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Maths · Further Algebra

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Algebraic Fractions and Proof

Simplifying algebraic fractions, solving equations with fractions, and proving algebraic statements.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Simplify algebraic fractions by factorising and cancelling common factors (Higher tier).
  2. 2Add, subtract, multiply and divide algebraic fractions (Higher tier).
  3. 3Solve equations that contain fractions with the unknown on the top.
  4. 4Prove algebraic statements, such as that the sum of two consecutive numbers is odd, and disprove others with a counter-example.

Fractions with letters, and showing why

Algebraic fractions follow exactly the same rules as number fractions, with brackets taking the place of numbers: cancel only what multiplies the whole top and the whole bottom, and use a common denominator to add. Proof is about showing that something is always true, using algebra instead of examples. Both topics use skills you already have, factorising and expanding, but they need tidy, careful working, and they are favourites for the last few marks of a Higher paper.

Simplifying with quadratics

(Higher tier) Simplify \(\dfrac{x^2 + 5x + 6}{x^2 + 3x + 2}\).

Show the solutionHide the solution
  1. 1 Factorise the top \(x^2 + 5x + 6 = (x + 2)(x + 3)\).
  2. 2 Factorise the bottom \(x^2 + 3x + 2 = (x + 1)(x + 2)\).
  3. 3 Cancel the common bracket \(\dfrac{(x + 2)(x + 3)}{(x + 1)(x + 2)} = \dfrac{x + 3}{x + 1}\).
  4. 4 Check Put \(x = 1\) in: \(\dfrac{12}{6} = 2\) and \(\dfrac{4}{2} = 2\).

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

Adding, multiplying and dividing (Higher tier)

Use the rules for number fractions, with brackets for the denominators.

  • Multiply

    Multiply the tops and the bottoms, and cancel before or after: \(\dfrac{x}{3} \times \dfrac{6}{x + 1} = \dfrac{6x}{3(x + 1)} = \dfrac{2x}{x + 1}\).

  • Divide

    Turn the second fraction upside down and multiply.

  • Add or subtract

    Find a common denominator. \(\dfrac{1}{x} + \dfrac{1}{x + 1} = \dfrac{x + 1}{x(x + 1)} + \dfrac{x}{x(x + 1)} = \dfrac{2x + 1}{x(x + 1)}\).

  • Keep the brackets

    Leave the denominator factorised, which makes later cancelling easier.

Equations with fractions

Multiply every term by the lowest common denominator to remove the fractions.

  • Find the common multiple

    For denominators 3 and 6, use 6.

  • Multiply every term

    Every term, including any whole number on the other side.

  • Solve the new equation

    Then check by substituting.

  • Brackets on the top

    Treat \(2(x - 1)\) as one thing when multiplying.

An equation with fractions

Solve \(\dfrac{x - 1}{3} + \dfrac{x + 2}{6} = 2\).

Show the solutionHide the solution
  1. 1 Multiply every term by 6 \(2(x - 1) + (x + 2) = 12\).
  2. 2 Expand \(2x - 2 + x + 2 = 12\).
  3. 3 Collect terms \(3x = 12\).
  4. 4 Solve and check \(x = 4\), and \(\dfrac{3}{3} + \dfrac{6}{6} = 1 + 1 = 2\).

Answer\(x = 4\)

Algebraic proof

To prove a statement for every number, write it with letters. An example is not a proof.

  • Consecutive numbers

    Call them \(n\) and \(n + 1\). Even numbers are \(2n\), and odd numbers are \(2n + 1\).

  • Prove: the sum of two consecutive numbers is odd

    \(n + (n + 1) = 2n + 1\), which is odd.

  • Prove: the sum of three consecutive numbers is a multiple of 3

    \(n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)\).

  • Prove: \((n + 1)^2 - (n - 1)^2\) is a multiple of 4

    \(n^2 + 2n + 1 - (n^2 - 2n + 1) = 4n\).

Counter-examples and identities

To show that a statement is false, one example that fails is enough.

  • Counter-example

    "\(n^2 + n + 1\) is always prime" fails for \(n = 4\), because \(16 + 4 + 1 = 21 = 3 \times 7\).

  • Identity

    The sign \(\equiv\) means two expressions are equal for every value, such as \((x + 3)^2 \equiv x^2 + 6x + 9\).

  • Conclude

    End a proof with a sentence: "which is a multiple of 3, so the sum is always a multiple of 3".

  • Pick numbers carefully

    A counter-example must make the statement false, not just fail to prove it.

Test yourself

  1. 1

    What may you cancel in an algebraic fraction?

    Show answerHide answer

    Factors that multiply the whole top and the whole bottom.

  2. 2

    How do you write an odd number using \(n\)?

    Show answerHide answer

    \(2n + 1\).

  3. 3

    What is the sum of three consecutive numbers, written with \(n\)?

    Show answerHide answer

    \(3n + 3\).

  4. 4

    What is the first step in solving \(\dfrac{x}{2} + \dfrac{x}{3} = 5\)?

    Show answerHide answer

    Multiply every term by 6.

  5. 5

    What does one counter-example show?

    Show answerHide answer

    That the statement is not always true.

Exam technique: fractions and proof

The working is the answer in these questions, so lay it out neatly.

  • Factorise before cancelling

    Cancelling terms instead of factors is the commonest error.

  • Show every step

    One method mark per step is common.

  • End a proof with a conclusion

    State what you have shown, in words.

  • Do not just test numbers

    Examples alone score nothing for a proof.

Summary and exam focus

  • Simplify an algebraic fraction by factorising the top and bottom and cancelling common factors (Higher tier).
  • Add fractions over a common denominator, and multiply or divide as with number fractions.
  • To solve an equation with fractions, multiply every term by the common denominator.
  • To prove, use letters such as \(n\) and \(n + 1\), and finish with a sentence. One counter-example disproves a claim.

Exam focus

Show that the sum of any three consecutive whole numbers is a multiple of 3. (3 marks) (3 marks)

Call the numbers \(n\), \(n + 1\) and \(n + 2\). Their sum is \(3n + 3 = 3(n + 1)\), which has a factor of 3. Finish with a sentence saying so, because the conclusion is a mark.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Algebraic fraction
A fraction whose top or bottom contains letters.
Cancel
Divide the top and bottom of a fraction by the same factor.
Common denominator
A shared multiple of the denominators, used to add or subtract fractions.
Proof
A logical argument that shows a statement is always true.
Counter-example
A single example that shows a statement is false.
Identity
An equation that is true for all values, written with \(\equiv\).
Consecutive
Following one after another, such as \(n\) and \(n + 1\).
Factor
An expression that divides exactly into another.
Multiple
A number in the times table of another number.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Prove 3 marks Stretch

Prove that the sum of two consecutive odd numbers is a multiple of 4.

Mark scheme — 3 marks available

  • \(2n + 1\) and \(2n + 3\) — M1
  • \(4n + 4\) — M1
  • \(4(n + 1)\) with a conclusion — C1

Model answer

Let the odd numbers be \(2n + 1\) and \(2n + 3\). Their sum is \(4n + 4 = 4(n + 1)\), which is a multiple of 4.

2. Exam question Simplify 2 marks Core

Simplify \(\dfrac{x^2 - 16}{x - 4}\).

Mark scheme — 2 marks available

  • \((x - 4)(x + 4)\) — M1
  • \(x + 4\) — A1

Model answer

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

3. Exam question Solve 3 marks Core

Solve \(\dfrac{x + 3}{2} + \dfrac{x - 1}{4} = 5\).

Mark scheme — 3 marks available

  • \(2(x + 3) + (x - 1) = 20\) — M1
  • \(3x + 5 = 20\) — M1
  • \(x = 5\) — A1

Model answer

Multiply every term by 4: \(2(x + 3) + (x - 1) = 20\). Then \(3x + 5 = 20\), so \(x = 5\).

4. Exam question Simplify 3 marks Stretch

Simplify \(\dfrac{x^2 + x - 6}{x^2 - 9}\).

Mark scheme — 3 marks available

  • \((x + 3)(x - 2)\) or \((x - 3)(x + 3)\) — M1
  • Both factorised — M1
  • \(\dfrac{x - 2}{x - 3}\) — A1

Model answer

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

5. Exam question Show that 3 marks Stretch

Show that \((n + 3)^2 - (n - 3)^2\) is a multiple of 12 for every integer \(n\).

Mark scheme — 3 marks available

  • \(n^2 + 6n + 9\) or \(n^2 - 6n + 9\) — M1
  • \(12n\) — M1
  • States that \(12n\) is a multiple of 12 — C1

Model answer

\((n + 3)^2 = n^2 + 6n + 9\) and \((n - 3)^2 = n^2 - 6n + 9\). The difference is \(12n\), which is a multiple of 12.

6. Exam question Write 3 marks Stretch

Write \(\dfrac{1}{x + 2} + \dfrac{1}{x - 1}\) as a single fraction, in its simplest form.

Mark scheme — 3 marks available

  • Common denominator \((x + 2)(x - 1)\) — M1
  • Numerator \((x - 1) + (x + 2)\) — M1
  • \(\dfrac{2x + 1}{(x + 2)(x - 1)}\) — A1

Model answer

Use the common denominator \((x + 2)(x - 1)\): \(\dfrac{(x - 1) + (x + 2)}{(x + 2)(x - 1)} = \dfrac{2x + 1}{(x + 2)(x - 1)}\).

7. Multiple choice 1 mark Easier

What may be cancelled in an algebraic fraction?

  1. A Any terms that appear on the top and bottom
  2. B Factors that multiply the whole top and the whole bottom Correct
  3. C Only numbers
  4. D Only letters

Why: Terms that are added or subtracted cannot be cancelled.

8. Multiple choice 1 mark Core

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

  1. A \(x - 3\) Correct
  2. B \(x + 3\)
  3. C \(x - 9\)
  4. D \(\dfrac{x - 9}{1}\)

Why: \(\dfrac{(x - 3)(x + 3)}{x + 3} = x - 3\).

9. Multiple choice 1 mark Core

Which statement about \(\dfrac{x + 3}{3}\) is correct?

  1. A It simplifies to \(x\)
  2. B It simplifies to \(x + 1\)
  3. C It simplifies to 1
  4. D The 3s cannot be cancelled because the 3 on top is added Correct

Why: Only factors can be cancelled, not terms.

10. Multiple choice 1 mark Core

Solve \(\dfrac{x - 1}{3} + \dfrac{x + 2}{6} = 2\).

  1. A \(x = 3\)
  2. B \(x = 5\)
  3. C \(x = 4\) Correct
  4. D \(x = 6\)

Why: Multiply by 6: \(2(x - 1) + (x + 2) = 12\), so \(3x = 12\).

11. Multiple choice 1 mark Easier

Which expression is an odd number for any whole number \(n\)?

  1. A \(2n\)
  2. B \(2n + 1\) Correct
  3. C \(n + 1\)
  4. D \(n^2\)

Why: \(2n\) is even, so adding 1 makes it odd.

12. Multiple choice 1 mark Core

What is the sum of three consecutive whole numbers \(n\), \(n + 1\) and \(n + 2\)?

  1. A \(3n + 3\) Correct
  2. B \(3n\)
  3. C \(3n + 2\)
  4. D \(n + 3\)

Why: \(n + n + 1 + n + 2 = 3n + 3 = 3(n + 1)\).

13. Multiple choice 1 mark Stretch

Which value of \(n\) is a counter-example to “\(n^2 + n + 1\) is always prime”?

  1. A \(n = 1\)
  2. B \(n = 2\)
  3. C \(n = 3\)
  4. D \(n = 4\) Correct

Why: \(16 + 4 + 1 = 21 = 3 \times 7\), which is not prime.

14. Multiple choice 1 mark Stretch

Simplify \(\dfrac{x^2 + 5x + 6}{x^2 + 3x + 2}\).

  1. A \(\dfrac{x + 2}{x + 1}\)
  2. B \(\dfrac{x + 3}{x + 2}\)
  3. C \(\dfrac{x + 3}{x + 1}\) Correct
  4. D \(\dfrac{5x + 6}{3x + 2}\)

Why: \(\dfrac{(x + 2)(x + 3)}{(x + 1)(x + 2)} = \dfrac{x + 3}{x + 1}\).

15. Multiple choice 1 mark Stretch

Expand and simplify \((n + 1)^2 - (n - 1)^2\).

  1. A \(2\)
  2. B \(4n\) Correct
  3. C \(2n\)
  4. D \(4n + 2\)

Why: \(n^2 + 2n + 1 - n^2 + 2n - 1 = 4n\).