Operations with Fractions

Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core

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1. Finding a fraction of an amount

Three routes to the same answer

  • Dividing by the denominator and then multiplying by the numerator keeps every number whole for as long as possible
    • For 3/8 of 56: 56 ÷ 8 = 7, then 7 × 3 = 21
  • Converting the fraction to a decimal and multiplying suits fractions with a short decimal form
    • For 3/4 of 26: 0.75 × 26 = 19.5
  • Writing the amount over a denominator of 1 turns the problem into a fraction multiplication
    • For 2/9 of 45: 2/9 × 45/1 = 90/9 = 10
  • The word "of" between a fraction and a quantity always means multiply

Shading a fraction of a shape

  • Shading works the same way: count the parts, divide by the denominator, then shade as many groups as the numerator asks for
    • To shade 5/7 of a grid of 21 squares, 21 ÷ 7 = 3, and 3 × 5 = 15 squares are shaded
  • Which particular squares are shaded does not matter, though grouping them together makes the fraction easier to read
  • Where the shape is not already divided, cut it into the number of equal parts the denominator names first

Worked example

Finding a fraction of a quantity

A theatre has successfully sold 5/6 of its 342 seats. Work out how many seats were sold.

Solution:

  • Divide by the denominator first, since 342 divides exactly by 6
  • 342 ÷ 6 = 57, so one sixth is 57 seats
  • Multiply by the numerator: 57 × 5 = 285
  • 285 seats were sold

2. Adding and subtracting fractions

Working with a common denominator

  • Fractions can only be added or subtracted once they share a denominator
  • Find the lowest common denominator, which is the lowest common multiple of the two denominators
  • Scale each fraction up so that both sit over that shared denominator
  • Combine the numerators and keep the denominator unchanged
    • Denominators are never added together
  • Simplify the result, returning it to mixed-number form where the question demands that

When a mixed number appears

  • Convert every mixed number to an improper fraction before starting
  • Carry out the addition or subtraction as usual, then convert back at the end

Exam tip

Each step of a mixed-number sum scores

A calculation with two fractions or mixed numbers was set in 12 of the 22 papers, including every Paper 1. When mixed numbers are added or subtracted, the schemes give one mark for the improper fractions, one for both fractions written over a common denominator, and the last for the answer in the form asked. Stopping at 17/10 when the question wants a mixed number in its simplest form (1 7/10) costs that last mark.

Worked example

Subtracting mixed numbers

Work out 4 1/6 − 1 3/4, giving the answer as a mixed number.

Solution:

  • Convert both to improper fractions: 4 1/6 = 25/6 and 1 3/4 = 7/4
  • The lowest common multiple of 6 and 4 is 12
  • Rewrite over 12: 25/6 = 50/12 and 7/4 = 21/12
  • Subtract the numerators: 50 − 21 = 29, giving 29/12
  • Convert back: 29 ÷ 12 = 2 remainder 5, so the answer is 2 5/12

3. Multiplying and dividing fractions

Multiplying

  • Cancel any common factor between a numerator and a denominator before multiplying, taking factors from either fraction
  • Multiply the numerators together, then multiply the denominators together
  • Cancelling first keeps the numbers small, which matters on the non-calculator papers

Dividing

  • Dividing by a fraction is the same as multiplying by its reciprocal, so invert the second fraction and change the sign to multiplication
  • Cancel and multiply exactly as above
  • Only the second fraction is inverted, and only after the division sign has been changed

Mixed numbers

  • Convert any mixed number to an improper fraction before multiplying or dividing
  • Attempting to multiply the whole parts and the fraction parts separately does not work

Worked example

Multiplying fractions by cancelling first

Work out 9/20 × 8/15, giving the answer in its simplest form.

Solution:

  • Cancel 9 against 15 using their common factor 3, leaving 3/20 × 8/5
  • Cancel 8 against 20 using their common factor 4, leaving 3/5 × 2/5
  • Multiply the numerators: 3 × 2 = 6
  • Multiply the denominators: 5 × 5 = 25
  • The answer is 6/25, which has no remaining common factor

Worked example

Dividing by a fraction

Work out 2 1/3 ÷ 7/9.

Solution:

  • Convert the mixed number: 2 1/3 = 7/3
  • Invert the second fraction and change the operation: 7/3 × 9/7
  • Cancel 7 against 7, leaving 1/3 × 9/1
  • Cancel 9 against 3, leaving 1/1 × 3/1
  • The answer is 3