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Ratios & FDP (OCR GCSE Maths)

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Roger

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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 3 over 40 of the people in the school are staff members
      • And students represent 37 of those parts, so 37 over 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 3 over 40 of the people in the school who are members of staff, 2 over 3 of them wore Christmas jumpers on a particular day
      • That means 3 over 40 space cross times space 2 over 3 space equals space 6 over 120 space equals space 1 over 20 of the people in the school that day were members of staff wearing Christmas jumpers
      • 1 over 20 space equals space 0.05, so this means 5% of the people in the school that day were members of staff wearing Christmas jumpers
    • And of the 37 over 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
      • 37 over 40 space cross times space 0.88 space equals space 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 fraction numerator 50 plus 814 over denominator 1000 end fraction space equals space 864 over 1000 space equals space 0.864 space equals space 86.4 percent sign 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.
2 over 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.

fraction numerator 7 space over denominator 7 plus 3 end fraction space equals space 7 over 10 space are space Stilton space Surprise fraction numerator 3 space over denominator 7 plus 3 end fraction space equals space 3 over 10 space are space Pickled space 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.

7 over 10 space cross times space 0.3 space equals space 7 over 10 space cross times space 3 over 10 space equals space 21 over 100 space 21 over 100 space equals space 0.21 So space 21 percent sign space are space regular space sized space Stilton space 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.

3 over 10 space cross times space 2 over 5 space equals space 6 over 50 6 over 50 space equals space 12 over 100 space equals space 0.12 So space 12 percent sign space are space regular space sized italic space Pickled space Haggis

Add those together to find the percentage that are regular sized.

21 space plus space 12 space equals space 33 percent sign space regular space sized

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

100 space minus space 33 space equals space 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 space divided by space 10 space equals space 10 space 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 space cross times space 10 space equals space 70 space are space Stilton space Surprise 3 space cross times space 10 space equals space 30 space are space Pickled space 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.

table row cell 70 space cross times space 0.3 space end cell equals cell space 70 space cross times space 3 over 10 space space end cell row blank equals cell space 210 over 10 end cell row blank blank blank row blank equals cell space 21 space space are space regular space sized space Stilton space Surprise end cell end table

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.

table row cell 30 space cross times space 2 over 5 space end cell equals cell space 60 over 5 end cell row blank blank blank row blank equals cell 12 space space are space regular space sized italic space Pickled space Haggis end cell end table

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

21 space plus space 12 space equals space 33 space space are space regular space sized

Subtract 33 from 100 to find how many are jumbo sized.

100 space minus space 33 space equals space 67 space space are space jumbo space sized

And finally turn that into a percentage out of 100.

67 over 100 space equals space 0.67

67% are jumbo sized

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    Roger

    Author: Roger

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.