Fractions, decimals and percentages: identify and work with fractions in ratio problems

The idea

Ratio compares quantities part-to-part or part-to-whole, while a fraction expresses a part relative to a whole. To identify fractions in a ratio problem, determine the whole: for a ratio 3 : 5 the whole is 3 + 5 = 8 equal parts, making the shares 3/8 and 5/8 of the total. To work with fractions, convert between forms. A share given as a fraction of the total becomes a ratio using a common denominator. A ratio becomes fractions by dividing each part by the total parts. For example, if money is split in the ratio 2 : 3 and the smaller share is £40, then 2 parts = £40, so 1 part = £20. The larger share is 3 × £20 = £60, and the total is £100, representing 2/5 and 3/5 of the whole. Always verify that the fractions sum to 1.

What earns the marks

• Recognises that in a ratio a : b the whole is a + b equal parts, so the shares are a/(a + b) and b/(a + b) of the total.
• Converts a ratio to fractions of the total by dividing each part by the sum of the parts, and converts fractions of a total back to a ratio using a common denominator.
• Uses one known share to find the value of one part, then scales to find the other share and the total.
• Checks that the fractions found sum to 1 and that the shares are in the stated ratio.

Common mistakes and how to put them right

• Writing the ratio 3.

  Put it right: Divide each part by the total number of parts, a + b.

• Treating the first number of the ratio as the numerator over the second number as the denominator.

  Put it right: Remember that a fraction of the total uses the sum of the parts as the denominator.

• Forgetting to add the parts to find the whole before forming fractions.

  Put it right: Calculate a + b first and use it as the denominator.

• Assuming a fraction of a total and a ratio share are the same thing without checking the whole.

  Put it right: State clearly what the whole represents before converting.

Exam tips

• Underline the word total in the question, because fractions in ratio problems are almost always fractions of the total, not of the other share.
• Write the sum of the parts as the denominator before doing any arithmetic, so the fractions are formed correctly from the start.
• Where a share is given in pounds or items, find the value of one part first, then multiply by the number of parts required.