Ratios & FDP

How are ratios and fractions linked?

Fractions and ratios are closely linked and are used to compare a part of a whole
- A fraction compares a part to the whole
- A ratio compares one part to another part
For example, a pizza is sliced into 8 pieces, and shared between two people such that the first person receives 5 slices, and the second person receives 3 slices
- As a fraction (of the whole)
  - The first person receives $\frac{5}{8}$ of the pizza
  - The second person receives $\frac{3}{8}$ of the pizza

- The ratio of slices of the first person to the second person is 5 : 3
  - The 8 does not appear in the ratio but is obtained by adding the parts together (5 + 3 = 8)
Unlike ratios, fractions can be converted/expressed as percentages or decimals

How do I answer questions involving ratios along with fractions and/or percentages?

In harder ratio questions, the information you are given may also involve fractions and/or percentages
Remember that you can use multiplication to find fractions or percentages of something
- E.g. Finding a fraction of a fraction
  - Of the $\frac{3}{40}$ of the people in the school who are members of staff, $\frac{2}{3}$ of them wore Christmas jumpers on a particular day
  - That means $\frac{3}{40} \times \frac{2}{3} = \frac{6}{120} = \frac{1}{20}$ of the people in the school that day were members of staff wearing Christmas jumpers
- E.g. Finding a percentage of a fraction
  - Of the $\frac{37}{40}$ of the people in the school who are students, 88% of them wore Christmas jumpers on that same day
  - To find 88% of something, you can multiply it by 0.88
  - $\frac{37}{40} \times 0.88 = 0.814$ , so this means 81.4% of the people in the school that day were students wearing Christmas jumpers
An alternative method is to 'pick a number' for the total number of things in the question
- E.g. Assume there are a 1000 people in the school (this is not necessarily the actual total)
- You can calculate the percentages or fractions using real numbers
- Remember to turn your final answer back into the correct form

Examiner Tip

In these types of questions it is really important to read the question carefully, so that you are clear about which bits are referring to what!
Give your final answer in the form asked for by the question (fraction, ratio, or percentage).

Worked example

A shop sells only two flavours of crisps - Stilton Surprise and Pickled Haggis. Packages of those two flavours of crisps only occur in two sizes - regular and jumbo.

The ratio of packages of Stilton Surprise crisps to packages of Pickled Haggis crisps in the shop is 7:3.

30% of the packages of Stilton Surprise crisps are regular sized.
$\frac{2}{5}$ of the packages of Pickled Haggis crisps are regular sized.

What percentage of all the packages of crisps in the shop are jumbo sized?

Method 1 (Pure fractions and percentages method)

First find the fractions of each type of crisp
The ratio divides the crisps into 7+3=10 parts
7 of those parts are Stilton Surprise, and 3 of them are Pickled Haggis

$\frac{7}{7 + 3} = \frac{7}{10} are Stilton Surprise \frac{3}{7 + 3} = \frac{3}{10} are Pickled Haggis$

30% of the Stilton Surprise are regular sized
So multiply 7/10 by 0.3 to find the percentage that are Stilton Surprise and regular sized

$\frac{7}{10} \times 0.3 = \frac{7}{10} \times \frac{3}{10} = \frac{21}{100} \frac{21}{100} = 0.21 So 21 % are regular sized Stilton Surprise$

2/5 of the Pickled Haggis are regular sized
So multiply 3/10 by 2/5 to find the percentage that are Pickled Haggis and regular sized

$\frac{3}{10} \times \frac{2}{5} = \frac{6}{50} \frac{6}{50} = \frac{12}{100} = 0.12 So 12 % are regular sized Pickled Haggis$

Add those together to find the percentage that are regular sized

$21 + 12 = 33 % regular sized$

And finally subtract 33% from 100% to find the percentage that are jumbo sized

$100 - 33 = 67$

67% are jumbo sized

Method 2 ('Pick a number' method)

Start by assuming (for example) that there are 100 packages of crisps in the shop
The ratio divides the crisps into 7+3=10 parts
So divide 100 by 10 to find how many packages are in 1 part

$100 \div 10 = 10 packages$

7 of those parts are Stilton Surprise, and 3 of them are Pickled Haggis
So multiply 10 by 7 and by 3 to find out how many packages of each type there are

$7 \times 10 = 70 are Stilton Surprise 3 \times 10 = 30 are Pickled Haggis$

30% of the Stilton Surprise are regular sized
So multiply 70 by 0.3 to find how many are Stilton Surprise and regular sized

$\begin{array}{rcl} 70 \times 0.3 & = & 70 \times \frac{3}{10} \\ = & \frac{210}{10} \\ = & 21 are regular sized Stilton Surprise \end{array}$

2/5 of the Pickled Haggis are regular sized
So multiply 30 by 2/5 to find how many are Pickled Haggis and regular sized

$\begin{array}{rcl} 30 \times \frac{2}{5} & = & \frac{60}{5} \\ = & 12 are regular sized Pickled Haggis \end{array}$

Add those together to find how many in total are regular sized

$21 + 12 = 33 are regular sized$

Subtract 33 from 100 to find how many are jumbo sized

$100 - 33 = 67 are jumbo sized$

And finally turn that into a percentage out of 100

$\frac{67}{100} = 0.67$

67% are jumbo sized

Ratios & FDP (Edexcel GCSE Maths: Foundation)

Revision Note

Ratios & FDP

How are ratios and fractions linked?

How do I answer questions involving ratios along with fractions and/or percentages?

Examiner Tip

Worked example

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Author: Naomi C