Fractions, decimals and percentages: how to link them

Transcription

1 Fractions decimals and percentages: how to link them Mathematics Skills Guide This is one of a series of guides designed to help you increase your confidence in handling mathematics In this guide you will find brief theory and exercises designed to help you bridge the work in Fractions Decimals and Percentages It brings out the essential connections that should help you to work with confidence in all three There are often different ways of doing things in mathematics and the methods suggested in the guides may not be the ones you were taught If you are successful and happy with the methods you use it may not be necessary for you to change them If you have problems or need help in any part of the work then there are a number of ways you can get help For students at the University of Hull Ask your lecturers You can contact a math Skills Adviser from the Skills Team on the shown below Access more maths Skills Guides and resources at the website below Look at one of the many textbooks in the library Web: wwwhullacuk/skills skills@hullacuk

2 Most people's experience of fractions decimals and percentages is of different topics learned at school at different times between the ages of eight and fourteen In consequence many do not realise the way in which these topics can be linked and when linked can be much more powerful as tools for handling number This worksheet shows the ways these links can be made The important mathematical principle known as the Equivalence of Fractions is the means we can use to link the three topics Most people will know about this principle though they may not have used this name for it nor have realised its general importance to mathematics One example of it is given by and we can use it to facilitate easy conversion from any one of fractions decimals and percentages to both the other two You will probably recognise the set of equalities and will be used to it either in a similar form to this or when cancelling fractions or reducing them to their lowest terms Specifically when we say that a half is equal to two quarters we are doing it by multiplying the top and bottom numbers in the fraction by two; similarly in the other cases we have multiplied top and bottom by other numbers or diagrammatically we can see that: Most people will know that a half is 0 as a decimal and that 0 over 00 represents 0% So we have established that is the same as 0 and 0% Let us use this to make a conversion table before using it to show how to change quickly and easily from one to the other Fractions Decimals Percentages Fractions Decimals Percentages 0 0% 0 0% 0 % 0 % 00 % 00 % and so on There are several things to notice about this table: 0 0% 0 00 % 0 00 % 0 00 % 0 and so on the process of halving has been used to build up the table; this is often a useful device for calculating if you look at the decimal columns and compare them with the percentage columns you may be able to notice that the whole number part of the percentage format is the first two decimal places of the decimal format This is not surprising if you remember that percentage means out of a hundred But you can further notice that where there is a fraction in the percentage this is reflected in the third and fourth decimal places in the decimal format

3 if you look at the forms for a half a quarter etc and then go to those for a twentieth a fortieth etc you will see another example of the principle mentioned above because a twentieth is a half divided by 0 and a fortieth is a quarter divided by 0 All of these things draw your attention to some useful expedients for calculating of all kinds and in particular for any connected with fractions decimals and percentages But we can also use similar procedures for some other initially less simple than the above: fractions such as a third a sixth a ninth etc Here is the table for such fractions: % % % In this case we are going into the idea of a recurring decimal ie one in which a figure or group of figures are repeated indefinitely and each of which can be seen to represent an exact fraction (See later section for some notes on recurring decimals) To become competent you should try to memorise the first three of each of the parts of this table and then use the device of dividing to get the others from them But it is also possible to derive other equivalencies such as which is made up of and added together So the decimal form of is ie 0 which is in turn % and could have been derived directly because most people recognise it as being You should try out a few such examples for yourself using the leaflet on fractions to help you if you have forgotten how to add fractions but in all cases you will see that the decimal and percentage equivalencies are very easily found from each other and can often be the way into such a conversion problem So let us now look at converting from any one of these to the other two As might be expected from the above decimals and percentages convert most easily into each of the other two; there is just a little more to converting fractions Starting from a decimal Take any decimal 0 say Clearly this represents % because the first two places always give us the whole number part of the percentage with the other places bringing in a decimal part of the percentage Watch people on TV talking about percentages and you will notice that they are always using this way of talking about them To complete we can either say that 0 is divided by 000 (from the same number principle about multiplying and dividing by 0 mentioned above) and so is the fraction or it is divided by 00 (a percentage) and this is the same as if we multiply both of the terms and 00 by 0

4 Starting from a percentage This is also simple Take the percentage %; this is 0 as a decimal and 00 as a fraction Note that in the fraction form here and in the section above we could reduce the fractions obtained by dividing top and bottom by the same number In the example of 00 we can divide top and bottom by and get ; in the example of we can divide top and bottom by and get Some percentages look as if they might be complicated but in reality are not For example % can be split up into % and %; their decimal and fraction equivalents are 0 and = (see the Fractions leaflet if you are unsure about this); or can be got directly from 0 because this is three times 0 which the conversion table tells us is Starting from a fraction Many fractions are easy to convert but some are slightly more complex In the latter cases we would recommend using a calculator after you have taken the steps advocated below Some simple examples: : is 0 so is 0 0 and 0% : If you can remember it is 00 so is 0 divided by = 0 divided by = 0 or use If you get an awkward fraction which cannot be dealt with in this way such as look at the numbers in the fraction to see if you can see an easier fraction near to them in value in order to get an estimate for your answers For example is < which is (0 and %) also is > which is (0 and 0%) So you now know that the decimal form of 0 lies between 0 and 0 and the percentage between % and 0% It is at this stage that you use your calculator keying in divided by to give the answer 00 which is 0% to decimal places an answer that satisfies the boundaries found above This is a somewhat more complex procedure and to avoid it you could go straight to your calculator to get an answer but as always when you use a calculator it is important to have an estimate against which to check the calculator answer so in going through this procedure you both practise your estimating skills and provide an essential check for your calculator answer

5 Being able to change quickly and easily between fractions decimals and percentages will provide you with a very powerful skill in dealing with numbers because it will enable you to work at will in any of the three media Exercise In each example given change to the other two forms: % % % % % % % % Some notes on recurring decimals Fractions which are written with a dot over one or more of the digits in them are known as recurring decimals For example 0 means 0 and goes on to an infinite number of terms In expressing it to a specific number of decimal places two say we would express it as 0 The is here because when we correct to a given number of decimals we look at the next figure and if it is < we give the existing figure while if it is or > we correct the next figure to the figure one higher than it is This procedure is called rounding up and rounding down and comes into all examples of numbers where an answer is required to a specific number of places 0 means 0 and goes on as before 0 means 0 and goes on as before Note that in this case we only put the dots over the first and last of the recurring figures With this in mind we want now to move to finding out what such recurring decimals represent in fractional and percentage forms As in the earlier part of this handout the percentage form is easy to write down 0 represents % to one decimal place % to two places 0 represents % to one place of decimals % to two places 0 represents % to one decimal place and % to two places To find the fraction represented by 0 recurring we first multiply the whole decimal by 0 (because it is only one figure which is recurring) and we get to We can write this as + 0 = + 0 We see that 0 times the recurring fraction has the same value as the number together with itself In other words This gives 0 Therefore the recurring decimal itself is equivalent to divided by or If you find it difficult to get your mind round the argument written in this way you may prefer an algebraic way of expressing the same thing: Let us give the value x to 0 (0); then we have 0 x = = + 0 = + x So using whatever rule you know for solving such an equation we get x leading to x

6 For 0 recurring =0 with two figures recurring we multiply by 00 to give us which by the same argument as above is + the recurring decimal we started with So times the decimal is and the decimal is equivalent to Or done as an equation we find that 00x 0= so that x giving us x x For 0 or 0 recurring = 0 because there are three figures recurring we multiply the decimal by 000 and find that this gives us times the fraction to be so that the decimal is equivalent to In equation form we have 000x x so that x and hence x Try these for yourself: Exercise Having tried all of these - and perhaps made up a few for yourself - you might like to see if you can formulate a rule which you could use for any similar recurring decimals so that you do not have to go through the whole process every time Note: For some specific recurring decimals such as 0 it is possible to or derive its value by realising that it comes from either halving 0 recurring dividing 0 by Alternatively using a similar procedure as for other recurring decimals and multiplying by 0 we get = + 0 = + and from this dividing by 0 again we have 0 = divided by 0 which is 0 or If you can become fluent in changing between fractions decimals and percentages and can remember the procedures and the rule you abstracted from these procedures you will find that you can approach examples of all of them with more confidence and competence

7 ANSWERS Exercise %= %= %= %= %= % = %= % = = 0 = 0% = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = % 0 = 0 = % 0 = %= 0 = %= 0 =%= 0 = % = 0 = % = = %= 0= %= = 0 = % ( to decimal places) Exercise 0 = %(to dec places)= 0 = %(to dec places)= 0 =%(to dec places)= 0 =0000%(to dec places)= = = 0 = % ( dp ) = 0 = ( dp ) = 0 = % ( dp ) = 00 = 0% ( dp ) 0 =%(to dec places)= 0 =%(to dec places)= We would appreciate your comments on this worksheet especially if you ve found any errors so that we can improve it for future use Please contact the Maths Skills Adviser by at skills@hullacuk The information in this leaflet can be made available in an alternative format on request using the above