Maths practice paper

Answers and working

Mark the paper with your child, not for them. The working matters more than the mark — it says which mistake each wrong answer comes from.

1. D

   All five numbers have the same amount of digits, so comparing them digit by digit from the left is the quickest way. They all begin 74 ten-thousands, so that column settles nothing; the thousands column is where they start to differ. 742,234 is the smallest. Count the digits first: a number with more digits is bigger, whatever the digits are.

2. A

   Adding moves right along the number line. From -12, going up 12 reaches zero, and there are 3 left to travel, so the answer is 3. Splitting the jump at zero is what stops the sign going wrong. Split the jump at zero. Getting to zero, then going on, keeps the sign right every time.

3. B

   Counting in 12s gives 12, 24, 36, 48 and so on; counting in 8s gives 8, 16, 24, 32. The first number that appears in both lists is 24. Multiplying 12 × 8 gives 96, which is a common multiple but not the lowest one. That is the trap in this question. Count in the larger number and check each one against the smaller. It is far quicker than listing both.

4. D

   Split 27 into 20 and 7, and multiply 556 by each part. 556 × 7 = 3,892. 556 × 20 = 11,120 — note the zero on the end, because this row is 556 × 2 tens, not 556 × 2. Adding the two rows: 3,892 + 11,120 = 15,012. Keep the columns straight. Most wrong answers in long multiplication are neat arithmetic added up crooked.

5. B

   264 ÷ 19 = 13 remainder 17. The question asks for full packets, so the remainder is left over rather than rounded up: 17 pencils are not enough to fill another one. The answer is 13. This is the part of division that costs marks — the arithmetic is usually right and the last step is not. A question asking how many packets are *needed* would have the answer 14. Ask whether the leftovers still need carrying. If they do, round up; if they are simply left, round down.

6. A

   The power comes first: 7² = 49. Then multiply: 49 × 2 = 98. Finally add: 4 + 98 = 102. Working straight across from left to right would give 242, which is the wrong answer and the one this question is testing for. Underline the part that has to be done first before writing anything. It stops the left-to-right habit.

7. D

   Look at the bottoms first: 4 has become 28, so it has been multiplied by 7 (28 ÷ 4 = 7). Whatever is done to the bottom must be done to the top, so the top becomes 1 × 7 = 7. The fraction 7/28 is the same amount as 1/4. Work out what happened to the denominator before touching the numerator. The bottom tells you the rule.

8. C

   Fractions can only be added when the bottoms match, so the first job is a common denominator. The lowest number both 8 and 3 divide into is 24. 5/8 becomes 15/24 and 2/3 becomes 16/24. Now the tops can be added: 15 + 16 = 31, giving 31/24, which is already in its simplest form and is written 1 7/24 as a mixed number. The bottom never changes during the addition itself — 8 + 3 = 11 is the wrong move, and it is the one most often made. The denominators tell you the size of the pieces. You cannot add pieces of different sizes until you cut them to match.

9. A

   Multiplying two fractions is the one case where you simply multiply straight across: 1 × 1 = 1 on the top, 4 × 2 = 8 on the bottom, giving 1/8. No common denominator is needed — that is only for adding and subtracting, and looking for one here wastes time. The word "of" means the same thing: 1/4 of 1/2 is this same calculation. Cancel before multiplying if you can. It keeps the numbers small and the simplifying is then already done.

10. C

    This works backwards from the part to the whole. If 40 is 4/5, then 1/5 is 40 ÷ 4 = 10. The whole is 5 of those: 10 × 5 = 50. Checking forwards: 4/5 of 50 is 40, which is what the question said. Multiplying 40 by 5 straight away gives 200 and skips the step that matters. Find one part first. Getting to the unit fraction is the move that makes every reverse question work.

11. D

    Money is decimals with two places, so line the amounts up by the decimal point rather than by the last digit. £8.94 − £3.03 = £5.91. Working in pence avoids the point altogether: 894 − 303 = 591 pence, which is £5.91. Line up the decimal points, not the ends of the numbers. That one habit removes most decimal errors.

12. A

    They cannot be compared while they are in different forms, so turn them all into decimals first: 8% = 0.08, 1/5 = 0.2, 0.35 = 0.35, 1/2 = 0.5. Now the order is obvious, and the largest is 1/2. Comparing a fraction against a percentage by eye is guesswork, however confident it feels. Convert everything to decimals. It is the common language of all three forms.

13. D

    Write it as a fraction of the whole first: 160 out of 200, or 160/200. To turn a fraction into a percentage, divide and multiply by 100: 160 ÷ 200 = 0.8, and 0.8 × 100 = 80%. The other way round — 200 ÷ 160 — answers a different question entirely. If the fraction simplifies to something with 100, 50, 25, 20 or 10 on the bottom, the percentage can be read straight off.

14. D

    A ratio of 4:6 means the conkers are split into 4 + 6 = 10 equal parts. 140 ÷ 10 = 14, so one part is 14 conkers. The larger share is 6 parts: 14 × 6 = 84. Checking the two shares add back to the total — 84 + 56 = 140 — is worth the few seconds. Always check the shares add back up to what you started with.

15. C

    3a means 3 × a, not 3 next to a — the multiplication sign is left out to save writing, and forgetting it is the commonest mistake in the whole topic. With a = 7, 3a = 3 × 7 = 21. 2b = 2 × 7 = 14, so the expression is 21 + 14 − 5 = 30. Replacing each letter with its number, in brackets, before doing any arithmetic keeps it honest. A number written against a letter always means multiply. 3a is three lots of a.

16. B

    Let n be the number of packs. Each pack holds 10, so the packs give 10n, and the 13 already at home are added on: 10n + 13 = 83. Take the 13 off both sides: 10n = 83 − 13 = 70. Then divide by 10: n = 70 ÷ 10 = 7. Dividing 83 by 10 straight away gives 8 and quietly counts the 13 she already had as though they came in packs. Look for the amount that is there whatever happens — it gets added on, not multiplied.

17. D

    The rule works for any position, so there is no need to continue the sequence term by term. Put n = 50 into 2n + 8: 2 × 50 = 100, then add 8, giving 108. Checking the rule on a term you can see is always worth doing first: n = 1 gives 2 × 1 + 8 = 10, which matches the sequence. The number in front of n is always the step between terms. Spotting it is most of the work.

18. C

    Area is length × width, so finding a missing side means dividing rather than multiplying: 240 ÷ 15 = 16. The other side is 16 cm. Subtracting gives 225, which comes from treating the area as though it were a perimeter. Checking forwards: 15 × 16 = 240, which is the area given. Watch the units: cm² is an area, cm is a length. The answer here is a length.

19. C

    The base area is 5 × 5 = 25 cm². Volume is base area × height, so the height is volume ÷ base area: 125 ÷ 25 = 5 cm. Doing it in two steps — base first, then divide — is safer than trying to divide by both numbers at once. Checking forwards: 5 × 5 × 5 = 125 cm³. Work out the base area first. It turns a three-number problem into a two-number one.

20. D

    Angles around a point add up to 360°, because that is a full turn. The angles given come to 65 + 46 + 57 = 168°. 360 − 168 = 192°. Checking: 65 + 46 + 57 + 192 = 360°. Sense-check against a right angle: does the answer look bigger or smaller than 90°?

21. B

    Coordinates are written across first, then up: (x, y). Moving 2 right and 6 up changes the x from 3 to 5 and the y from 7 to 13, giving (5, 13). Left and down make the number smaller even when that means going further below zero. Right and up add; left and down subtract. Negatives follow the same rule.

22. C

    Time is not decimal, so subtracting the digits as though they were ordinary numbers goes wrong. Count on instead. From 17:10 to 18:00 is 50 minutes. From 18:00 to 18:00 is 0 hours. Then 45 more minutes to 18:45. Altogether that is 1 hour 35 minutes. An hour is 60 minutes, not 100. Column subtraction on times is the classic trap.

23. C

    Better value means cheaper per item, so work out the cost of one from each pack. Pack A: £0.58 ÷ 3 = 19.33p each. Pack B: £0.90 ÷ 5 = 18p each. Pack B works out cheaper per apple, so it is the better value. The bigger pack is not automatically the better buy, and the cheaper pack certainly is not. Work out the price of one item from each pack. Comparing anything else is guesswork.

24. A

    The two measurements are in different units, so one has to change before anything can be subtracted. 8 m is 800 cm. The pieces use 8 × 44 = 352 cm. 800 − 352 = 448 cm left. Subtracting 352 from 8 directly would be comparing centimetres with metres. Give the answer in the unit the question asks for, which is not always the one you worked in.

25. B

    Put them in order first, whichever is being asked for: 1, 11, 14, 14, 14, 15, 20. The median is the middle one. With seven numbers the middle is the fourth, which is 14. Finding the middle of the unsorted list is the mistake to avoid. Median is the middle, mode is the most common, range is the spread. Mode and most both start with m.