Solutions — Full Answer Key

Mathematics

Year 8 · 8.5 Directed Number and Algebra Basics

Solutions · Full Answer Key

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

Pack A — Answers

Bronze

-12

2.7

-24

-5

5x + 4y

-2a + 3b

; coefficient of

b

is 3

5a^3

8.7

3a^2

10.

15 + 5x

Silver

11.17

12.

-4

13.

(-5)^2 = 25

;

-5^2 = -25

14.

9x + 2

15.

-3x + 15

(or

15 - 3x

)

16.

12pq

17.

9x^2

18.

-6

19.

-3x + 5y - 2

20.

-8, -7, -2, 0, 3, 5

Gold

21.

6x^2 - 8x

22.

8x - 2

23.22

24.20

25.10

26.

x + 8y = 5 - 8 = -3

27.

2x + 8y

28.

P = 4(-2x + 5) = -8x + 20

29.

-64

30.

12 x^3 y^2

Platinum

31.

-11x + 6

32.

-64

33.

-14°C

34.

2x^2 + 5x - 12

35.17

36.

-£30

37.They differ unless

a = 0

38.

7x + 13y

39.4

40.

7x^2 + 4xy - 4y^2

Pack B — Answers

Bronze

-3

2.3

3.21

4.4

10x - 2y

2a + 4b

; coefficient of

b

is 4

7a^3

8.4

5a^3

10.

28 + 4x

Silver

11.2

12.5

13.

36

and

-36

14.

5x + 1

15.

-4x + 28

16.

35pq

17.

25x^2

18.14

19.

-7x + 2y - 4

20.

-9, -6, -1, 0, 4, 7

Gold

21.

10x^2 - 35x

22.

6x - 7

23.34

24.

-48

25.8

26.

x - 10y = 3 - 10 = -7

27.

5x + 9y

28.

P = 4(3x - 7) = 12x - 28

29.

-125

30.

10 x^3 y^2

Platinum

31.

-19x + 7

32.

-4

33.

-11°C

34.

5x^2 + 13x - 6

35.5

36.

-£105

37.See Pack A.

38.

2x + 16y

39.4

40.

9x^2 + 3xy - 6y^2

Problem-solving — Worked Solutions

1Problem 1

Answer

(a) +5°C (b) 6°C drop (2 pm → 6 pm) (c)

-2.2°C

Full working

(a)

-2 - (-7) = -2 + 7 = +5°C

. Rose by 5°C.

(b) Consecutive changes: 6→10: +5; 10→2: +7; 2→6:

-6

; 6→10:

-5

. Largest drop: 6°C (2 pm → 6 pm).

(c) Mean =

\frac{-7 + (-2) + 5 + (-1) + (-6)}{5} = \frac{-11}{5} = -2.2°C

2Problem 2

Answer

(a) 180,

-20

, 25,

-35

-50

(b) Yes — overdrawn twice (c) £10

Full working

(a) Start £100. After £80 deposit: £180. After £200 withdrawal:

-£20

. After £45 deposit: £25. After £60 withdrawal:

-£35

. After £15 fee:

-£50

.

(b) Account overdrawn after transactions 2 (by £20), 4 (by £35), 5 (by £50).

(c) £5/day × 2 days = £10 fee.

3Problem 3

Answer

(a) 5 (b) 24 (c)

-3

(d)

-2

Full working

(a)

-2 + 3 - (-4) = -2 + 3 + 4 = 5

.

(b)

(-2)(3)(-4) = -2 \times 3 = -6; -6 \times -4 = 24

.

(c)

a^2 = 4, b^2 = 9, c^2 = 16

4 + 9 - 16 = -3

.

(d) Numerator:

ab - c = (-2)(3) - (-4) = -6 + 4 = -2

. Denominator:

a + b = -2 + 3 = 1

. Hmm so

-2/1 = -2

✓.

4Problem 4

Answer

(a)

10x - 7

(b)

8x + 1

(c)

-5x - 0y = -5x

(d)

7a - 5b

Full working

(a)

6x - 15 + 4x + 8 = 10x - 7

.

(b)

-2x + 6 + 10x - 5 = 8x + 1

.

(c)

-3x - 2y - 2x + 2y = -5x + 0y = -5x

.

(d)

8a - 12b - 2a + 6b + a + b = (8 - 2 + 1)a + (-12 + 6 + 1)b = 7a - 5b

5Problem 5

Answer

(a)

3x - 2

; coeff 3 (b)

8x^2 - 2x

; coeff

-2

(c)

-2a + 3b + 5

Full working

(a)

x

terms:

5 + 2 - 4 = 3 \Rightarrow 3x

. Constants:

-3 + 1 = -2

. Result:

3x - 2

.

(b)

x^2

7 + 2 - 1 = 8 \Rightarrow 8x^2

x

-3 + 1 = -2 \Rightarrow -2x

. Result:

8x^2 - 2x

.

(c)

a

1 - 3 = -2

b

2 + 1 = 3

. Constants: 5. Result:

-2a + 3b + 5

6Problem 6

Answer

(a) 13 (b) 4 (c) Undefined (0 in denominator) (d)

-16

16

Full working

(a)

(-2)^2 = 4

4 \times 4 = 16

-3 + 16 = 13

.

(b) Brackets:

(-3 + 4) = 1

(-2)^2 = 4

1 \times 4 = 4

.

(c) Numerator:

-12 + 12 = 0

. Denominator:

4 - 4 = 0

. **Undefined** (

0/0

form).

(d)

-2^4 = -(2^4) = -16

(the minus is **outside** the exponent).

(-2)^4 = 16

(the minus is inside, and the exponent is even).

7Problem 7

Answer

(a)

5a

(b)

a^3 b

(c)

2x^2

(d)

3 a^2 b

(e)

4a^2

Full working

(a) Drop the × between a number and variable:

5a

.

(b) Three factors of

a

and one of

b

a^3 b

(no

\times

symbol needed).

(c) Two factors of

x

2x^2

.

(d) Two factors of

a

, one of

b

, coefficient 3:

3 a^2 b

.

(e)

(2a)^2 = 2^2 \times a^2 = 4a^2

8Problem 8

Answer

(a)

P = 6x + 4

(b) 34 cm (c)

x = 4

Full working

(a)

P = 2(2x + 3) + 2(x - 1) = 4x + 6 + 2x - 2 = 6x + 4

.

(b)

P(5) = 30 + 4 = 34

cm.

(c)

6x + 4 = 28 \Rightarrow 6x = 24 \Rightarrow x = 4

9Problem 9

Answer

(a)

-2x + 3y - 4

(b)

-2x - 6

(c)

-8x - 5

(d)

0x - 6y

(

= -6y

)

Full working

(a) Multiply each by

-1

-2x + 3y - 4

.

(b)

4 - 2x - 10 = -2x - 6

.

(c)

-6x + 3 - 2x - 8 = -8x - 5

.

(d)

5x - 10y - 6x + 3y + x + y = (5 - 6 + 1)x + (-10 + 3 + 1)y = 0x - 6y = -6y

10Problem 10

Answer

(a) 0 (b) See working

Full working

(a) Sum of all 9 numbers:

-4 -3 -2 -1 +0 +1 +2 +3 +4 = 0

. There are 3 rows, each adds to the magic sum, so

3 \times \text{magic sum} = 0

, giving magic sum **= 0**.

(b) Sum of all 4 lines through the centre (2 diagonals + middle row + middle column) =

4 \times 0 = 0

. Each off-centre cell appears in exactly one of these lines, contributing once; the centre appears in all four lines, contributing

4c

where

c

is the centre value. The 8 non-centre cells sum to

0 - c = -c

(since total is 0). So the 4 lines contribute:

4c + (\text{non-centre sum used once}) = 4c + ((-c) - \text{but each non-centre cell is in exactly one of these 4 lines})

... cleaner argument: by symmetry of magic squares using

\{-n, \ldots, n\}

, the centre must equal the mean = 0.

11Problem 11

Answer

(a)

12 a^3

(b)

3 a^2

(c)

-10 x^4

(d)

4 a^2 b^4

Full working

(a)

3 \times 4 = 12

;

a^2 \times a = a^3

. Result:

12 a^3

.

(b)

12/4 = 3

;

a^3 / a = a^2

. Result:

3 a^2

.

(c)

5 \times -2 = -10

;

x \times x^3 = x^4

. Result:

-10 x^4

.

(d)

(2 a b^2)^2 = 2^2 \times a^2 \times (b^2)^2 = 4 a^2 b^4

12Problem 12

Answer

(a)

-2

(basement 2) (b) +9 (after the third move) (c)

-3

(after the fourth move)

Full working

(a) Sum:

+5 -3 +7 -12 +1 = -2

. Final position: 2 floors below ground.

(b) After move 1: +5. After move 2: +2. After move 3: +9 (highest). After move 4:

-3

. After move 5:

-2

. **Max: +9**.

(c) Lowest:

-3

(after move 4).