Algebra: formulae, sequences and missing numbers – Year 6 maths worksheet answers to download as PDF

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1. Answer to Question 1

   1. Replace \(n\) with 5. Multiplication comes before addition: \(3\times5=15\), then \(15+2=\) 17.
   2. Replace \(n\) with 6: \(4\times6=24\), then \(24-3=\) 21.
   3. Work out \(2n\) first: \(2\times7=14\). Then subtract from 20: \(20-14=\) 6. (Doing 20 − 2 first would give the wrong answer.)
   4. Brackets first: \(3+1=4\). Then multiply by 5: \(5\times4=\) 20.
   5. \(\frac{n}{2}\) means \(n\div2\): \(10\div2=5\). Then \(5+9=\) 14.

2. Answer to Question 2

   1. Write \(P=4s\) and replace \(s\) with 7: \(P=4\times7=\) 28 cm. A perimeter is a length, so the unit is cm.
   2. \(lw\) means \(l\times w\): \(A=9\times6=\) 54 cm². Area is measured in square units.
   3. Multiply before adding: \(2\times8=16\) and \(2\times5=10\), so \(P=16+10=\) 26 cm.
   4. \(V=4\times3\times5=12\times5=\) 60 cm³. Volume is measured in cubic units.
   5. Replace \(h\) with 4: \(3\times4=12\), then \(5+12=17\). Four hours cost £17: the £5 fixed charge plus £3 for each hour.

3. Answer to Question 3

   1. Add 5 each time: 24, 29.
   2. Subtract 7 each time: 22, 15.
   3. Add 0.5 each time: 3.5, 4.
   4. Add 3 each time: 5, 8.
   5. Sequence (b) goes down each time, so it is the decreasing sequence.

4. Answer to Question 4

   1. The difference is 4 and the first term is 4, so the nth term is \(4n\).
   2. The difference is 3. \(3n\) gives 3, 6, 9, 12. We need 2, 5, 8, 11, which is 1 less each time. The nth term is \(3n-1\).
   3. The difference is 5. \(5n\) gives 5, 10, 15, 20. We need 2 more each time. The nth term is \(5n+2\).
   4. \(3\times20+2=60+2=\) 62.
   5. \(5\times12-4=60-4=\) 56.

5. Answer to Question 5

   1. \(n+8=21\). Subtract 8 from both sides: \(n=\) 13. Check: 13 + 8 = 21.
   2. Divide both sides by 3: \(a=\) 8. Check: 3 × 8 = 24.
   3. Subtract 5: \(2a=12\). Divide by 2: \(a=\) 6. Check: 2 × 6 + 5 = 17.
   4. Add 3: \(4b=24\). Divide by 4: \(b=\) 6. Check: 4 × 6 − 3 = 21.
   5. Multiply both sides by 5: \(x=\) 45. Check: 45 ÷ 5 = 9.

6. Answer to Question 6

   1. For example (1, 9), (2, 8), (5, 5) and (7, 3). Any pair of whole numbers that add up to 10 is correct.
   2. If \(x=0\) then \(y=12\). Each time \(x\) goes up by 1, \(y\) goes down by 2: (0, 12), (1, 10), (2, 8), (3, 6), (4, 4), (5, 2), (6, 0).
   3. There are 7 pairs. \(x\) cannot be bigger than 6, because then \(2x\) would be more than 12 and \(y\) would be negative. So \(x=0,1,2,\ldots,6\) gives every possible pair.

7. Answer to Question 7

   1. Replace \(m\) with 7: \(C=2\times7+3=14+3=\) £17.
   2. Replace \(m\) with 10: \(C=2\times10+3=20+3=\) £23.
   3. Set the cost equal to 25: \(2m+3=25\). Undo the +3 by subtracting 3: \(2m=22\). Undo the ×2 by dividing by 2: \(m=\) 11 miles. Check: \(2\times11+3=25\).
   4. \(C=1.5\times10+5=15+5=\) £20. Compare with £23 from part (b): the second company is cheaper by £3 for 10 miles.

8. Answer to Question 8

   1. \(C=5+3\times4=\) £17.
   2. Yes. The costs 8, 11, 14, 17 go up by £3 each time, which is the same each time, so the sequence is linear.
   3. \(5+3h=29\) gives \(3h=24\) and \(h=8\). At 8 hours the cost is £29, the same as the day ticket. So the day ticket is cheaper for more than 8 hours.
   4. \(C=5+3\times6=\) £23. The day ticket costs £29, so it is more expensive. She does not save any money: the hourly rate is £6 cheaper.

9. Answer to Question 9

   1. The sequence goes up by 3: 4, 7, 10, 13, 16.
   2. The difference is 3, so start with \(3n\) (3, 6, 9, ...). We need 4, 7, 10, which is 1 more. The nth term is \(3n+1\).
   3. \(3\times20+1=\) 61 sticks.
   4. \(3n+1\le100\) means \(3n\le99\), so \(n\le33\). With 33 squares she uses 100 sticks exactly. She can make 33 squares.

10. Answer to Question 10

    1. \(S=12+5\times8=12+40=\) £52.
    2. 17, 22, 27, 32.
    3. \(12+5w=100\) gives \(5w=88\) and \(w=17.6\). Ella cannot save for part of a week, so she needs 18 weeks.
    4. After 6 weeks she has \(12+30=42\), plus £20 gives £62. She needs £38 more: \(38\div5=7.6\), so 8 more weeks, which is 14 weeks altogether.

11. Answer to Question 11

    1. 12, 14, 16, 18, 20.
    2. The difference is 2, so start with \(2n\) (2, 4, 6, ...). We need 12, 14, 16, which is 10 more. The nth term is \(2n+10\).
    3. \(2\times15+10=\) 40 seats.
    4. \(2n+10=36\) gives \(2n=26\) and \(n=13\). It is row 13.
    5. \(2n+10=45\) gives \(2n=35\) and \(n=17.5\). The row number must be a whole number, so no. (All the rows have an even number of seats.)

12. Answer to Question 12

    1. \(C=8+0.05\times120=8+6=\) £14.
    2. \(C=8+0.05\times0=\) £8.
    3. \(8+0.05m=20\) gives \(0.05m=12\) and \(m=240\). He can use 240 minutes.
    4. The second plan is \(12+0.02m\). Set the two plans equal: \(8+0.05m=12+0.02m\). Subtract \(0.02m\) and 8 from both sides: \(0.03m=4\), so \(m=133.3...\). The plans cost the same at about 133 minutes (a little over 133).

13. Answer to Question 13

    1. \(20+6\times4=\) 44 cm.
    2. \(20+6\times10=\) 80 cm.
    3. \(20+6\times30=200\) cm = 2 m. Leo is right.
    4. \(20+6w>100\) gives \(6w>80\) and \(w>13.3\). After 13 weeks it is \(20+78=98\) cm. After 14 weeks it is 104 cm, which is the first time it is taller than 1 m.

14. Answer to Question 14

    1. Replace \(g\) with 15: \(6\times15=90\), then \(40+90=\) £130.
    2. Set up \(40+6g=160\). Subtract 40 from both sides: \(6g=120\). Divide by 6: \(g=\) 20 guests. Check: \(40+6\times20=160\).
    3. Only the fixed hall charge changes, from 40 to 55; the cost per guest stays £6. The new formula is \(C=55+6g\).
    4. \(C=55+6\times15=55+90=\) £145. Difference: \(145-130=\) £15, which is exactly the rise in the hall price.

15. Answer to Question 15

    1. The output is \(3n-4\).
    2. \(3\times5-4=15-4=11\). This matches the table.
    3. \(3n-4=23\) gives \(3n=27\) and \(n=\) 9. This matches the table.
    4. The output is \(2(n+7)=2n+14\).
    5. The outputs are equal when \(3n-4=2n+14\). Subtract \(2n\) from both sides: \(n-4=14\). Add 4: \(n=\) 18. Check: \(3\times18-4=50\) and \(2\times25=50\).

16. Answer to Question 16

    1. \(2\times12+2w=40\), so \(24+2w=40\) and \(2w=16\), so \(w=\) 8 m.
    2. \(A=12\times8=\) 96 m².
    3. \(40\times6.50=\) £260.
    4. New width = 10 m. New perimeter = \(2\times12+2\times10=\) 44 m. New area = \(12\times10=\) 120 m².
    5. 120 − 96 = 24 m².

17. Answer to Question 17

    1. The sequence adds 3 each time: 14 tiles.
    2. The difference is 3, so start with \(3n\) (3, 6, 9). We need 5, 8, 11, which is 2 more. The nth term is \(3n+2\).
    3. \(3\times12+2=\) 38 tiles.
    4. \(3n+2=50\) gives \(3n=48\) and \(n=\) 16. It is pattern 16.
    5. Pattern 20 uses \(3\times20+2=62\) tiles. \(62\times15\text{p}=930\text{p}=\) £9.30.

18. Answer to Question 18

    1. If \(a=3b\) then \(a+b=3b+b=4b=20\), so \(b=5\) and \(a=\) 15. Check: 15 + 5 = 20.
    2. Son = 1 part, Mum = 4 parts, so 5 parts = 45. One part = 9. The son is 9 and Mum is 36.
    3. If the numbers are \(x\) and \(y\) with \(x+y=15\) and \(x-y=3\), then \(2x=18\), so \(x=9\) and \(y=\) 6. The numbers are 9 and 6.
    4. Three pencils cost \(3\times0.40=1.20\). The two rubbers cost \(2.40-1.20=1.20\). One rubber costs 60p.

19. Answer to Question 19

    1. \(C_A=5+2d\).
    2. \(C_B=11+d\).
    3. Alpha: 9, 13, 17, 21. Beta: 13, 15, 17, 19.
    4. \(5+2d=11+d\). Subtract \(d\) from both sides: \(5+d=11\). So \(d=\) 6 GB. At 6 GB both cost £17.
    5. Alpha: \(5+2\times9=23\). Beta: \(11+9=20\). Beta is cheaper, by £3 a month. Over a year, he saves \(3\times12=\) £36.

20. Answer to Question 20

    1. Replace \(n\) with 1, 2, 3, 4: 2, 8, 14, 20.
    2. \(6n-4=62\) gives \(6n=66\) and \(n=\) 11. It is the 11th term.
    3. \(6n-4=100\) gives \(6n=104\) and \(n=17.33...\). This is not a whole number, so 100 is not a term. (All the terms are 4 less than a multiple of 6, which means they are all 2 more than a multiple of 6, and 100 is 4 more than a multiple of 6.)
    4. \(6n-4>200\) gives \(6n>204\) and \(n>34\). For \(n=34\), \(6\times34-4=200\), which is not bigger than 200. For \(n=35\): \(6\times35-4=\) 206.