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Algebraic Fractions

discia.org 10 questions · 32 marks
Name Date
  1. Either paper 2 marks

    A bottling plant fills \(m\) crates of bottles per minute. Each crate holds 40 bottles.

    Show that the plant fills \(\frac{ 2m }{ 3 }\) bottles per second.

    Show answer
    Per minute: \(40m\) bottles. There are 60 seconds in one minute, so per second it is \(\frac{ 40m }{ 60 }\), and dividing top and bottom by 20 gives \(\frac{ 2m }{ 3 }\).
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  2. Either paper 3 marks

    Reuben is asked to simplify \(\frac{ 6n^2 - 5n }{ n }\).

    Reuben's method is to divide the second term of the numerator by \(n\), then add the first term.

    (a)

    State the expression Reuben gets.

    (b)

    Simplify \(\frac{ 6n^2 - 5n }{ n }\) correctly.

    (c)

    Explain why Reuben's answer is wrong.

    Show answer
    (a) \(6n^2 - 5\)
    (b) \(6n - 5\)
    (c) Reuben divided only the second term by \(n\). The whole of the numerator is divided by \(n\), so the other term must be divided as well: \(\frac{ 6n^2 }{ n } - \frac{ 5n }{ n } = 6n - 5\), not \(6n^2 - 5\).
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  3. Either paper Higher1 mark

    Circle the expression that is equivalent to

    \(\frac{ 15x^2 }{ 10x^2 + 15 }\)

    A  \(\frac{ 3x }{ 2x + 3 }\)

    B  \(\frac{ 3x^2 }{ 10x^2 + 3 }\)

    C  \(\frac{ 3x^2 }{ 2x^2 + 3 }\)

    D  \(\frac{ 3x^2 }{ 2x^2 + 15 }\)

    Show answer
    option C: \(\frac{ 3x^2 }{ 2x^2 + 3 }\)
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  4. Either paper Higher6 marks
    (a)

    Write \(\frac{ 5x + 1 }{ 4 } + \frac{ 3x + 4 }{ 6 }\) as a single fraction.

    Give your answer in its simplest form.

    (b)

    Write \(\frac{ 5x - 1 }{ 4 } - \frac{ 4x + 1 }{ 7 }\) as a single fraction.

    Give your answer in its simplest form.

    Show answer
    (a) \(\frac{ 21x + 11 }{ 12 }\)
    (b) \(\frac{ 19x - 11 }{ 28 }\)
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  5. Either paper Higher2 marks

    Write \(\frac{ 6 }{ x^3 } - \frac{ 1 }{ 2x }\) as a single fraction.

    Give your answer in its simplest form.

    Show answer
    \(\frac{ 12 - x^2 }{ 2x^3 }\)
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  6. Either paper Higher3 marks

    Simplify fully \(\frac{ 2(3x + 1)^{ 3 } }{ (3x + 1) }\)

    Your answer must be in the form \(ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are integers.

    Show answer
    \(18x^2 + 12x + 2\)
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  7. Either paper Higher3 marks

    \(a\) and \(b\) are positive numbers.

    Show that \(\frac{ 21a + 6b - 3a }{ 14a + 9b + a - 4b }\) always simplifies to the same value.

    Show answer
    Top \(18a + 6b = 6(3a + b)\), bottom \(15a + 5b = 5(3a + b)\); the bracket cancels, leaving \(\frac{ 6 }{ 5 }\) whatever \(a\) and \(b\) are.
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  8. Either paper Higher3 marks

    Simplify fully \(\frac{x^2 - 4x + 3 }{x^2 - 3x + 2 }\)

    Show answer
    \(\frac{x - 3 }{x - 2 }\)
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  9. Either paper Higher3 marks

    Simplify \(\frac{x^2 - 16 }{x^2 - 2x - 24 }\) as far as possible.

    Show answer
    \(\frac{x - 4 }{x - 6 }\)
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  10. Either paper Higher6 marks
    (a)

    Write \(\frac{ 3 }{x - 2 } + \frac{ 7 }{x + 1 }\) as a single fraction.

    Give your answer in its simplest form.

    (b)

    Write \(\frac{ 9 }{x - 1 } - \frac{ 8 }{x - 3 }\) as a single fraction.

    Give your answer in its simplest form.

    Show answer
    (a) \(\frac{ 10x - 11 }{(x - 2)(x + 1)}\)
    (b) \(\frac{ x - 19 }{(x - 1)(x - 3)}\)
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discia.org Algebraic Fractions · sheet 1
Answers Algebraic Fractions · sheet 1
  1. Per minute: \(40m\) bottles. There are 60 seconds in one minute, so per second it is \(\frac{ 40m }{ 60 }\), and dividing top and bottom by 20 gives \(\frac{ 2m }{ 3 }\).
  2. (a) \(6n^2 - 5\)
    (b) \(6n - 5\)
    (c) Reuben divided only the second term by \(n\). The whole of the numerator is divided by \(n\), so the other term must be divided as well: \(\frac{ 6n^2 }{ n } - \frac{ 5n }{ n } = 6n - 5\), not \(6n^2 - 5\).
  3. option C: \(\frac{ 3x^2 }{ 2x^2 + 3 }\)
  4. (a) \(\frac{ 21x + 11 }{ 12 }\)
    (b) \(\frac{ 19x - 11 }{ 28 }\)
  5. \(\frac{ 12 - x^2 }{ 2x^3 }\)
  6. \(18x^2 + 12x + 2\)
  7. Top \(18a + 6b = 6(3a + b)\), bottom \(15a + 5b = 5(3a + b)\); the bracket cancels, leaving \(\frac{ 6 }{ 5 }\) whatever \(a\) and \(b\) are.
  8. \(\frac{x - 3 }{x - 2 }\)
  9. \(\frac{x - 4 }{x - 6 }\)
  10. (a) \(\frac{ 10x - 11 }{(x - 2)(x + 1)}\)
    (b) \(\frac{ x - 19 }{(x - 1)(x - 3)}\)
discia.org Algebraic Fractions · sheet 1