Solutions — Full Answer Key

Mathematics

Year 8 · Algebra

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze

1.19

28x

4x + 12

3x^2 - 2x

5(2x + 3)

3x(2x + 3)

3x

3x + 12

-3

10.

7p + 7q

11.

x = 12

12.

x = 13

13.

x = 13

14.

x = 28

15.

x = 5

16.

x = 3

17.

x = 6

18.Yes

19.

x = 8

20.

x = -5

Silver

21.41

22.

9x + 2

23.

4x(5x - y)

24.

3x

25.

2x + 3

26.

-5x + 2

27.

\dfrac{2b}{3}

28.

c = 8, d = -2

29.

x(x + 6)

30.(a)

x - 120

(b)

3(x - 120)

31.

x = 5

32.

x = 4

33.

x = 5

34.

x = 18

35.

x = 9

36.

x = 16

37.

n = 4

38.

x = -5

39.

x = 6

40.

x = 6

Gold

41.

-8x + 6

42.

8x(3x + 2)

43.

2x + 3

44.Both equal

x^2 + 6x

✓.

45.

ab(a + b)

46.40

47.

(x + 3)(4 + y)

48.

\dfrac{8}{x}

49.

\dfrac{7x}{12}

50.

P = 6x + 4

;

P(4) = 28

51.

x = 11

52.

x = 22

53.

x = 30

54.220

55.

x = 20

56.31, 32, 33

57.

x = 20

58.

x = 5

59.

x = -10

60.

x = 0

Platinum

61.

6x^2 - 2x + 10

62.

3(x^2 + 4x + 3) = 3(x + 1)(x + 3)

63.

\dfrac{2x - 3}{2(x + ?)}

— see working

64.(a)

y = \tfrac{3}{2}

(b)

z = 2

65.

x = 8

66.

x = 20

67.

(a - b)(a + b)

68.

A = x^2 + 7x + 10

;

A(8) = 130

m²

69.

x = 4

70.

\dfrac{x + 11}{6}

71.

x = -23

72.

x = -8

73.

x = 1000

chf

74.Pen 1.20 chf; pencil 1.00 chf

75.Son 8, man 32

76.

x = 2

77.11 and 12

78.

x = 12

79.

x = \tfrac{7}{6}

80.

x = 3

Pack B — Answers

Bronze

-2

30x

6x + 30

5x^2 - 4x

6(2x + 3)

5x(2x + 5)

3x

5x + 35

-10

10.

13p + 2q

11.

x = 8

12.

x = 15

13.

x = 9

14.

x = 30

15.

x = 5

16.

x = 4

17.

x = 17

18.Yes

19.

x = -6

20.

x = -4

Silver

21.

-1

22.

5x + 1

23.

3x(5x - 3y)

24.

3x

25.

2x + 3

26.

-8x + 7

27.

\dfrac{3x}{2}

28.

c = 6, d = 2

29.

x(x + 9)

30.(a)

x - 75

(b)

3(x - 75)

31.

x = 5

32.

x = 12

33.

x = 5

34.

x = 25

35.

x = 5

36.

x = 7

37.

n = 3

38.

x = -3

39.

x = 5

40.

x = 7

Gold

41.

-11x + 7

42.

9x(3x + 2)

43.

3x + 2

44.Both equal

x^2 + 10x

✓.

45.

pq(p - q)

46.37

47.

(x - 2)(5 - y)

48.

\dfrac{11}{x}

49.

\dfrac{7x}{10}

50.

P = 8x + 4

;

P(4) = 36

51.

x = 7

52.

x = 14

53.

x = 24

54.240

55.

x = 500

56.39, 40, 41

57.

x = 10

58.

x = 7

59.

x = -5

60.Same — 0

Platinum

61.

4x^2 - 7x + 21

62.

2(x^2 + 5x + 4) = 2(x + 1)(x + 4)

63.See working

64.(a)

y = \tfrac{3}{2}

(b)

z = 2

65.

x = 4

66.

x = 500

67.

(5 - x)(5 + x)

68.

A = x^2 + 10x + 21

;

A(8) = 165

m²

69.Same.

70.

\dfrac{x + 23}{12}

71.

x = -11

72.

x = 11

73.

x = 1000

chf

74.Pen 0.90 chf; pencil 0.90 chf

75.Son 6, man 30

76.

x = 8

77.12 and 13

78.

x = 18

79.

x = 2

80.

x = 4

Problem-solving — Worked Solutions

1Problem 1

Answer

(a) Equivalent (b) NOT equivalent (c) Equivalent (d) Equivalent

Full working

(a)

3(x + 4) = 3x + 12

✓.

(b)

(x + 2)^2 = x^2 + 4x + 4

, not

x^2 + 4

.

(c)

2x - 6 - x + 5 = x - 1

✓.

(d)

\tfrac{6x}{2} = 3x

✓.

2Problem 2

Answer

(a) 0 (b) 5 (c)

\tfrac{-1}{-5} = \tfrac{1}{5}

(d) 64

Full working

(a)

-6 + 6 = 0

.

(b)

9 - 4 = 5

.

(c) Numerator

-1

, denominator

-5

. Result

\tfrac{1}{5}

.

(d)

(2 \times -3 - 2)^2 = (-8)^2 = 64

3Problem 3

Answer

(a)

6(x + 2)

(b)

4x(x - 2)

(c)

6ab(2a - 3b)

(d)

(x - 1)(5 + y)

Full working

(a) HCF 6.

(b) HCF

4x

.

(c) HCF

6ab

.

(d) Common factor

(x - 1)

4Problem 4

Answer

(a)

\tfrac{2a}{b}

(b)

2x - 3

(c)

\tfrac{9x}{10}

(d)

\tfrac{x - 11}{12}

Full working

(a) Cancel

4ab

from numerator and denominator.

(b) Factorise numerator:

3x(2x - 3)

. Divide by

3x

2x - 3

.

(c) Common denominator 10:

\tfrac{5x + 4x}{10}

.

(d) Common denominator 12:

\tfrac{3(x - 3) - 2(x + 1)}{12} = \tfrac{x - 11}{12}

5Problem 5

Answer

(a)(i)

x - 120

(ii)

3(x - 120)

(b) Elena = 220 songs

Full working

(a) Sabrina =

x - 120

. Pete =

3 \times

Sabrina =

3(x - 120)

.

(b)

3(x - 120) = 300 \Rightarrow x - 120 = 100 \Rightarrow x = 220

6Problem 6

Answer

(a)

P = 6x + 8

;

A = (2x + 1)(x + 3)

(b)

A = 2x^2 + 7x + 3

(c)

P = 32

;

A = 63

Full working

(a)

P = 2(2x + 1) + 2(x + 3) = 6x + 8

A = (2x + 1)(x + 3)

.

(b) Expand:

A = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3

.

(c) At

x = 4

P = 32

;

A = 32 + 28 + 3 = 63

7Problem 7

Answer

(a)

x^2 + 8x + 15

(b)

x^2 + 3x - 28

(c)

2x^2 + 5x - 3

(d)

x^2 + 4x + 4

Full working

Use the area model / FOIL.

(a)

x^2 + 3x + 5x + 15 = x^2 + 8x + 15

.

(b)

x^2 + 7x - 4x - 28 = x^2 + 3x - 28

.

(c)

2x^2 + 6x - x - 3 = 2x^2 + 5x - 3

.

(d)

(x + 2)^2 = x^2 + 4x + 4

8Problem 8

Answer

E = \tfrac{P}{3} + d

Full working

Sabrina =

P/3

. Sabrina = Elena

- d

, so Elena

= P/3 + d

9Problem 9

Answer

(a)

6x = 180

x = 30

(b)

60°, 100°, 20°

Full working

(a) Sum 180°:

2x + 3x + 10 + x - 10 = 6x = 180

x = 30

.

(b) Angles:

60°, 100°, 20°

. Check

60 + 100 + 20 = 180°

✓.

10Problem 10

Answer

(d)

52^2 = 50^2 + 2 \cdot 50 \cdot 2 + 2^2 = 2500 + 200 + 4 = 2704

Full working

(a)

(a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2

✓.

(b)

(a - b)(a - b) = a^2 - ab - ab + b^2 = a^2 - 2ab + b^2

.

(c)

(a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2

.

(d) Use

52 = 50 + 2

52^2 = 2500 + 200 + 4 = 2704

. ✓

11Problem 11

Answer

(a) Test

x = 6

(6+6)/2 = 6

(12+6)/3 = 6

. They are equal at

x = 6

(b)

x = 6

Full working

Set them equal:

\tfrac{x + 6}{2} = \tfrac{2x + 6}{3}

. Cross:

3(x + 6) = 2(2x + 6)

3x + 18 = 4x + 12

x = 6

.

For

x > 6

, the second is larger; for

x < 6

, the first is larger.

12Problem 12

Answer

(a) See working (b)

\tfrac{4}{11}

(c)

\tfrac{1}{7}

Full working

(a)

x = 0.\overline{ab}

. Then

100x = ab.\overline{ab}

. Subtract

x

99x = ab

, so

x = \tfrac{ab}{99}

.

(b)

0.\overline{36} = \tfrac{36}{99} = \tfrac{4}{11}

(divide by 9).

(c)

0.\overline{142857} = \tfrac{142857}{999999} = \tfrac{1}{7}

(since

7 \times 142857 = 999999

13Problem 13

Answer

(a)

x = 6

(b)

x = 10

(c)

x = 7

(d)

x = 25

Full working

(a)

4x = 24 \Rightarrow x = 6

.

(b)

x - 4 = 6 \Rightarrow x = 10

.

(c)

3x = 21 \Rightarrow x = 7

.

(d)

x + 3 = 28 \Rightarrow x = 25

14Problem 14

Answer

(a)

8n + 6 = 38, n = 4

(b) Long 11 cm, short 8 cm (c)

2(11) + 2(8) = 38

✓

Full working

(a)

P = 2(3n - 1) + 2(n + 4) = 8n + 6 = 38 \Rightarrow n = 4

.

(b) Long:

3(4) - 1 = 11

cm. Short:

4 + 4 = 8

cm.

(c)

22 + 16 = 38

✓.

15Problem 15

Answer

x = 220

Full working

3(x - 120) = 300 \Rightarrow x - 120 = 100 \Rightarrow x = 220

16Problem 16

Answer

(a)

y = 3/2

(b)

z = 2

Full working

(a)

3y = y + 3 \Rightarrow y = \tfrac{3}{2}

, so long =

4.5

cm.

(b)

z = 3z - 4 \Rightarrow z = 2

, so short = 2 cm.

(c)

P = 2(4.5) + 2(2) = 9 + 4 = 13

cm.

17Problem 17

Answer

(a) 797.30 chf (b) New price 743.40; difference 53.90 chf

Full working

(a)

0.74 x = 590 \Rightarrow x = 590/0.74 = 797.30

chf.

(b)

590 \times 1.26 = 743.40

. Difference

= 797.30 - 743.40 = 53.90

chf.

18Problem 18

Answer

Number = 5

Full working

\tfrac{3x - 7}{2} = 4 \Rightarrow 3x - 7 = 8 \Rightarrow 3x = 15 \Rightarrow x = 5

19Problem 19

Answer

(a)

x = 15

chf (b) Effective rises from 15.20 to 15.40 chf/ticket

Full working

(a)

25x + 5 = 380 \Rightarrow 25x = 375 \Rightarrow x = 15

chf.

(b) Doubling fee to 10: total

375 + 10 = 385

chf, per ticket

= 385/25 = 15.40

chf. Before, per-ticket effective was

380/25 = 15.20

20Problem 20

Answer

(a)

4s + 4 = 3(s + 4)

(b) Son 8, man 32

Full working

(a) Currently: son

s

, man

4s

. In 4 years: son

s + 4

, man

4s + 4

. Equation:

4s + 4 = 3(s + 4)

.

(b)

4s + 4 = 3s + 12 \Rightarrow s = 8

. Man = 32.

21Problem 21

Answer

5 L

Full working

Acid mass

= 1

L (20% of 5). After adding

w

L water, total volume

= 5 + w

. Acid is still 1 L → fraction

= 1/(5 + w) = 0.10

. So

5 + w = 10 \Rightarrow w = 5

22Problem 22

Answer

(a)

4x + 8 = 36, x = 7

(b) 10, 13, 13 cm (c)

10 + 13 > 13

✓ — valid

Full working

(a)

(x + 3) + (2x - 1) + (x + 6) = 4x + 8 = 36 \Rightarrow x = 7

.

(b) 10, 13, 13.

(c) Pairs:

10 + 13 = 23 > 13

✓;

13 + 13 = 26 > 10

✓. Valid (isosceles).

23Problem 23

Answer

(a) Small:

3/n

; Large:

11/(5n - 4)

(b)

n = 3

Full working

(a) Small:

3/n

chf/bag. Large:

11/(5n - 4)

chf/bag.

(b) Equal value:

\tfrac{3}{n} = \tfrac{11}{5n - 4} \Rightarrow 3(5n - 4) = 11n \Rightarrow 15n - 12 = 11n \Rightarrow 4n = 12 \Rightarrow n = 3

.

(c) Try

n = 4

: small

0.75

chf/bag; large

11/16 \approx 0.69

chf/bag. Large is better.

24Problem 24

Answer

x \approx 5.0

(exact: 5)

Full working

x = 5

25 + 5 = 30

✓. So

x = 5

exactly. (If a non-integer solution is wanted, e.g.

x^2 + x = 31

gives

x \approx 5.1

.)

For the actual equation:

f(5) = 30

exactly, so

x = 5

is the positive solution.