Ratio, proportion and rates of change: express one quantity as a fraction of another, where the fraction is less than 1 or greater than 1

The idea

The phrase "A as a fraction of B" means A/B, so the quantity that follows the word "of" goes on the bottom. Put both quantities into the same unit first, then write the fraction and cancel it. For example 45 minutes as a fraction of 2 hours is 45/120, which cancels to 3/8. When the first quantity is the larger one the answer is greater than 1, and that is correct rather than a mistake: 2 hours as a fraction of 45 minutes is 120/45, which cancels to 8/3. Leave it as an improper fraction unless the question asks for a mixed number. The comparison is often hidden in wording such as "what fraction of the class", so decide which quantity is the whole before you write anything.

What earns the marks

• Converting both quantities into the same unit before the fraction is formed.
• The fraction written the right way round, with the quantity named after "of" as the denominator.
• Dividing both numbers by a common factor, even where the cancelling is not taken all the way.
• The fraction in its simplest form, which is greater than 1 whenever the first quantity is the larger of the two.

Common mistakes and how to put them right

• Writing the fraction upside down, so 20 g as a fraction of 50 g is given as 50/20.

• Comparing 30 cm with 2 m without converting, giving 30/2 rather than 30/200.

• Assuming an answer above 1 must be wrong and inverting it to force a proper fraction.

• Cancelling part way and stopping at a fraction that still has a common factor.

Exam tips

• Underline the words that come after "of"; that quantity is the denominator every time.
• Convert the units before you write the fraction, so the one you cancel is already the right one.
• Leave the answer as a fraction unless the question asks for a decimal or a percentage.