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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 11 · 11.11 Algebraic Fractions

Solutions · Full Answer Key

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

Pack A — Answers

Bronze
1.23\dfrac{2}{3}
2.2x+32x + 3
3.8x\dfrac{8}{x}
4.43\dfrac{4}{3}
5.x=20x = 20
6.2
7.xx
8.x=5x = 5
9.2
10.x=0x = 0 and x=3x = 3
Silver
11.10+3x5x\dfrac{10 + 3x}{5x}
12.x+3x(x+1)\dfrac{x + 3}{x(x + 1)}
13.43x\dfrac{4}{3x}
14.x=12x = 12
15.x=2x = 2
16.x+3x\dfrac{x + 3}{x}
17.3x+1x(x+1)\dfrac{3x + 1}{x(x + 1)}
18.x−2x−5\dfrac{x - 2}{x - 5}; x≠−3x \neq -3 (and x≠5x \neq 5 for the simplified expression)
19.x=7x = 7
20.x+7(x−2)(x+1)\dfrac{x + 7}{(x - 2)(x + 1)}
Gold
21.2x(x+1)\dfrac{2}{x(x + 1)}
22.x+2x−2\dfrac{x + 2}{x - 2}
23.x+1x−1\dfrac{x + 1}{x - 1}
24.x=−12x = -\dfrac{1}{2} or x=2x = 2
25.k=5k = 5
26.x+4x + 4
27.x=−13x = -13
28.1+1x1 + \dfrac{1}{\sqrt{x}}
29.x=7x = 7
30.x−3x+3\dfrac{x - 3}{x + 3}; x≠−3x \neq -3
Platinum
31.x=2±7x = 2 \pm \sqrt{7}
32.A=2A = 2, B=1B = 1
33.1x\dfrac{1}{x}
34.Identity (true for all x≠±1x \neq \pm 1)
35.x=5x = 5
36.x=3x = 3
37.x=2x = 2, y=3y = 3
38.3x+5x2−1\dfrac{3x + 5}{x^2 - 1}
39.A=−1A = -1, B=2B = 2, C=2C = 2
40.v≈54.5v \approx 54.5 km/h

Pack B — Answers

Bronze
1.23\dfrac{2}{3}
2.2x+32x + 3
3.9x\dfrac{9}{x}
4.52\dfrac{5}{2}
5.x=21x = 21
6.2
7.xx
8.x=32x = \dfrac{3}{2}
9.13\dfrac{1}{3}
10.x=−4x = -4 and x=2x = 2
Silver
11.8+3x4x\dfrac{8 + 3x}{4x}
12.x+4x(x+1)\dfrac{x + 4}{x(x + 1)}
13.14x\dfrac{1}{4x}
14.x=18x = 18
15.x=32x = \dfrac{3}{2}
16.x+4x\dfrac{x + 4}{x}
17.2xx2−1\dfrac{2x}{x^2 - 1}
18.x+1x+6\dfrac{x + 1}{x + 6}; x≠4x \neq 4 and x≠−6x \neq -6
19.x=−7x = -7
20.x−13(x+3)(x−1)\dfrac{x - 13}{(x + 3)(x - 1)}
Gold
21.2x+1\dfrac{2}{x + 1}
22.x−2x+2\dfrac{x - 2}{x + 2}
23.x+2x−2\dfrac{x + 2}{x - 2}
24.x=2±6x = 2 \pm \sqrt{6}
25.k=2k = 2
26.x−6x - 6
27.x=4x = 4
28.x−2\sqrt{x} - 2
29.x=−132x = -\dfrac{13}{2}
30.x+4x+2\dfrac{x + 4}{x + 2}; x≠±2x \neq \pm 2
Platinum
31.x=1±4x = 1 \pm \sqrt{4} — i.e. x=−1x = -1 or x=3x = 3; but x=3x = 3 makes a denominator zero, so reject. x=−1x = -1.
32.A=34A = \dfrac{3}{4}, B=54B = \dfrac{5}{4}
33.2−x2+x\dfrac{2 - x}{2 + x}
34.Solve: (x−2)+1=(x+2)⇒−1=0(x - 2) + 1 = (x + 2) \Rightarrow -1 = 0, false. No solution.
35.x=−1x = -1
36.x=5x = 5
37.x=3x = 3, y=4y = 4
38.2x+12x2−4\dfrac{2x + 12}{x^2 - 4}
39.A=−1A = -1, B=1B = 1, C=1C = 1
40.v≈50.0v \approx 50.0 km/h

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) x(x+1)x(x + 1). (b) 5x+3x(x+1)\dfrac{5x + 3}{x(x + 1)}. (c) x≠0x \neq 0 and x≠−1x \neq -1.
Full working
(b) 3(x+1)x(x+1)+2xx(x+1)=5x+3x(x+1)\frac{3(x + 1)}{x(x + 1)} + \frac{2x}{x(x + 1)} = \frac{5x + 3}{x(x + 1)}.
2Problem 2
Answer
(a) x+11(x−1)(x+2)\dfrac{x + 11}{(x - 1)(x + 2)}. (b) x≠1x \neq 1, x≠−2x \neq -2.
Full working
(a) LCD: (x−1)(x+2)(x - 1)(x + 2). Num: 4(x+2)−3(x−1)=4x+8−3x+3=x+114(x + 2) - 3(x - 1) = 4x + 8 - 3x + 3 = x + 11.
3Problem 3
Answer
(a) 2. (b) xx. (c) 1 (with restrictions x≠±2x \neq \pm 2).
Full working
(a) Cancel (x+1)(x + 1) and xx. (b) x2x+1⋅x+1x=x\frac{x^2}{x + 1} \cdot \frac{x + 1}{x} = x. (c) Factor numerator: (x−2)(x+2)(x−2)(x+2)=1\frac{(x - 2)(x + 2)}{(x - 2)(x + 2)} = 1.
4Problem 4
Answer
x+3x+2\dfrac{x + 3}{x + 2}, with x≠3x \neq 3 and x≠−2x \neq -2.
Full working
Numerator: (x−3)(x+3)(x - 3)(x + 3). Denominator: (x−3)(x+2)(x - 3)(x + 2). Cancel (x−3)(x - 3).
5Problem 5
Answer
x=2+7x = 2 + \sqrt{7} or x=2−7x = 2 - \sqrt{7}.
Full working
(a) 2(x+2)+3(x−1)=(x−1)(x+2)2(x + 2) + 3(x - 1) = (x - 1)(x + 2). (b) 5x+1=x2+x−2⇒x2−4x−3=0⇒x=2±75x + 1 = x^2 + x - 2 \Rightarrow x^2 - 4x - 3 = 0 \Rightarrow x = 2 \pm \sqrt{7}. (c) Both ≠ 1, −2-2, so both valid.
6Problem 6
Answer
11.
Full working
Numerator: (x+1)−xx(x+1)=1x(x+1)\frac{(x + 1) - x}{x(x + 1)} = \frac{1}{x(x + 1)}. So whole expression =1/(x(x+1))1/(x(x+1))=1= \frac{1/(x(x + 1))}{1/(x(x + 1))} = 1.
7Problem 7
Answer
A=32A = \dfrac{3}{2}, B=72B = \dfrac{7}{2}.
Full working
5x+1=A(x+3)+B(x−1)5x + 1 = A(x + 3) + B(x - 1). x=1x = 1: 6=4A⇒A=3/26 = 4A \Rightarrow A = 3/2. x=−3x = -3: −14=−4B⇒B=7/2-14 = -4B \Rightarrow B = 7/2.
8Problem 8
Answer
(a) 60v+40v−10=2\frac{60}{v} + \frac{40}{v - 10} = 2. (b) v2−60v+300=0v^2 - 60v + 300 = 0. (c) v≈54.5v \approx 54.5 km/h.
Full working
(a) Time = distance/speed. (b) Multiply by v(v−10)v(v - 10): 60(v−10)+40v=2v(v−10)⇒v2−60v+300=060(v - 10) + 40v = 2v(v - 10) \Rightarrow v^2 - 60v + 300 = 0. (c) v=30±106v = 30 \pm 10\sqrt{6}; take positive (and > 10): v≈54.5v \approx 54.5.
9Problem 9
Answer
(a) A: 18\frac{1}{8}; B: 112\frac{1}{12}. (b) 18+112=1T\frac{1}{8} + \frac{1}{12} = \frac{1}{T}. (c) T=4.8T = 4.8 h.
Full working
(a) Per-hour fractions. (b) Combined rate. (c) 1T=524\frac{1}{T} = \frac{5}{24}, T=4.8T = 4.8.
10Problem 10
Answer
A=3A = 3, k=−1k = -1.
Full working
Combine RHS: A(x−2)+kx−2=Ax+(k−2A)x−2\frac{A(x - 2) + k}{x - 2} = \frac{Ax + (k - 2A)}{x - 2}. Compare numerators: A=3A = 3, k−2A=−7⇒k=−1k - 2A = -7 \Rightarrow k = -1.
11Problem 11
Answer
(a) x−1x+1\dfrac{x - 1}{x + 1}; x≠−1x \neq -1. (b) x−2x+2\dfrac{x - 2}{x + 2}; x≠±2x \neq \pm 2. (c) xx; x≠±1x \neq \pm 1.
Full working
(a) (x−1)(x+1)/(x+1)2(x - 1)(x + 1) / (x + 1)^2. (b) (x−2)2/[(x−2)(x+2)](x - 2)^2 / [(x - 2)(x + 2)]. (c) x(x−1)(x+1)/[(x−1)(x+1)]x(x - 1)(x + 1) / [(x - 1)(x + 1)].
12Problem 12
Answer
(a) See working. (b) n≥20n \geq 20. (c) Approaches a=0.40a = 0.40 from above.
Full working
(a) na+bn=a+bn\frac{na + b}{n} = a + \frac{b}{n}. (b) 0.40+6n≤0.70⇒6n≤0.30⇒n≥200.40 + \frac{6}{n} \leq 0.70 \Rightarrow \frac{6}{n} \leq 0.30 \Rightarrow n \geq 20. (c) As n→∞n \to \infty, bn→0\frac{b}{n} \to 0, so average → aa.