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Ecolint Campus des NationsMathematics
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Pack A — Fluency

Ecolint Campus des NationsMathematics
Year 11 · 11.11 Algebraic Fractions

Pack A · Fluency

Name: _________________________________
Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10
  1. 1.
    Simplify axbx\dfrac{ {a}x}{ {b}x} (assume x≠0x \neq 0).
     
  2. 2.
    Simplify ax+9c\dfrac{ {a}x + 9}{ {c}}.
     
  3. 3.
    Simplify ax+bx\dfrac{ {a}}{x} + \dfrac{ {b}}{x}.
     
  4. 4.
    Simplify ax⋅xb\dfrac{ {a}}{x} \cdot \dfrac{x}{ {b}}.
     
  5. 5.
    Solve xa=5\dfrac{x}{ {a}} = 5.
     
  6. 6.
    Simplify ax÷bx\dfrac{ {a}}{x} \div \dfrac{ {b}}{x}.
     
  7. 7.
    Simplify x2x\dfrac{x^2}{x} (assume x≠0x \neq 0).
     
  8. 8.
    State the value of xx for which 1x−5\dfrac{1}{x - 5} is undefined.
     
  9. 9.
    Evaluate x+1x−1\dfrac{x + 1}{x - 1} at x=3x = 3.
     
  10. 10.
    State the values of xx for which x+2x(x−3)\dfrac{x + 2}{x(x - 3)} is undefined.
     
SilverQuestions 11–20
  1. 11.
    Simplify 2x+3a\dfrac{2}{x} + \dfrac{3}{ {a}} as a single fraction.
     
  2. 12.
    Simplify ax−bx+1\dfrac{ {a}}{x} - \dfrac{ {b}}{x + 1} as a single fraction.
     
  3. 13.
    Simplify axb⋅cdx2\dfrac{ {a}x}{ {b}} \cdot \dfrac{ {c}}{ {d}x^2}.
     
  4. 14.
    Solve x2+x3=10\dfrac{x}{2} + \dfrac{x}{3} = 10.
     
  5. 15.
    Solve xx+1=ab\dfrac{x}{x + 1} = \dfrac{ {a}}{ {b}}.
     
  6. 16.
    Simplify x2−a2x2−ax\dfrac{x^2 - {a}^2}{x^2 - {a}x}.
     
  7. 17.
    Simplify 1x+2x+1\dfrac{1}{x} + \dfrac{2}{x + 1} as a single fraction.
     
  8. 18.
    Simplify and state restrictions: (x−2)(x+3)(x+3)(x−5)\dfrac{(x - 2)(x + 3)}{(x + 3)(x - 5)}.
     
  9. 19.
    Solve 2x−1=3x+2\dfrac{2}{x - 1} = \dfrac{3}{x + 2}.
     
  10. 20.
    Simplify 3x−2−2x+1\dfrac{3}{x - 2} - \dfrac{2}{x + 1} as a single fraction.
     
GoldQuestions 21–30
  1. 21.
    Simplify 1x−1x+1+1x(x+1)\dfrac{1}{x} - \dfrac{1}{x + 1} + \dfrac{1}{x(x + 1)}.
     
  2. 22.
    Simplify x2+5x+6x2+x−6\dfrac{x^2 + 5x + 6}{x^2 + x - 6}.
     
  3. 23.
    Simplify 1x+11−1x\dfrac{\dfrac{1}{x} + 1}{1 - \dfrac{1}{x}}.
     
  4. 24.
    Solve 2x+3x+1=2\dfrac{2}{x} + \dfrac{3}{x + 1} = 2.
     
  5. 25.
    Find kk such that 2x+kx−3=2+11x−3\dfrac{2x + k}{x - 3} = 2 + \dfrac{11}{x - 3}.
     
  6. 26.
    Simplify (x+2)2−4x\dfrac{(x + 2)^2 - 4}{x}.
     
  7. 27.
    Solve x−23−x+12=1\dfrac{x - 2}{3} - \dfrac{x + 1}{2} = 1.
     
  8. 28.
    Simplify x(x+1)x\dfrac{\sqrt{x}(\sqrt{x} + 1)}{x}.
     
  9. 29.
    Solve x+3x−2=2\dfrac{x + 3}{x - 2} = 2.
     
  10. 30.
    Simplify x2−9x2+6x+9\dfrac{x^2 - 9}{x^2 + 6x + 9} and state restrictions on xx.
     
PlatinumQuestions 31–40
  1. 31.
    Solve 2x−1+3x+2=1\dfrac{2}{x - 1} + \dfrac{3}{x + 2} = 1 and verify each solution against the original equation.
     
  2. 32.
    Find constants AA and BB such that 3x+5(x+1)(x+2)=Ax+1+Bx+2\dfrac{3x + 5}{(x + 1)(x + 2)} = \dfrac{A}{x + 1} + \dfrac{B}{x + 2}.
     
  3. 33.
    Simplify xx−1−11x−1+1\dfrac{\dfrac{x}{x - 1} - 1}{\dfrac{1}{x - 1} + 1}.
     
  4. 34.
    Solve 1x−1−2x2−1=1x+1\dfrac{1}{x - 1} - \dfrac{2}{x^2 - 1} = \dfrac{1}{x + 1} and state restrictions.
     
  5. 35.
    Solve x2−4x−2+x2−9x−3=14\dfrac{x^2 - 4}{x - 2} + \dfrac{x^2 - 9}{x - 3} = 14.
     
  6. 36.
    Pipe A fills a tank in xx hours; pipe B fills it in x+3x + 3 hours. Together they fill it in 2 hours. Find xx.
     
  7. 37.
    Solve   1x+1y=56,  1x−1y=16  \;\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{5}{6}, \; \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{1}{6}\; for x,yx, y.
     
  8. 38.
    Simplify 2x−1−3x+1+4xx2−1\dfrac{2}{x - 1} - \dfrac{3}{x + 1} + \dfrac{4x}{x^2 - 1}.
     
  9. 39.
    Find AA, BB, CC such that 3x2+1x(x−1)(x+1)=Ax+Bx−1+Cx+1\dfrac{3x^2 + 1}{x(x - 1)(x + 1)} = \dfrac{A}{x} + \dfrac{B}{x - 1} + \dfrac{C}{x + 1}.
     
  10. 40.
    A car drives the first 60 km at vv km/h then the next 40 km at v−10v - 10 km/h. The total time is 2 hours. Write the equation and find vv to 3 s.f.