer from the 5 options given:
How many ml of water is the jug holding?
Answer and Explanation: E, 225 ml. This question requires looking at a scale, and understanding intervals. Between 100 and 200 there is 1 scaled dash, representing half way between 200 and 100, therefore 150. This is also true for 300 and 200, where the scaled dash represents 250. The water line is half way between 250 and 200. This would be 225ml, and therefore our answer, E.
Tips: Look at your numbers and the number of intervals between them. This will give you an idea of what the scale may be. Scales could be going up in intervals of 5, 10, 15 or even fractions such as 1/3. If you are stuck, look at your options and try to eliminate those that simply do not make sense, such as 260ml in the sample question above.
Sample 11 Plus Maths Question: Solve the following rounding problem. Choose your answer from the 5 options given:
What is 1/4 of 37 rounded to the nearest whole number?
Answer and Explanation: D, 9. A Quarter of 37? 36 is the closest number to 37 that is a factor of 4. 4 goes into 4 9 times, with a remainder of 1. Therefore, it can be deduced that the nearest whole number that a quarter of 37 can be rounded to is 9. It cannot be 10, because 40 is not as close to 37 as 36 is. All the other options are not whole numbers and therefore can be eliminated.
Tips: Division is the easy bit - for 'quartering' something, you can divide it by 2, then do the same again as a short cut. For rounding, remember the rules! If the first number after the decimal point is 5 or greater it rounds up (eg. 9.5, 9.50, 9.6), and if it is 4 or less it rounds down (eg. 9.49, 9.4, 9.2). Remember that for subsequemt digits after the one you are dealing with, there is no 'knock-on effect' - for example, 9.49 rounds to 9 to the nearest number or 9.5 to one decimal place. You cannot round 9.49 to 9.5, then re-round it to 10, as successive rounding introduces errors.
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