All 40 Maths syllabus points for the 11 plus independent schools are listed, with the corresponding 40 sample paper questions, fully explained answers and helpful tips.
Sample 11 Plus Question: Solve the following volume and capacity problem. Choose your answer from the 5 options given:
Erik uses 2cm3 blocks to make a box. What is the volume of his box?
Answer and Explanation: D, 48 cm3. Each block is 2cm3. Therefore by finding how many blocks makes up this shape, we can find the total volume of the box. The box has a length of 4 blocks, a height of 2 blocks and a depth of 3 blocks. 4 x 2 x 3 gives 24 BLOCKS. This is NOT the end of the question, as each block has a volume of 2cm3. Therefore 24 x 2 = 48cm3 and the answer is D.
Tips: A frustrating mistake is to do the calculation for the volume, only to forget to read the question carefully enough to know that each block is 2cm3. Try to be concise of these tricks examiners love to use, and do not be caught out! Read all the information carefully, even if you think you have seen the question before!
22) 11 Plus Means, Medians and Ranges
Sample 11 Plus Question: Solve the following problem. Choose your answer from the 5 options given:
A class of 10 students take a maths test. Here are their scores out of 20.
16, 11, 15, 17, 11, 20, 14, 11, 15, 18
Which of the following options shows the correct mode and range of the scores?
Answer and Explanation: C, Mode: 11; Range: 9. This question has a number of shortcuts. Mode is means the ‘most’ and therefore simply look for which number is most frequent in your set. After some checking 11 appears 3 times and is therefore the Mode. Therefore the answer could be either C or D. Now look for the range, which is the difference between the largest and smallest number. The largest number is 20 and the smallest number is in fact 11. Therefore the Range is 9, and the answer is C, with a Mode of 11 and Range of 9.
Tips: Most is the ‘most’ common number. Median is the ‘middle’ number of the data set and the range is the difference between the largest and smallest number. The mean is the average of all the numbers. Know the definitions of these words, and practice taking the average of large number, by adding those that add to give factors of 10, e.g. 8 + 12 = 20 to make the final sum easy to solve.
23) Squaring, Cubing and Prime Numbers
Sample 11 Plus Question: Solve the following problem involving number factors. Choose your answer from the 5 options given:
Solve: 152 ÷ 32
Answer and Explanation: B, 25. One could attempt to do 15 x 15 which is 225. 225 divided by 9 (32) is 25. Or you could see that 15 is 3 x 5. Therefore 3 x 5 x 3 x 5 is being divided by 3 x 3. Crossing out the repeated 3s on either side gives you 5 x 5 and therefore 25. This maths may not have been taught but by looking for shortcuts may help differentiate a child from others by simply speed.
Tips: You can see that one method given above is much faster than the other. When you are posed with a question that makes you do seemingly long amounts of calculations, look to see if there is a hidden trick or short cut that might be available to use. Often the question writers are looking to credit students that can work around the problem efficiently!
Sample 11 Plus Question: Solve the following problem involving percentages. Choose your answer from the 5 options given:
Wilbert buys 9 pens costing £1.20 each. He gets a 40% discount on his purchase. How much money does Wilbert save from the purchase?
Answer and Explanation: E, £4.32. 9 pens at £1.20 each means 9 x £1.20. A quicker way would be doing 10 pens at £1.20 x 10 = £12 and take £1.20 which would be £10.80. 40% is a hard number to work with, so break 40% into 10%. 10% of £10.80 is simple, just take a 0 off the total number of pennies to make 108p = £1.08. Multiple this number by 4 to get £4.32 (spilt £1 and times by £4 and 8p times 4 to get 32p and add together to get £4.32). Therefore, the answer is £4.32, E, without a calculator.
Tips: Breaking up questions into sizeable chunks means that once complicated numbers like 40% of £10.80 can now be easily solved, without the need for a calculator, and within your head. Having said that, do not forget to write out your working so that you do not make a mistake whilst stressed under exam conditions!
25) Classification Trees
Sample 11 Plus Question: Solve the following classification tree problem. Choose your answer from the 5 options given:
The classification tree above is used to sort musical instruments. What is missing from the empty box?
Answer and Explanation: “Is it electronic?” D. This is a classification tree, and tests the ability of your child to understand classifications and how to sort information logically. In this case, the missing box is after the person chooses ‘No’ to ‘Is it a string instrument?” Therefore it is not a musical instrument. Looking beyond the blank box, the options are yes for Electronic drum, and no for acoustic drum kit. What question would you ask to get an answer to this options? It would have to differentiate the electrical drum kit from a non-electrical, acoustic drum kit, and therefore the most appropriate question to ask would be D, “Is it electronic?”
Tips: For classification trees, imagine you are doing the sorting yourself and try to work out the most appropriate option to pick from the 5 options you are given, and see if they fit into the blank effectively.
Sample 11 Plus Question: Solve the following probability problem. Choose your answer from the 5 options given:
Two fair dice are rolled together and their points are added to give a total.
What is the probability of scoring 7? The probability table has been provided to help you.
Answer and Explanation: C, 1/6. If two die are rolled, and their points added together to give a total, the probability table shows all the different ways you could get to 7. You could get 6 and 1, a 1 and 6 (as either dice would show a 1 or a 6), a 5 and 2 (and 2 and 5), 3 and 4 (4 and 3). This added to 6 out of a total of 36 options. 6/36 simplifies to 1/6 and therefore the answer is C, 1/6.
Tips: Sometimes you will not be provided the table to help you, therefore get into the habit of knowing the different similar rolls, eg. how you may get 1 and 6 rolled on two die, but also 6 and 1.
Sample 11 Plus Question: Solve the following folding net problem. Choose your answer from the 5 options given:
Which of the following 5 open nets represents the box seen below?
Answer and Explanation: E. Nets can be tricky. Visualising images and rotating them in your head can be confusing, so try to imagine each of the 5 options as 3D shapes, and see which one may correspond to your cube. A cannot as the base of the bulb is not next to the scissors. The same can be said for B and C. For D, the grass is the wrong way when joined to the scissors, whilst E works, as the scissors, bulb and grass correspond to the correct positions as indicated on the cube, and therefore the correct answer.
Tip: Start by looking at the nets. Find 2 shapes on a net that have similar shapes to the cube, and think about how they will fold. If they are compatible with the cube, then try and do this again with other pairs of sides of the cube using the same net. This method saves you from having to picture the whole folded net all at once!
28) Spatial Perception
Sample 11 Plus Question: Solve the following spatial reasoning problem. Choose your answer from the 5 options given:
The following shape has been made out of 3 blocks. Which of the 5 options show the 3 blocks used to make the shape?
Answer and Explanation: E. These questions are less easy to practice for, and require an understanding of who blocks fit into each other in 3D. In this question, firstly there is clearly a 3 blocked box. Therefore, C cannot be correct. There also needs to be a single box, therefore A, B and D cannot be correct either. The answer is therefore E, as the most appropriate answer with the information provided to you.
Tips: Some spatial reasoning questions may be particularly ambiguous, and you may find yourself stuck between two options. Think what would be the most appropriate answer, with the information provided to you, and make your decision that way.
29) Negative Numbers and Magnitude
Sample 11 Plus Question: Solve the following problem involving negative numbers. Choose your answer from the 5 options given:
Following is a record of temperatures in various countries around the world taken on the same day and time.
Netherlands: 2°C; Germany: 0°C; Slovakia: -2°C; Romania: -3°C; Finland: -6°C; Russia: -12°C; Ireland: 3°C
What is the difference between the temperatures in Romania and Ireland?
Answer and Explanation: D, 6°C. Romania is -3°C and Ireland is 3°C. The difference between these two can be calculated using a number line.
Counting from -3, there are 6 numbers before you reach 3, and therefore the difference between 3 and -3 is 6. Another, quicker way of solving this us by knowing that 3 - - 3 = 6. ( a minus minus a minus is a pulse: - - = +)
Tips: Number lines are an easy way to understand negative numbers. Once you have mastered this, you can start to understand the mathematical rules behind negative numbers, such as 3 - - 3 = 6 and -3 - - 3 = 0.
30) Exchange Rates
Sample 11 Plus Question: Solve the following problem involving exchange rates. Choose your answer from the 5 options given:
Kristopher and his friends are on holiday in China. The exchange rate is £1 = 8.80 Yuan. A jumper costs 220 Yuan and a scarf costs 132 Yuan. If Kristopher buys one jumper and two scarves, what is the total cost in pounds?
Answer and Explanation: D, £55. A jumper costs 220 and a scarf cost 132 Yuan. Two scarves therefore are 2 x 132 = 264 Yuan, and plus one jumper 220 + 264 = 484 Yuan. If 8.80 Yuan = £1, then we have to divide 484 by 8.8 to convert into pounds. Multiple both numbers by 10, giving 4840 and 88, and do a long division sum. If you do this correctly you should receive £55, the correct answer.
Tips: When the answers are very close together, it would be advisable to do long division rather than estimating an answer, as you need an exact answer here, since there is only a £1 difference between option D, the correct answer, and option C.
31) Number Puzzles
Sample 11 Plus Question: Solve the following number puzzle. Choose your answer from the 5 options given:
Add up the numbers below and select the correct answer in figures.
• Twelve million
Answer and Explanation: A, 12031049. This is a number puzzle, so read the question carefully. Twelve million = 12,000,000. Use commas to avoid being confused by how many zeros you have. Thirty thousand = 30,000. Therefore 12,000,000 + 30,000 = 12,030,000. 12,030,000 + 1,000 = 12,031,000. 12,031,000 + 49 = 12,031,049. Therefore the answer is A.
Tips: Doing these questions systematically, one at a time, will make sure you do not make any mistakes. Remember a million has 6 zeros, eg 3 million = 3,000,000. A thousand has 3 zeros, e.g 3 thousand = 3,000.
Sample 11 Plus Question: Solve the following angle problem. Choose your answer from the 5 options given:
Triangle PQR sits inside the rectangle QRST such that a side QR of the triangle forms the line QR of the rectangle. Given the angles below, what is the size of angle c?
a = 15°
Answer and Explanation: D, 50°. We need to find angle c. Therefore since angle c is within a triangle, if we find the sizes of the other two angles, we could add them together, and take the answer away from the total amount of degrees in a triangle, 180, to get an answer. Angle a = 15°. Since angle TQR is a right angle, angle E = 90-15. Therefore E = 75°. Angle QRS is also a right angle, and since angle b is 35°, Angle d is 55°. 55°+ 75° = 130°. 180° - 130° = 50°. Therefore the answer is D.
Tips: The number of degrees at a right angle is 90, and within a triangle is 180. Within a square it is 360 (4 right angles), as is a rectangle. A circle is 360°. Being aware of these mathematical facts is a necessity in order to excel in these sorts of questions.
Sample 11 Plus Question: Solve the following time problem. Choose your answer from the 5 options given:
The timetable shows times for a bus from Rodney Green to Wood Green.
Ella and Betty are two sisters living at Rodney Green. Ella’s school is on Queens Road and Betty’s on Wood Green. If both their schools start at 7:50 am, what is the time difference in minutes in the times they get their buses?
Answer and Explanation: B, 33 minutes. If both girls’ schools start at 07:50 (7:50am), then for Ella, she must arrive by at leans 07:44 at Queens Road, and Betty at 07:31 in order to be on time. Therefore, Betty must get the bus at 06:50 from Rodney Green, whilst Ella must get the 07:23 bus. Therefore, between 06:50 and 07:23 there is 33 minutes, and the correct answer is B.
Tips: Knowing how bus schedules are formatted, moving down represents the route a single bus takes, and the time it reaches one particular stop. Moving across represents the same route, taken by different buses, at different times.
Sample 11 Plus Question: Solve the following addition problem. Choose your answer from the 5 options given:
Shelley buys two gift sets costing £31 each and a perfume for £27 and other groceries for £23 from her supermarket. How much is that in total?
Answer and Explanation: A, £112. This is a relatively simple question, £31 + £31 + £27 + £23. Start with £27 and £23 as they add up nicely to £50. £50 + £62 = £121.
Tips: These should be easy question you must get right every time. If you need to do a long addition sum, ensure you write it out clearly so as to not make any mistakes!
Sample 11 Plus Question: Solve the following problem involving decimals. Choose your answer from the 5 options given:
Solve: 27.86 + 4.29
Answer and Explanation: E, 32.15. Whilst you may choose to write this out as a written sum, a useful tip may be of use in this case. 27 + 4 = 31. 0.86 + 0.29 is greater than 1, therefore add a 1 to 31 to get 32. There is only one option that has 32 whole numbers, followed by a decimal, therefore the answer is E, 32.15.
Tips: For the conventional question where such a short cut may not be possible, you should complete a full written sum. Ensure that you correctly line up the written numbers by aligning the decimal points vertically, and be absolutely certain that the correct digits line up. It is very easy to make a mistake otherwise!
Sample 11 Plus Question: Solve the following problem involving distance and time. Choose your answer from the 5 options given:
A flight travels at a constant speed of 710 miles per hour for thirteen hours. Another flight travels at a constant speed of 840 miles per hour for eleven hours. What is the difference in the distance travelled by the two flights?
Answer and Explanation: B, 10 miles. Flight A travels at 710 miles per hour for 13 hrs. Distance = Speed x Time. Therefore 710 x 13 = 9230 miles. Flight B travels 840 miles for 11 hrs. Therefore 840 x 11 = 9240. 9240 – 9230 = 10 miles and therefore the answer is 10 miles.
Tips: You should know the basic formulas of speed, distance and time. Speed = Distance ÷ Time. Distance = Speed x Time. Time = Distance ÷ Speed.
Sample 11 Plus Question: Solve the following problem involving speed. Choose your answer from the 5 options given:
Kuran’s car travelled 210 miles in the last 210 minutes. What was the speed of the car during the journey?
Answer and Explanation: D, 60 miles per hour. Speed = Distance ÷ Time. Therefore 210 ÷ 210 = 1 mile PER MINUTE. Therefore, in 60 minutes (1 hour) Kuran would have travelled 60 miles. Therefore 60 miles per HOUR, and the answer is D.
Tips: Beware not to get tricked between miles per minute and per hour. Understand that there are 60 minutes in an hour, 1000 metres in a kilometre and these questions should be straightforward!
38) Verbal Reasoning / BODMAS Questions
Sample 11 Plus Question: The following problem involves BODMAS. Read the question and choose your answer from the following 5 options given:
9 + 8 ÷ 4 = 44 ÷ ?
Answer and Explanation: A, 4. Using BODMAS, we start with division and 8 ÷ 4 = 2. 9 + 2 = 11. 44 ÷ 4 = 11 and therefore the answer is 4, A.
Tips: These questions will continually test your understanding of BODMAS or BIDMAS, (Brackets, Orders or Indices, Division, Multiplication, Addition, Subtraction).
39) Ordering, Combinations and Permutations
Sample 11 Plus Question: Solve the following problem. Choose your answer from the 5 options given:
Analyse these numbers:
Calculate the difference between the smallest and the largest numbers of the following numbers.
Answer and Explanation: D, 0.60. Start by converting these numbers into one single format. 46% = 0.46. 21/25 = 0.84 (21x4 = 84. This is because 25 is a quarter of 100, and a fraction out of 100 is the number as a decimal). 0.50 stays as it is. 55% = 0.55. 6/25 = 0.24 (by the same principle as before) and 0.81 stays the same. Our numbers are now 0.46 (46%), 0.84 (21/25 ), 0.50, 0.55 (55%), 0.24(6/25 ) and 0.81. Therefore, the smallest is 0.24 and the largest is 0.84. 0.84 – 0.24 = 0.60. Therefore, the answer is D, 0.60.
Tips: Converting percentages to decimals is relatively simple, as a percentage is just a number out of 100. Therefore 84% = 84 out of 100. A decimal could be considered similar as 0.05 is technically the same as 5 out of 100, ie 5%. As you can see, fractions, decimals and percentages are all interchangeable and similar, therefore understand how to move from one to the other seamlessly to truly master these questions.
40) Verbal Reasoning
Sample 11 Plus Question: Solve the following verbal reasoning problem. Choose your answer from the 5 options given:
If A = 15, B = 40, C = 4, D = 60, E = 10, which of the letters completes the series?
D ÷ C x E = __ x E
Answer and Explanation: A. These questions vary tremendously, but ultimately use letters or words to convey mathematics. In this case, it is algebra. Replacing letters for numbers gives the following: 60 ÷ 4 x 10 = __ x 10. Simplifying this, USING BODMAS, gives 150 = __ x 10. This can then be seen to be 15, as 15 x 10 = 150. Another way to reach this is to divide 150 by 10, to get 15. As a letter, 15 = A, and therefore our answer.
Tips: Do not be phased by letters or words, as it is still a maths paper! Remember your rules about maths, and ensure you try to replace as many of the words or letters with numbers to attempt to make sense of your problem. Good luck!