Printable · GCSE Higher · ages 14-16
Inverse operations and priority of operations worksheet — GCSE Higher
Fifteen questions on "inverse operations and priority of operations" — DfE statement N3. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Inverse operations and priority of operations worksheet — GCSE Higher
MathsUKwww.geekhero.co.uk
- 1.A shop buys boxes of pens for £4 each. It sells each box on to a school for £3 more than it paid, and charges an extra £2 delivery fee for the whole order. A school orders 5 boxes. Work out the total cost of the order.
- 2.Work out (−6) + (−4) × 3
- 3.Write down the reciprocal of 0.2
- 4.Find the missing number: 17 × ▢ = 391
- 5.Insert one pair of brackets into 2 + 3 × 5 − 1 so that the calculation is equal to 24. Which calculation is correct?
- 6.Work out √144 − 2 × 3 + √25
- 7.Work out 100 − 4 × 5²
- 8.Work out √25 + 4² − 12 ÷ 3
- 9.Work out (−3) × 4 + 2 × (−5)
- 10.Work out (−36) ÷ (−6) × 2
- 11.Work out 3 × (−2)² − 5
- 12.Grace works out 7 × 99 by writing 99 as 100 − 1. Use her method to work out 7 × 99.
- 13.Work out (−2)² − 3
- 14.Work out (−2)³ + (−3)² − (−4)
- 15.Work out the reciprocal of (2 + 3)
Answer key
- (b) £37 — One box costs £4 + £3 = £7. Five boxes cost 5 × £7 = £35. Adding the single £2 delivery fee gives £35 + £2 = £37. A candidate who added the £2 delivery fee to each box instead of once for the whole order worked out 5 × (£7 + £2) = 5 × £9 = £45. A candidate who forgot the £3 markup and used the shop's buying price worked out 5 × £4 + £2 = £22. A candidate who added the £3 markup only once, after multiplying the buying price by 5, worked out 5 × £4 + £3 + £2 = £25.
- (b) −18 — Method: the multiplication is carried out before the addition, and a negative multiplied by a positive is negative. Working: (−4) × 3 = −12, so the calculation becomes (−6) + (−12) = −18. Answer: −18. The distractors: −30 comes from adding first and multiplying afterwards, giving (−6 + −4) × 3 = −10 × 3 = −30; 6 comes from treating (−4) × 3 as +12 on the grounds that a minus sign makes a product positive, giving −6 + 12 = 6; 18 comes from ignoring both minus signs and working out 6 + 4 × 3 = 18.
- (c) 5 — 0.2 = 1/5, and turning the fraction upside down gives the reciprocal 5/1 = 5. Writing −0.2 mistakes the reciprocal for the negative of the number. Working out 1 − 0.2 = 0.8 mistakes the reciprocal for subtracting the number from 1. Writing 0.5 is the reciprocal of 2, not of 0.2 — a place-value slip that drops the decimal, since 1 ÷ 0.2 = 5 while 1 ÷ 2 = 0.5.
- (a) 23 — Division undoes multiplication, so the missing number is 391 ÷ 17 = 23. Writing down 17 repeats the number already given instead of solving for the missing one. Subtracting instead of dividing gives 391 − 17 = 374. Multiplying instead of dividing gives 391 × 17 = 6647.
- (b) (2 + 3) × 5 − 1 — 2 + 3 = 5, then 5 × 5 = 25, then 25 − 1 = 24, so the brackets belong around 2 + 3. Placing them around 5 − 1 instead gives 5 − 1 = 4, then 3 × 4 = 12, then 2 + 12 = 14. Leaving the multiplication bracketed instead changes nothing, because it already had priority: 3 × 5 = 15, then 2 + 15 = 17, then 17 − 1 = 16. Bracketing both 2 + 3 and 5 − 1 uses two pairs instead of the one asked for: 2 + 3 = 5, 5 − 1 = 4, then 5 × 4 = 20.
- (b) 11 — Method: roots and the multiplication are worked out before the addition and subtraction, and what is left is then worked through from left to right. Working: √144 = 12, √25 = 5 and 2 × 3 = 6, so the calculation becomes 12 − 6 + 5, which gives 6 + 5 = 11. Answer: 11. The distractors: 1 comes from carrying out the addition before the subtraction, giving 12 − (6 + 5) = 12 − 11 = 1; 35 comes from working from left to right with no priority, giving 12 − 2 = 10, then 10 × 3 = 30 and 30 + 5 = 35; 7 comes from combining the two roots as √(144 + 25) = √169 = 13 and then subtracting the product, giving 13 − 6 = 7.
- (a) 0 — Method: BIDMAS works through the index first, then the multiplication, then the subtraction. Working: 5² = 25, then 4 × 25 = 100, and finally 100 − 100 = 0. Answer: 0. The distractors: 2400 comes from working from left to right and subtracting first, giving (100 − 4) × 25 = 96 × 25 = 2400; −300 comes from multiplying before applying the index, giving (4 × 5)² = 20² = 400 and then 100 − 400 = −300; 60 comes from reading 5² as 5 × 2 = 10, so that 4 × 10 = 40 and 100 − 40 = 60.
- (b) 17 — Roots and powers are worked out first: √25 = 5 and 4² = 16. Division comes next: 12 ÷ 3 = 4. Then addition and subtraction, left to right: 5 + 16 − 4 = 17. A candidate who treated 4² as 4 × 2 = 8, multiplying the base by the exponent instead of squaring it, worked out 5 + 8 − 4 = 9. A candidate who did not evaluate the root and used 25 itself worked out 25 + 16 − 4 = 37. A candidate who ignored the priority of division and worked through 5 + 16 − 12 ÷ 3 strictly left to right got 5 + 16 = 21, then 21 − 12 = 9, then 9 ÷ 3 = 3.
- (a) −22 — Method: both multiplications are carried out before the addition, and a positive multiplied by a negative is negative. Working: (−3) × 4 = −12 and 2 × (−5) = −10, so the calculation becomes −12 + (−10) = −22. Answer: −22. The distractors: 22 comes from ignoring the minus signs and working out 3 × 4 + 2 × 5 = 22; 50 comes from working from left to right with no priority at all, giving −12 + 2 = −10 and then −10 × (−5) = 50; −2 comes from taking 2 × (−5) as +10, so that −12 + 10 = −2.
- (c) 12 — Method: division and multiplication have equal priority, so they are carried out in the order they are written, from left to right; a negative divided by a negative is positive. Working: (−36) ÷ (−6) = 6, and then 6 × 2 = 12. Answer: 12. The distractors: 3 comes from carrying out the multiplication first, (−6) × 2 = −12 followed by (−36) ÷ (−12) = 3; −12 comes from treating a negative divided by a negative as negative, giving −6 and then −6 × 2 = −12; 6 comes from stopping at the division and never carrying out the multiplication by 2.
- (a) 7 — Method: BIDMAS deals with the index first, then the multiplication, then the subtraction. Working: (−2)² = (−2) × (−2) = 4, then 3 × 4 = 12, and finally 12 − 5 = 7. Answer: 7. The distractors: −17 comes from squaring only the 2 and keeping the minus sign, giving 3 × (−4) = −12 and then −12 − 5 = −17; 31 comes from multiplying before applying the index, giving (3 × (−2))² = (−6)² = 36 and then 36 − 5 = 31; −3 comes from carrying out the subtraction before the multiplication, giving 3 × (4 − 5) = 3 × (−1) = −3.
- (c) 693 — Method: multiplying a bracket by a number multiplies every term inside it, so 7 × (100 − 1) = 7 × 100 − 7 × 1. Working: 7 × 100 = 700 and 7 × 1 = 7, so the calculation becomes 700 − 7 = 693. Answer: 693. The distractors: 699 comes from subtracting the 1 itself rather than 7 lots of it, giving 700 − 1 = 699; 707 comes from adding the second product instead of subtracting it, giving 700 + 7 = 707; 700 comes from rounding 99 up to 100 and then offering the estimate 7 × 100 as an exact value.
- (b) 1 — Method: BIDMAS deals with the index before the subtraction, and a negative number multiplied by itself gives a positive result. Working: (−2)² = (−2) × (−2) = 4, so the calculation becomes 4 − 3 = 1. Answer: 1. The distractors: −7 comes from squaring only the 2 and leaving the minus sign outside the index, giving −(2²) − 3 = −4 − 3 = −7; −1 comes from subtracting the square from 3 instead of 3 from the square, giving 3 − 4 = −1; 25 comes from carrying out the subtraction before the index, giving (−2 − 3)² = (−5)² = 25.
- (a) 5 — Method: each index is worked out first, and subtracting a negative number is the same as adding the positive. Working: (−2)³ = (−2) × (−2) × (−2) = −8 and (−3)² = (−3) × (−3) = 9, while − (−4) becomes + 4, so the calculation becomes −8 + 9 + 4 = 5. Answer: 5. The distractors: −13 comes from taking (−3)² as −9, giving −8 − 9 + 4 = −13; −3 comes from reading − (−4) as − 4, giving −8 + 9 − 4 = −3; 21 comes from treating every power of a negative number as positive, so that (−2)³ is taken as 8 and the calculation becomes 8 + 9 + 4 = 21.
- (a) 1/5 — Work out the bracket first: 2 + 3 = 5. The reciprocal of 5 is 1/5. A candidate who forgot to take the reciprocal and just gave the value of the bracket wrote 5. A candidate who took the reciprocal but made a sign error wrote −1/5. A candidate who found the reciprocal of each number separately and added them, treating reciprocal as if it distributes over addition, worked out 1/2 + 1/3 = 5/6.
Build your own mix at the worksheet builder.