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Algebra

Algebra

Algebraic fractions

Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Simplifying
  4. Multiplying and dividing
  5. Adding and subtracting
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 2.3: algebraic fractions
  • 2.3Simplify algebraic fractions by factorising and cancelling common factors.
  • 2.3Multiply and divide algebraic fractions.
  • 2.3Add and subtract them, including algebraic denominators, as a single fraction.

Before you start: Fractions and Factorising. Next: Equations leading to quadratics.

Formula bank

CancellingLearn it\(\dfrac{a\times c}{b\times c}=\dfrac ab\)cancel a FACTOR (something multiplied), never a term (something added).Use it when simplify.
Multiply / divideLearn it\(\dfrac ab\times\dfrac cd=\dfrac{ac}{bd}\), \(\dfrac ab\div\dfrac cd=\dfrac ab\times\dfrac dc\)Use it when × or ÷ of fractions.
Add / subtractLearn it\(\dfrac ab+\dfrac cd=\dfrac{ad+bc}{bd}\)use the lowest common denominator if there is one.Use it when 'as a single fraction'.

Simplifying

\(\tfrac{12}{18}=\tfrac23\) because 6 is a factor of both. Algebra works the same way: factorise the top and the bottom, then cancel any factor (often a whole bracket) that is in both.

\(\dfrac{x+3}{x+5}\) is NOT \(\dfrac35\): \(x\) is added (a term), not multiplied (a factor). Test with \(x=1\): \(\tfrac46\ne\tfrac35\).

General methodHow to simplify an algebraic fraction

  1. Factorise the top fully.
  2. Factorise the bottom fully.
  3. Cancel brackets (or numbers, letters) that appear in both.
  4. Leave the answer factorised.

Example Simplify \(\dfrac{x^{2}+3x}{x^{2}-9}\).

Top: common factor. Bottom: difference of two squares.

  1. \(\dfrac{x(x+3)}{(x+3)(x-3)}\)
  2. cancel \((x+3)\)

Answer \(\dfrac x{x-3}\)

Try it yourself

1
Simplify \(\dfrac{6x+9}3\).
Show answer

Answer \(2x+3\)

2
Simplify \(\dfrac{4x^{2}}{6x}\).
Show answer

Answer \(\dfrac{2x}3\)

3
Simplify \(\dfrac{x^{2}-25}{x+5}\).
Show answer

Answer \(x-5\)

Multiplying and dividing

General methodHow to multiply or divide algebraic fractions

  1. Dividing: keep the first fraction, flip the second, change ÷ to ×.
  2. Factorise everything you can.
  3. Cancel factors across the tops and bottoms.
  4. Multiply the tops and multiply the bottoms.

Example Simplify \(\dfrac{5b}6\div\dfrac{10b}9\).

Flip the second fraction and multiply, cancelling first.

  1. \(\dfrac{5b}6\times\dfrac9{10b}\)
  2. \(=\dfrac{45b}{60b}\)

Answer \(\dfrac34\)

Try it yourself

1
Simplify \(\dfrac{5b}6\times\dfrac9{10b^{2}}\).
Show answer

Answer \(\dfrac3{4b}\)

Adding and subtracting

General methodHow to write as a single fraction

  1. Find a common denominator: the LCM of the numbers, or multiply the algebraic denominators together (unless they share a factor).
  2. Multiply each numerator by what its denominator was multiplied by - IN BRACKETS.
  3. Expand the top and collect like terms (careful with a minus in front of a bracket).
  4. Leave the bottom factorised. Check whether anything cancels.

Example Write \(\dfrac2{x+1}+\dfrac3{x-2}\) as a single fraction.

Common denominator \((x+1)(x-2)\).

  1. \(\dfrac{2(x-2)+3(x+1)}{(x+1)(x-2)}\)
  2. \(=\dfrac{2x-4+3x+3}{(x+1)(x-2)}\)

Answer \(\dfrac{5x-1}{(x+1)(x-2)}\)

Try it yourself

1
Write \(\dfrac x2+\dfrac x5\) as a single fraction.
Show answer

Answer \(\dfrac{7x}{10}\)

2
Write \(\dfrac x3+\dfrac{x-4}2\) as a single fraction.
Show answer

Answer \(\dfrac{5x-12}6\)

Ways they test it

Type 1 Simplify fully

"Simplify \(\dfrac{x^{2}+2x-15}{x^{2}-9}\)." (3 marks)

Spot it a fraction with quadratics: both factorise.

  1. Factorise top and bottom.
  2. Cancel the common bracket.

For example \(\dfrac{(x+5)(x-3)}{(x+3)(x-3)}\)

    Answer \(\dfrac{x+5}{x+3}\)

    Type 2 Single fraction, number denominators

    "Write \(\dfrac{x+1}3-\dfrac{x-2}4\) as a single fraction." (2 marks)

    Spot it two fractions, numbers underneath.

    1. Denominator 12.
    2. Numerators in brackets: \(4(x+1)-3(x-2)\).

    For example \(\dfrac{4x+4-3x+6}{12}\)

      Answer \(\dfrac{x+10}{12}\)

      Type 3 Single fraction, algebraic denominators

      "Write \(\dfrac3{x+4}+\dfrac5{x-2}\) as a single fraction." (3 marks)

      Spot it x in the denominators.

      1. Multiply the denominators.
      2. Cross-multiply the tops, expand, collect.

      For example \(\dfrac{3(x-2)+5(x+4)}{(x+4)(x-2)}\)

        Answer \(\dfrac{8x+14}{(x+4)(x-2)}\)

        Type 4 Shared factor in the denominators

        "Simplify \(\dfrac3{x^{2}-9}-\dfrac1{x-3}\)." (3 marks)

        Spot it one denominator factorises and contains the other.

        1. Factorise: \(x^{2}-9=(x+3)(x-3)\).
        2. The LCD is just \((x+3)(x-3)\).

        For example \(\dfrac{3-(x+3)}{(x+3)(x-3)}\)

          Answer \(\dfrac{-x}{(x+3)(x-3)}\)

          Mistakes to avoid

          • ✗ \(\dfrac{x+3}{x+5}=\dfrac35\)✓ it doesn't simplifyCancel factors, never terms.
          • ✗ \(\dfrac{6x+9}3=6x+3\)✓ \(2x+3\)The 3 must divide BOTH terms.
          • ✗ \(\dfrac x2+\dfrac x5=\dfrac{2x}7\)✓ \(\dfrac{7x}{10}\)Never add the bottoms.
          • ✗ \(\ldots-3(x-2)=\ldots-3x-6\)✓ \(-3x+6\)A minus in front of a bracket changes both signs.

          Summary

          • Factorise top and bottom, then cancel common FACTORS.
          • Dividing: flip the second fraction and multiply.
          • Adding/subtracting: common denominator, numerators in brackets.
          • Leave denominators factorised.

          If you see …

          • "simplify" a fraction with quadraticsfactorise both, cancel the matching bracket.
          • "as a single fraction"common denominator, then expand the top.

          Practice

          Practice: easy, medium, hard

          Easy

          1
          Simplify \(\dfrac{x^{2}+3x}{x}\).
          Show answer

          Answer \(x+3\)

          2
          Write \(\dfrac x4+\dfrac x6\) as a single fraction.
          Hint
          Denominator 12.
          Show answer

          Answer \(\dfrac{5x}{12}\)

          3
          Simplify \(\dfrac{3x}{8}\times\dfrac{4}{9x}\).
          Hint
          Cancel x, then 3 with 9 and 4 with 8.
          Show answer

          Answer \(\dfrac16\)

          Medium

          1
          Simplify \(\dfrac{x^{2}+2x-15}{x^{2}-9}\).
          Guide me

          What is it asking? a simpler fraction

          What do we know? both quadratics factorise

          Plan cancel \((x-3)\)

          Show working
          1. \(\dfrac{(x+5)(x-3)}{(x+3)(x-3)}\)

          Answer \(\dfrac{x+5}{x+3}\)

          2
          Write \(\dfrac{x+1}3-\dfrac{x-2}4\) as a single fraction.
          Guide me

          What is it asking? one fraction

          What do we know? denominator 12

          Plan \(4(x+1)-3(x-2)\)

          Show working
          1. \(\dfrac{4x+4-3x+6}{12}\)

          Answer \(\dfrac{x+10}{12}\)

          3
          Write \(\dfrac3{x+4}+\dfrac5{x-2}\) as a single fraction.
          Guide me

          What is it asking? one fraction

          What do we know? denominator \((x+4)(x-2)\)

          Plan cross-multiply the tops

          Show working
          1. \(\dfrac{3(x-2)+5(x+4)}{(x+4)(x-2)}\)
          2. \(=\dfrac{8x+14}{(x+4)(x-2)}\)

          Answer \(\dfrac{8x+14}{(x+4)(x-2)}\)

          Hard

          1
          Simplify \(\dfrac{2x^{2}-8}{x^{2}+4x+4}\).
          Guide me

          What is it asking? a simpler fraction

          What do we know? top \(2(x+2)(x-2)\), bottom \((x+2)^{2}\)

          Plan cancel one \((x+2)\)

          Show working
          1. \(\dfrac{2(x+2)(x-2)}{(x+2)(x+2)}\)

          Answer \(\dfrac{2(x-2)}{x+2}\)

          PartAnswer / workingMarksGuidance
          \(2(x+2)(x-2)\)B1
          \((x+2)^{2}\)B1
          \(\dfrac{2(x-2)}{x+2}\)B1
          Total: 3 marks
          2
          Simplify \(\dfrac{x^{2}-x-12}{3x^{2}-11x-4}\).
          Guide me

          What is it asking? a simpler fraction

          What do we know? both factorise

          Plan common bracket \((x-4)\)

          Show working
          1. \(\dfrac{(x-4)(x+3)}{(3x+1)(x-4)}\)

          Answer \(\dfrac{x+3}{3x+1}\)

          PartAnswer / workingMarksGuidance
          \((x-4)(x+3)\)B1
          \((3x+1)(x-4)\)B1
          \(\dfrac{x+3}{3x+1}\)B1
          Total: 3 marks
          3
          Write \(\dfrac4{x-1}-\dfrac3{x+2}\) as a single fraction.
          Guide me

          What is it asking? one fraction

          What do we know? denominator \((x-1)(x+2)\)

          Plan careful: − 3(x − 1)

          Show working
          1. \(\dfrac{4(x+2)-3(x-1)}{(x-1)(x+2)}\)
          2. \(=\dfrac{4x+8-3x+3}{(x-1)(x+2)}\)

          Answer \(\dfrac{x+11}{(x-1)(x+2)}\)

          PartAnswer / workingMarksGuidance
          common denominatorM1
          \(4(x+2)-3(x-1)\)M1
          \(\dfrac{x+11}{(x-1)(x+2)}\)A1
          Total: 3 marks

          Can I do this?

          Anything unticked: revise that section first. All ticked: take the quiz.

          Quiz

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