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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
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
1.−12-12
2.7
3.−24-24
4.−5-5
5.5x+4y5x + 4y
6.−2a+3b-2a + 3b; coefficient of bb is 3
7.5a35a^3
8.7
9.3a23a^2
10.15+5x15 + 5x
Silver
11.17
12.−4-4
13.(−5)2=25(-5)^2 = 25; −52=−25-5^2 = -25
14.9x+29x + 2
15.−3x+15-3x + 15 (or 15−3x15 - 3x)
16.12pq12pq
17.9x29x^2
18.−6-6
19.−3x+5y−2-3x + 5y - 2
20.−8,−7,−2,0,3,5-8, -7, -2, 0, 3, 5
Gold
21.6x2−8x6x^2 - 8x
22.8x−28x - 2
23.22
24.20
25.10
26.x+8y=5−8=−3x + 8y = 5 - 8 = -3
27.2x+8y2x + 8y
28.P=4(−2x+5)=−8x+20P = 4(-2x + 5) = -8x + 20
29.−64-64
30.12x3y212 x^3 y^2
Platinum
31.−11x+6-11x + 6
32.−64-64
33.−14°C-14°C
34.2x2+5x−122x^2 + 5x - 12
35.17
36.−£30-£30
37.They differ unless a=0a = 0.
38.7x+13y7x + 13y
39.4
40.7x2+4xy−4y27x^2 + 4xy - 4y^2

Pack B — Answers

Bronze
1.−3-3
2.3
3.21
4.4
5.10x−2y10x - 2y
6.2a+4b2a + 4b; coefficient of bb is 4
7.7a37a^3
8.4
9.5a35a^3
10.28+4x28 + 4x
Silver
11.2
12.5
13.3636 and −36-36
14.5x+15x + 1
15.−4x+28-4x + 28
16.35pq35pq
17.25x225x^2
18.14
19.−7x+2y−4-7x + 2y - 4
20.−9,−6,−1,0,4,7-9, -6, -1, 0, 4, 7
Gold
21.10x2−35x10x^2 - 35x
22.6x−76x - 7
23.34
24.−48-48
25.8
26.x−10y=3−10=−7x - 10y = 3 - 10 = -7
27.5x+9y5x + 9y
28.P=4(3x−7)=12x−28P = 4(3x - 7) = 12x - 28
29.−125-125
30.10x3y210 x^3 y^2
Platinum
31.−19x+7-19x + 7
32.−4-4
33.−11°C-11°C
34.5x2+13x−65x^2 + 13x - 6
35.5
36.−£105-£105
37.See Pack A.
38.2x+16y2x + 16y
39.4
40.9x2+3xy−6y29x^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-2.2°C
Full working
(a) −2−(−7)=−2+7=+5°C-2 - (-7) = -2 + 7 = +5°C. Rose by 5°C.

(b) Consecutive changes: 6→10: +5; 10→2: +7; 2→6: −6-6; 6→10: −5-5. Largest drop: 6°C (2 pm → 6 pm).

(c) Mean = −7+(−2)+5+(−1)+(−6)5=−115=−2.2°C\frac{-7 + (-2) + 5 + (-1) + (-6)}{5} = \frac{-11}{5} = -2.2°C.
2Problem 2
Answer
(a) 180, −20-20, 25, −35-35, −50-50 (b) Yes — overdrawn twice (c) £10
Full working
(a) Start £100. After £80 deposit: £180. After £200 withdrawal: −£20-£20. After £45 deposit: £25. After £60 withdrawal: −£35-£35. After £15 fee: −£50-£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) −3-3 (d) −2-2
Full working
(a) −2+3−(−4)=−2+3+4=5-2 + 3 - (-4) = -2 + 3 + 4 = 5.

(b) (−2)(3)(−4)=−2×3=−6;−6×−4=24(-2)(3)(-4) = -2 \times 3 = -6; -6 \times -4 = 24.

(c) a2=4,b2=9,c2=16a^2 = 4, b^2 = 9, c^2 = 16. 4+9−16=−34 + 9 - 16 = -3.

(d) Numerator: ab−c=(−2)(3)−(−4)=−6+4=−2ab - c = (-2)(3) - (-4) = -6 + 4 = -2. Denominator: a+b=−2+3=1a + b = -2 + 3 = 1. Hmm so −2/1=−2-2/1 = -2 ✓.
4Problem 4
Answer
(a) 10x−710x - 7 (b) 8x+18x + 1 (c) −5x−0y=−5x-5x - 0y = -5x (d) 7a−5b7a - 5b
Full working
(a) 6x−15+4x+8=10x−76x - 15 + 4x + 8 = 10x - 7.

(b) −2x+6+10x−5=8x+1-2x + 6 + 10x - 5 = 8x + 1.

(c) −3x−2y−2x+2y=−5x+0y=−5x-3x - 2y - 2x + 2y = -5x + 0y = -5x.

(d) 8a−12b−2a+6b+a+b=(8−2+1)a+(−12+6+1)b=7a−5b8a - 12b - 2a + 6b + a + b = (8 - 2 + 1)a + (-12 + 6 + 1)b = 7a - 5b.
5Problem 5
Answer
(a) 3x−23x - 2; coeff 3 (b) 8x2−2x8x^2 - 2x; coeff −2-2 (c) −2a+3b+5-2a + 3b + 5
Full working
(a) xx terms: 5+2−4=3⇒3x5 + 2 - 4 = 3 \Rightarrow 3x. Constants: −3+1=−2-3 + 1 = -2. Result: 3x−23x - 2.

(b) x2x^2: 7+2−1=8⇒8x27 + 2 - 1 = 8 \Rightarrow 8x^2. xx: −3+1=−2⇒−2x-3 + 1 = -2 \Rightarrow -2x. Result: 8x2−2x8x^2 - 2x.

(c) aa: 1−3=−21 - 3 = -2. bb: 2+1=32 + 1 = 3. Constants: 5. Result: −2a+3b+5-2a + 3b + 5.
6Problem 6
Answer
(a) 13 (b) 4 (c) Undefined (0 in denominator) (d) −16-16 vs 1616
Full working
(a) (−2)2=4(-2)^2 = 4. 4×4=164 \times 4 = 16. −3+16=13-3 + 16 = 13.

(b) Brackets: (−3+4)=1(-3 + 4) = 1. (−2)2=4(-2)^2 = 4. 1×4=41 \times 4 = 4.

(c) Numerator: −12+12=0-12 + 12 = 0. Denominator: 4−4=04 - 4 = 0. **Undefined** (0/00/0 form).

(d) −24=−(24)=−16-2^4 = -(2^4) = -16 (the minus is **outside** the exponent). (−2)4=16(-2)^4 = 16 (the minus is inside, and the exponent is even).
7Problem 7
Answer
(a) 5a5a (b) a3ba^3 b (c) 2x22x^2 (d) 3a2b3 a^2 b (e) 4a24a^2
Full working
(a) Drop the × between a number and variable: 5a5a.

(b) Three factors of aa and one of bb: a3ba^3 b (no ×\times symbol needed).

(c) Two factors of xx: 2x22x^2.

(d) Two factors of aa, one of bb, coefficient 3: 3a2b3 a^2 b.

(e) (2a)2=22×a2=4a2(2a)^2 = 2^2 \times a^2 = 4a^2.
8Problem 8
Answer
(a) P=6x+4P = 6x + 4 (b) 34 cm (c) x=4x = 4
Full working
(a) P=2(2x+3)+2(x−1)=4x+6+2x−2=6x+4P = 2(2x + 3) + 2(x - 1) = 4x + 6 + 2x - 2 = 6x + 4.

(b) P(5)=30+4=34P(5) = 30 + 4 = 34 cm.

(c) 6x+4=28⇒6x=24⇒x=46x + 4 = 28 \Rightarrow 6x = 24 \Rightarrow x = 4.
9Problem 9
Answer
(a) −2x+3y−4-2x + 3y - 4 (b) −2x−6-2x - 6 (c) −8x−5-8x - 5 (d) 0x−6y0x - 6y (=−6y= -6y)
Full working
(a) Multiply each by −1-1: −2x+3y−4-2x + 3y - 4.

(b) 4−2x−10=−2x−64 - 2x - 10 = -2x - 6.

(c) −6x+3−2x−8=−8x−5-6x + 3 - 2x - 8 = -8x - 5.

(d) 5x−10y−6x+3y+x+y=(5−6+1)x+(−10+3+1)y=0x−6y=−6y5x - 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=0-4 -3 -2 -1 +0 +1 +2 +3 +4 = 0. There are 3 rows, each adds to the magic sum, so 3×magic sum=03 \times \text{magic sum} = 0, giving magic sum **= 0**.

(b) Sum of all 4 lines through the centre (2 diagonals + middle row + middle column) = 4×0=04 \times 0 = 0. Each off-centre cell appears in exactly one of these lines, contributing once; the centre appears in all four lines, contributing 4c4c where cc is the centre value. The 8 non-centre cells sum to 0−c=−c0 - c = -c (since total is 0). So the 4 lines contribute: 4c+(non-centre sum used once)=4c+((−c)−but each non-centre cell is in exactly one of these 4 lines)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,…,n}\{-n, \ldots, n\}, the centre must equal the mean = 0.
11Problem 11
Answer
(a) 12a312 a^3 (b) 3a23 a^2 (c) −10x4-10 x^4 (d) 4a2b44 a^2 b^4
Full working
(a) 3×4=123 \times 4 = 12; a2×a=a3a^2 \times a = a^3. Result: 12a312 a^3.

(b) 12/4=312/4 = 3; a3/a=a2a^3 / a = a^2. Result: 3a23 a^2.

(c) 5×−2=−105 \times -2 = -10; x×x3=x4x \times x^3 = x^4. Result: −10x4-10 x^4.

(d) (2ab2)2=22×a2×(b2)2=4a2b4(2 a b^2)^2 = 2^2 \times a^2 \times (b^2)^2 = 4 a^2 b^4.
12Problem 12
Answer
(a) −2-2 (basement 2) (b) +9 (after the third move) (c) −3-3 (after the fourth move)
Full working
(a) Sum: +5−3+7−12+1=−2+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: −3-3. After move 5: −2-2. **Max: +9**.

(c) Lowest: −3-3 (after move 4).