Skip to content
Cambridge IGCSE0607

Algebraic fractions

International Mathematics 0607 Chapter Notes

What this chapter covers

Algebraic fractions
ShareWhatsAppPost
Algebraic fractions notes

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Algebraic fractions notes as text: skim, search, and jump between subtopics.

~5 min read

In this chapter

  1. 1Simplifying by cancelling a common factor
  2. 2Multiplying and dividing
  3. 3Adding and subtracting
  4. 4Factorise first, then cancel
  5. 5Difference of two squares and quadratics
  6. 6Factorising by grouping

1. Simplifying by cancelling a common factor

To simplify an algebraic fraction, divide the top and the bottom by the same common factor. Cancel numbers first, then powers of x.

One step is enough for the Core examples: x²/x = x and 3/6x = 1/2x.

When everything on the top cancels, a 1 is left there. Do not write 0.

x²/x = x

3/6x = 1/2x

Examiner insight

“Some candidates did not cancel fully but gained one mark for partial cancellation.”

Worked example 12 marks

Simplify 12x²/8x.

  1. 1

    divide top and bottom by the common factor 4x [1]

  2. 2

    12x²/8x = 3x/2 [1]

2. Multiplying and dividing

To multiply, multiply the tops and multiply the bottoms. Cancel any common factor, before or after.

To divide, turn the second fraction upside down and multiply. Then cancel fully.

3a/4 × 9a/10 = 27a²/40

3a/4 ÷ 9a/10 = 3a/4 × 10/9a = 30a/36a = 5/6

Examiner insight

“Some candidates managed to gain a mark by inverting the second fraction and multiplying but then did not know how to continue.”

Worked example 11 mark

Simplify. 5x/y × 3/p

  1. 1

    multiply the tops and multiply the bottoms: (5x × 3)/(y × p)

  2. 2

    = 15x/yp [1]

3. Adding and subtracting

Write each fraction over the same common denominator, combine the numerators, and give a single fraction.

Use the simplest common denominator. For x/3 + x/2 - 4 it is 6, so the answer is (5x - 24)/6.

For 1/(x - 2) + 1/(x - 3) the common denominator is (x - 2)(x - 3). Put brackets round a numerator that follows a minus sign.

1/(x - 2) + 1/(x - 3) = (2x - 5)/((x - 2)(x - 3))

Examiner insight

“There were some very good attempts at a common denominator but errors were seen in dealing with the subtraction of terms in the numerator.” “Not all candidates wrote the answer in its simplest form.”

Worked example 13 marks

Write as a single fraction in its simplest form. 2/(x - 3) - 5x/(x + 1)

  1. 1

    common denominator (x - 3)(x + 1) [1]

  2. 2

    (2(x + 1) - 5x(x - 3))/((x - 3)(x + 1)) [1]

  3. 3

    = (-5x² + 17x + 2)/((x - 3)(x + 1)) [1]

4. Factorise first, then cancel

A factor can be cancelled only if it multiplies the whole top and the whole bottom. So factorise the numerator and the denominator first.

A common factor comes out as a bracket: 10x - 2x² = 2x(5 - x). Take out -2x instead and the bracket becomes (x - 5), which can match a bracket below.

5 - x = -(x - 5)

Examiner insight

“To complete the question, candidates had to amend one of their signs before cancelling.” “Candidates who factorised the numerator using 2x were unable to cancel a factor, and if a factor of –2x was used then cancelling became straightforward.”

Worked example 13 marks

Simplify. (10x - 2x²)/(x² - 25)

  1. 1

    numerator: 10x - 2x² = -2x(x - 5) [1]

  2. 2

    denominator: x² - 25 = (x - 5)(x + 5) [1]

  3. 3

    cancel (x - 5): (-2x)/(x + 5) [1]

5. Difference of two squares and quadratics

A difference of two squares factorises as a² - b² = (a - b)(a + b). A quadratic such as 2x² + 5x - 12 factorises into two brackets.

Factorise top and bottom, cancel the matching bracket, and write what is left. Check the final answer has no common factor.

a² - b² = (a - b)(a + b)

Examiner insight

“Of those who did not reach the final answer, the most common error was in not being able to factorise the denominator correctly, if at all.”

Worked example 14 marks

Simplify. (4x² - 9)/(2x² + 5x - 12)

  1. 1

    numerator: 4x² - 9 = (2x - 3)(2x + 3) [1]

  2. 2

    denominator: 2x² + 5x - 12 = (2x - 3)(x + 4) [2]

  3. 3

    cancel (2x - 3): (2x + 3)/(x + 4) [1]

6. Factorising by grouping

When there are four terms, group them in pairs and take out a common factor from each pair.

If the two pairs then share a bracket, the bracket is a common factor of the whole expression. Take care with signs: w - 1 = -(1 - w).

Examiner insight

“There were also candidates who gave the incorrect sign after factorising the negative term from the expression.”

Worked example 14 marks

Simplify. (1 - w²)/(1 + wx - w - x)

  1. 1

    numerator: 1 - w² = (1 - w)(1 +w) [1]

  2. 2

    group the denominator: (1 -w) + x(w - 1) = (1 -w) - x(1 -w) [1]

  3. 3

    denominator = (1 - w)(1 -x) [1]

  4. 4

    cancel (1 - w): (1 + w)/(1 -x) [1]

Recap

  • Cancel only a factor of the whole numerator and the whole denominator.
  • Divide by inverting the second fraction and multiplying.
  • Add or subtract over the simplest common denominator, then simplify the single fraction.
  • Factorise top and bottom first, then cancel.
  • a² - b² = (a - b)(a + b). Four terms: group in pairs.
  • 5 - x = -(x - 5): watch the sign before cancelling.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Simplify 10x⁴/2x.2 marks
  2. Do not use a calculator. Simplify 6/9x.2 marks
  3. Do not use a calculator. Simplify (4x + 8)/4.2 marks
  4. Do not use a calculator. Simplify (x² - 3x)/(x² - 9).3 marks
  5. Do not use a calculator. Simplify (x² - 25)/(x² + x - 20).3 marks
  6. Simplify (6 - 2x)/(x² - 9).3 marks
  7. Simplify 5x/6 ÷ 15x²/8.3 marks
  8. Write 2x/3 - (3(x - 5))/2 as a single fraction in its simplest form.3 marks
  9. Do not use a calculator. A boat travels 12 km upstream at x km / h and then 12 km back downstream at (x + 3) km / h. The time for each journey, in hours, is the distance divided by the speed. (a) Write an expression for how many hours longer the upstream journey takes than the downstream journey. Give your answer as a single fraction in its simplest form. (b) Divide the upstream time by the downstream time. Give your answer in its simplest form.5 marks
  10. P = (x² - 4)/(x² - 5x + 6) and Q = (x² + 2x)/(9 - 3x). (a) Show that P ÷ Q = -3/x. (b) Use part (a) to write P ÷ Q + 3/(x - 1) as a single fraction in its simplest form.6 marks

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Types of numberSetsPowers and rootsFractions, decimals and percentagesOrderingThe four operations