Ratio, proportion and rates of change: use ratio notation, including reduction to simplest form

The idea

A ratio compares parts of the same kind and is written with a colon, so 12 : 18 means twelve of one thing for every eighteen of another. To reduce it, divide every part by their highest common factor: the highest common factor of 12 and 18 is 6, so 12 : 18 becomes 2 : 3. All parts must be in the same unit first, so 40 cm : 2 m becomes 40 : 200 and then 1 : 5. Three-part ratios work the same way, using a number that divides all three parts. You can also write a two-part ratio in the form 1 : n by dividing both parts by the first, and n need not be a whole number, so 4 : 10 becomes 1 : 2.5. That form makes two ratios quick to compare.

What earns the marks

• Converting the parts into a common unit before any cancelling.
• Dividing every part by the same number, even where the number chosen is not the highest common factor.
• The fully simplified ratio, written with a colon rather than as a fraction.
• A correct 1 : n form with the decimal kept, where the question asks for that form.

Common mistakes and how to put them right

• Dividing only one part, so 12.

  Put it right: 18 is written as 2 : 18.

• Stopping at 6 : 9 because the numbers look smaller, when 3 still divides both parts.

• Simplifying 30 minutes : 2 hours as 30 : 2 instead of 30 : 120.

• Reversing the order of the parts, so an answer of 3.

  Put it right: 5 is given for a question that named the quantities the other way round.

Exam tips

• Look for one number that divides every part at once; if you cannot see one, halve or divide by 3 repeatedly until nothing divides again.
• Check a simplified ratio by multiplying it back up to the numbers you started with.
• Keep the parts in the order the question names them, because 3 : 5 and 5 : 3 are different answers.