Ratios & FDP

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
In most of these questions the key is to remember that ratios can also be turned into fractions (or vice versa)
- For example, the ratio of staff to students in a school may be 3:37
  - That means the ratio divides the school into 3+37=40 parts
  - Staff members represent 3 of those parts, so $\frac{3}{40}$ of the people in the school are staff members
  - And students represent 37 of those parts, so $\frac{37}{40}$ of the people in the school are students
- Note that you don't need to know how many people are in the school to determine those fractions!
Another key concept is the fact that you can use multiplication to find fractions or percentages of something
- This includes finding something like a fraction of a fraction, or a percentage of a percentage, or a fraction of a percentage, or a percentage of a fraction
- For example, 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
  - $\frac{1}{20} = 0.05$ , so this means 5% of the people in the school that day were members of staff wearing Christmas jumpers
- And 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
- Adding those percentages together tells us that 5+81.4=86.4% of the people in the school that day were wearing Christmas jumpers
- Notice again that you don't need to know how many people are in the school to determine all of that
There is another method that can be used if you don't really like dealing with 'abstract' fractions and percentages
- You can pick a number for the total number of things in the question
  - So in the example above, you might assume that there are 1000 people in total in the school
  - This doesn't mean there are really 1000 people in the school (and it doesn't matter if there actually aren't 1000 people in the school)
  - It just gives you a 'real number' to work with, instead of abstract fractions and percentages
- Then do all the rest of the workings on that number, using your usual techniques for fractions, ratios, and percentages
  - For example if you assumed there were 1000 people in the school described above, you would find to start that there were 75 members of staff and 925 students
  - And then you would find that 50 members of staff and 814 students wore Christmas jumpers on that day
  - So $\frac{50 + 814}{1000} = \frac{864}{1000} = 0.864 = 86.4 %$ of people in the school were wearing Christmas jumpers
- You just need to make sure at the end that you turn your answer back into the form (fraction, ratio, or percentage) that the question asks for!
- The only problem with this method is you might pick a number that doesn't 'play nicely' with the ratios, fractions, and percentages in the question
  - With the question above, for example, you might find that the number was dividing up into numbers of staff members and students that weren't whole numbers
  - If this happens you might have to pick a new number and start over
  - For many questions, big numbers like 100 or 120, or 1000 or 1200, might be a good bet
- See the Worked Example below for an example of this method being used

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!
Be sure to 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 (OCR GCSE Maths)

Revision Note

Ratios & FDP

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

Examiner Tip

Worked example

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Author: Roger