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Cambridge IGCSE0580

Algebraic fractions

Mathematics 0580 Chapter Notes

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Algebraic fractions
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Algebraic fractions notes

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In this chapter

  1. 1Adding and subtracting: numbers underneath
  2. 2Adding and subtracting: brackets underneath
  3. 3Multiplying and dividing
  4. 4Factorise, then cancel
  5. 5Factorising four terms

1. Adding and subtracting: numbers underneath

To add or subtract fractions, write them over a common denominator. Multiply the top and bottom of each fraction by whatever makes the bottoms equal.

Then put the numerators together over that one denominator. Write each numerator in brackets first, so the sign in front of a bracket reaches every term inside it.

a/c ± b/c = (a ± b)/c

Examiner insight

“Care should be taken with all signs in the expression as there were many cases of a plus erroneously becoming a negative and vice versa.”

Worked example 13 marks

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

  1. 1

    common denominator = 6: (2(2x - 3))/6 - (3(x - 5))/6 [1]

  2. 2

    one numerator, brackets kept: (2(2x - 3) - 3(x - 5))/6 = (4x - 6 - 3x + 15)/6 [1]

  3. 3

    collect terms: (x + 9)/6 [1]

2. Adding and subtracting: brackets underneath

When the denominators contain x, the common denominator is their product, kept in bracket form. Multiply each numerator by the bracket it is missing.

Write the numerator as one expression with its brackets, expand it, then collect like terms. Leave the denominator factorised.

a/p + b/q = (aq + bp)/pq

Examiner insight

“The most common errors arose from the –3(x – 1) term, when candidates either did not have brackets or did not remove the brackets correctly, with the 3 frequently becoming –3 or –1.” “Candidates who did not score had often simply added the numerators and denominators.”

Worked example 13 marks

Write 8/(x + 2) + 3/(x - 4) as a single fraction in its simplest form.

  1. 1

    common denominator (x + 2)(x - 4) and numerators 8(x - 4) and 3(x + 2) [1]

  2. 2

    (8(x - 4) + 3(x + 2))/((x + 2)(x - 4)) = (8x - 32 + 3x + 6)/((x + 2)(x - 4)) [1]

  3. 3

    (11x - 26)/((x + 2)(x - 4)) or (11x - 26)/(x² - 2x - 8) [1]

3. Multiplying and dividing

To multiply fractions, multiply the numerators and multiply the denominators. No common denominator is needed.

To divide, turn the second fraction upside down and multiply. Cancel common factors, then give the answer as a fraction in its simplest form.

a/b × c/d = ac/bd

a/b ÷ c/d = a/b × d/c

Worked example 14 marks

Do not use a calculator. Simplify(a) 3a/4 × 9a/10 and(b) 3a/4 ÷ 9a/10.

  1. 1

    (a) multiply across: (3a × 9a)/(4 × 10) [1]

  2. 2

    = 27a²/40 [1]

  3. 3

    (b) invert the second fraction: 3a/4 × 10/9a [1]

  4. 4

    cancel 3a: 10/(4 × 3) = 10/12 = 5/6 [1]

4. Factorise, then cancel

To simplify a rational expression, factorise the numerator and the denominator fully.

Then cancel any common factor. Only a whole factor, such as a whole bracket, can be cancelled. A single term inside a sum cannot.

Look for a common factor such as 2x, a quadratic that splits into two brackets, or a difference of two squares.

(k A)/(k B) = A/B

x² - y² = (x - y)(x + y)

Examiner insight

“In attempts to factorise the numerator, some candidates only took out a factor of 2 rather than 2x” “In attempts to factorise the denominator the most common error was to have a sign error in one of the two brackets, for example (x – 5)(x– 5).”

Worked example 13 marks

Simplify (x² - 2x)/(x² - 5x + 6).

  1. 1

    numerator: x² - 2x = x(x - 2) [1]

  2. 2

    denominator: x² - 5x + 6 = (x - 2)(x - 3) [1]

  3. 3

    cancel the common factor (x - 2): x/(x - 3) [1]

5. Factorising four terms

Some expressions have four terms. Group them in pairs and take out a common factor from each pair. The same bracket then appears twice.

Factorise the whole numerator and the whole denominator before any cancelling. Do the cancelling as a separate last step.

Examiner insight

“Many were unable to correctly factorise the numerator often resulting in two brackets of the form (x + a)(x – b) rather than (x + a)(2x – b).” “It was common to see cancelling taking place in the same stage as the factorisation.”

Worked example 15 marks

Simplify (2x² + 3x - 20)/(ax + 4a - 3bx - 12b).

  1. 1

    split the middle term: 2x² + 8x - 5x - 20 [1]

  2. 2

    numerator = 2x(x + 4) - 5(x + 4) = (2x - 5)(x + 4) [1]

  3. 3

    group the denominator: a(x + 4) - 3b(x + 4) [1]

  4. 4

    denominator = (a - 3b)(x + 4) [1]

  5. 5

    cancel (x + 4): (2x - 5)/(a - 3b) [1]

Recap

  • Extended: add and subtract algebraic fractions over a common denominator.
  • Put each numerator in brackets so the sign reaches every term.
  • Multiply across. To divide, turn the second fraction upside down and multiply.
  • Extended: factorise the numerator and the denominator fully, then cancel whole common factors.

End-of-chapter exercise

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

  1. Do not use a calculator. Write x/5 + x/2 as a single fraction.2 marks
  2. Write (x + 1)/4 + (x - 2)/3 as a single fraction.3 marks
  3. Write (3x - 1)/2 - (x + 4)/5 as a single fraction in its simplest form.3 marks
  4. Do not use a calculator. Simplify 4a/9 × 3a/8.2 marks
  5. Do not use a calculator. Simplify 5p/6 ÷ 10p/9.3 marks
  6. Do not use a calculator. Simplify (x² + 3x)/(x² - 9).3 marks
  7. Simplify (2x² - 8x)/(x² - x - 12).3 marks
  8. Write 3/(x + 1) + 2/(x - 2) as a single fraction.3 marks
  9. Do not use a calculator. A rectangle has area (3x² - 10x - 8) cm² and length (px - 4p + 2qx - 8q) cm. Find an expression for the width of the rectangle, in cm, in its simplest form.5 marks
  10. A walker covers (x + 3) km at (x - 1) km / h, then (x - 3) km at (x + 1) km / h, where x > 3. Find how much longer the first part takes than the second part. Give your answer, in hours, as a single fraction in its simplest form.5 marks

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