Percentages. Mathematics Skills Guide

Percentages Mathematics Skills Guide This is one of a series of guides designed to help you increase your confidence in handling mathematics. This gu...
Author: Harriet Simmons
0 downloads 2 Views 272KB Size
Percentages Mathematics Skills Guide

This is one of a series of guides designed to help you increase your confidence in handling mathematics. This guide contains both theory and exercises which cover:1. 2. 3. 4.

Finding a percentage Increasing by a percentage Decreasing by a percentage Finding a percentage change

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 email shown below.  Access more maths Skills Guides and resources at the website below.  Look at one of the many textbooks in the library.

Web: www.hull.ac.uk/skills Email: [email protected]

1. Percentages The booklets on ‘Estimation and Mental Methods’ and ‘Fractions, Decimals and Percentages:how to link them’ show how to simplify some calculations by using the methods they recommend and, in particular, by being able to convert easily between fractions, decimals and percentages because this underpins the way this handout approaches the topic of percentages. In thinking about percentages it is important to realise that they pervade the whole of our lives but are often misunderstood by many people. Further than that, changes in percentage, either increases or decreases, occur almost daily in such areas of life as mortgage rates, bank rates and as ways of expressing changes in population and other human affairs. So this booklet should enable you to avoid the many mistakes frequently made in handling percentages in the media and should enable you to approach psychometric tests and percentages in your ordinary life confident that you can work accurately with them. If you have not read the ‘Estimation and Mental Methods’ and ‘Fractions, Decimals and Percentages:how to link them’ booklets you should do so because methods developed in it will be used in this one so that the three together provide a comprehensive whole for handling many of the everyday calculations we need to make. Principle for percentages The first thing to realise about percentages is that there are two quite distinct aspects of calculations to do with them. These are - finding a given percentage of something - finding one thing as a percentage of another and you will find that, bearing this in mind, the first thing about any question is to determine which of these two is required for any particular calculation. 2. Finding a given percentage of something To calculate a particular percentage of a given quantity, you have several methods of approach at your disposal. By realising that fractions, decimals and percentages are linked, you will see that finding a percentage can be done by means of either fractions or decimals. So 37% of something can be seen to be 37 times whatever 100

the quantity was or it can be seen as 0.37 times the quantity. In the decimal form a calculator is the obvious way forward but you have to remember that, if you use a calculator, you must do an estimate first to enable you to check you have pressed the right buttons. So in this case you can see that 37% must lie between a third and a half, (ie between 33% and 50%) both of which are easy amounts to check for any quantity. However, some percentages can easily be calculated by breaking the percentage into manageable components. For example, 75% can be broken up into 50% and 25% ie a half and a quarter, amounts that can easily be done mentally, 37 % can be broken up into 25% (a quarter) and 12 1 % (an eighth) while 57% can be broken up 2

into 50% (a half) and 5% (a tenth of a half) - and 2% (easily calculated by dividing by 100 and doubling). page 1

Example 1 Find 87 1 % of 236. 2

Method 1. 87 1 = 50 + 25 + 12 1

Method 2. Note 87 1 % is 12 1 % less than 100%.

50% of 236 25%

is 118 is 59

12 1 % is 1 ,

12 1 %

is 29.5

2

2

87 1 % of 2

2

2

2

2 8 1 87 % of 236 = 236 – 236 2 8

= 236 - 29.5 = 206.5

236 is 206.5

Method 3. Using your calculator 0.875  236  206.5 Example 2. Find 49% of 350. Method 1. Notice that 49 is 1 less than 50 50% of 350 is 175 (one half) 1% of 350 is 3.5 49% of 350

Method 2. Using your calculator 49% of 350 = 0.49 × 350 = 171.5, a little less than 12 of 350.

is 171.5

Example 3. Find 52 1 % of 376. 2

Method 1. Notice that 52 1  50  2 1 2

Method 2. Using your calculator 52 1 % of 376 = 0.525 × 376

2

50% of 376 is 188 (one half)

2

2 1 % of 376 is 9.4 ( 1 of 10%, ie 1 of 37.6) 2

4

4

52 1 % of 350 is 197.4

= 197.4, a little more than 12 of 350

2

Example 4. Find 57% of 342. Method 1. Split 57% into 50% (half) + 5% +2% 50% of 342 is 171 (one half) 5% of 342 is 17.1 (one tenth of one half) 2% of 342 is 6.84 (two hundredths) 57% of 342 is 194.94

Method 2. Using your calculator 57% of 342 = 0.57 × 342 = 194.94,

If you practise such examples, always using the simple fractions of 100 and of 10 such as 50, 25, 12 1 and 5, 2 12 , 1 14 , you will soon find that you become very efficient 2

in seeing ways to break up any given percentage into manageable fractions. This can be a useful skill for helping with psychometric tests as well as calculating percentages you encounter in ordinary life so it is worth persevering with it. Always remember that if you use a calculator to determine a percentage of something, you must find an estimate as well in order to check your answer.

page 2

Exercise 1 Find 37 %, 56%, 62 %, 76% of each of the following: 456, £560, 2345. For all of these you can use a calculator to do the calculation and this can serve as a check on your other methods but do not let yourself get into the habit of always using a calculator because this will tend to erode your developing mental calculating facility and in ordinary life as well as in the psychometric tests employers set you this is not necessarily the best way to operate. With other methods at your disposal you will be better able to operate with figures in ordinary life as well as in the tests. 3. Finding one number as a percentage of another For this aspect of percentages all you need to do is to make a fraction of the two

3 quantities. For example 45 as a percentage of 60: here 45 reduces to 4 so the 60

percentage is 75%. Some fractions are easily reducible to a fraction to which you can give the percentage from the table provided in the booklet Linking Fractions, Decimals and Percentages but some are slightly more cumbersome. For example if you want to see what percentage 37 is of 49 you make the fraction 37 and use the procedures for linking fractions, decimals and percentages to find out 49

what percentage this represents. Note that you can make a quick estimate by looking at 40 and seeing that this is four fifths ie 80%. For this one it is easier to us a 50

calculator since the arithmetic is quite heavy and with the estimate already made you are in a position to check the answer your calculator gives you. Using your calculator 37 = 0.755102 which represents 75.5% (to 1 decimal place). 49

However, many examples of finding one thing as a percentage of another can also be simplified using the ideas from the Estimation booklet to help you. Thus, while 37 as a percentage of 74 can be seen to be 50% because 37 is half of 74, there are many other sets of figures that can be calculated as percentages easily from the fractions they make. For example, 21 as a fraction of 35 is three fifths ie 60%, while 35 as a percentage of 21 is found by putting 35 which reduces to 53 which as a 21

decimal is 1.666... ie 167%. The ‘Fractions, Decimals and Percentages: how to link them’ booklet will enable you to see ways of simplifying any given fraction where this is possible Exercise 2 For the pairs of numbers given, find the smaller as a percentage of the larger and the larger as a percentage of the smaller. (Notice that in one case the percentages arrived at will be less than 100% while in the other the percentages arrived at will be more than 100% because in the first case the fractions will all be less than 1 and in the second they will be greater than 1. The significance of this will come out later.) 27, 81

56, 91

350, 490

121, 254

137, 423

page 3

4. Percentage change As already indicated, percentages in real situations fluctuate sometimes daily, so we need to be able to look at the two aspects of percentages in terms of possible changes and, having done so, be in a position to work competently with most of the situations that can occur relating to such changes. And because so many people are vague about their purpose and meaning it is easy to be deceived into believing statements made in the public domain which, on examination, can be seen to be false. This can be illustrated by a quote from a statement made by a BT spokesman in October 1991 about the BT Chairman having recently had an increase of 12 12 % bringing his salary to £469 000. He said that this rise was not excessive because BT workers had recently received a 7.3% rise. There is no indication about how this justification was received by those he was speaking to and there is no means of knowing whether the BT spokesman was himself ignorant about the way percentages work or whether he was relying on the ignorance of those to whom he was speaking. The flaws in his argument are brought out by the following: 10% rise for all; is it fair? Pay £5000 rise £500 Pay £10,000 rise £1000 Pay £50,000 rise £5000 Pay £100,000 rise £10,000

New pay £5500 New pay £11,000 New pay £55,000 New pay £110,000

It is easy to see from this table that an increase of 10% not only increases the actual rise received but also the differentials between the lower and higher paid. You can decide whether this is a fair way of proceeding. Whatever your view turns out to be a proper understanding about how percentage increases work enables you to see the flaws in arguments such as that of the BT spokesman. So let us proceed to looking at increasing by a given percentage. 5. Increasing by a given percentage For this it is always assumed that the starting figure represents 100%. Let us consider some simple increases in percentage: An increase of

50%

on 100% takes us to

150%

 1 1 or  1.5

2  1 1 or  1.25 4  1 3 or  1.75 4

25%

125%

75%

175%

100%

200%

125%

225%

2  2 1 or  2.25

200%

300%

3

4

300% 400%  4 For some examples the fraction form is the easiest to work with, but for others eg increasing by 73% the decimal form is much easier and for this it is sensible to use a calculator, always remembering to determine an estimate for the answer, in this case page 4

that it will be increased by something just less than three quarters. However, some fractions which do not appear to be easy fractions to handle can in fact be made simple. Examples of this are increasing by 12 1 %, 37 1 %, 87 1 % and the VAT rate of 17 1 %.

2 2 2 2 1 1 For these think about 12 as 10 + 2 , for which you could get a tenth and a quarter 2 2 of a tenth, quite easy to do in your head. Similarly 37 1 is 25 + 12 1 , ie a quarter and 2 2

a half of a quarter. Others like 57% can also be done as 50% (a half) plus 5% (a tenth of that) plus 2% which is easy (either a fifth of a tenth or twice a hundredth) So, when asked to increase by a given percentage, look at the numbers involved to see if you can make them into easy fractions or split them into easy fractions; if not then use the decimal form and use your calculator, always remembering that, with calculator use for anything, you should always first make an estimate of the answer in order to be able to judge whether the display answer is a reasonable one. Example 1 The price of something before VAT is added is £236; what will its price be including VAT? This can be done in two ways: Method 1: 17 1 % = 10% + 5% + 2 1 % , so the VAT will be £23.60 + £11.80 + £5.90, by getting 2

2

10% (a tenth) and then successively halving the amount to get 5% and 2 1 % and, by 2

adding all of these to the original price, the total price will be: £236 + £23.60 + £11.80 + £5.90 = £277.30 Method 2: To add VAT of 17 1 % find 1.175 x 236 using your calculator in the same way as 2

above for simple percentages of a given quantity. Notice that the answer will be something less than £236 + a fifth of £236 ie £236 + twice £23.60 (ie £47.20). That is it will be a little less than £283.20 which is the result of adding a fifth of £236 to £236. The calculator gives £277.30 when the answer is given to two decimal places because 1.175 × 236 = 277.3 Example 2 The population of a certain country is 270,000. Trends suggest a population growth of 13% will occur during the next 10 years. What would you expect the population be at the end of those ten years? Method 1: 13 = 10 + 1 + 2 so, after ten years, the population should be 270,000 + 27,000 + 2,700 + 5,400 = 305,100 Method 2: After ten years the population should be 113% of what it is now: 270,000 ×1.13 which, using a calculator, is 305,100. Increasing by 13% represents an increase of something under 15% which would be 27,000 + 13,500 which is 40,500; 40,500 more would be 310,000, a bit more than the actual answer. page 5

Example 3 Increase 357 by 37% Method 1 37% = 25% (a quarter) + 10% (a tenth) + 2% (1% × 2) so 357 increased by 37% = 357 + 89.25 + 35.7 + 7.14 = 489.09 Method 2 357 increased by 37% = 357 × 1.37 = 489.09 using a calculator Note: Some people find it quite difficult to adopt this way of working out increased percentages; they prefer to find the amount of the increase first and then add it onto the original. There is nothing at all wrong with this except that you would have to do two calculations rather than being able to encompass the calculation within a single process, besides the fact that the way given above builds directly on the work on changing between fractions, decimals and percentages and so, in the long term will be a more economical way of working. Exercise 3 Find the result of increasing the following by the percentages indicated: 1. Increase 3624 by 56%, 37%, 72%, 84%, 82% 2. Find the cost including VAT at 17 1 % on: £360, £2376, £23.50, £445.75 2

6. Decreasing by a given percentage Decreasing by a given percentage has some features which are radically different from that of increasing by a given percentage and understanding these differences is crucial to competent operation of percentages. As with increasing by a given percentage, the starting point is always 100%. If we decrease something by 50% we reduce it to 50% (a half or 0.5) If we decrease it by 25% we reduce it to 75% ( 34 or 0.75) by 75%

to 25% ( 14 or 0.25)

by 100%

to zero.

So the crucial difference here is that it is impossible to reduce anything by more than 100% unless the negative has meaning. So we cannot reduce a population or a weight or height by more than 100% but we can reduce an amount of money by more than 100% because it will bring us into a position of debt as many students know only too well. In this case we have to think rather differently about what this means. If reducing by 100% brings us to zero, then reducing by 200% brings us to a debt of the amount we had in the first place ie it is -1 times what we started from. Reducing by 300% brings us to a debt of twice what we started from ie we have -2 times what we started from.

page 6

So we could calculate any intermediate percentages by seeing what the resulting negative would be as either a negative fraction or decimal of what we started with. In practice, as was demonstrated above, the fraction forms are more cumbersome to handle than are the decimal forms which, in turn, are more easily done using a calculator. Exercise 4 Decrease the following numbers by each of the given percentages: 480 2376 5372 868 by 50% 75% 33 13 % 49% 7. Finding a percentage increase or decrease Numbers can be increased or decreased and it is useful to be able to calculate the percentage increase or decrease these changes represent. For example, when price increases are announced, it is useful to be able to see what percentage increase they represent so as to compare this with the current rate of inflation. Similarly when there is a sale and prices are reduced it is useful to be able to calculate the percentage reduction this represents. Although sales are usually announced by claiming there are particular percentage decreases in operation in such claims as 20%, 50% or even, sometimes, as much as 75% off, they can just be given as discounts of a particular amount so it is useful to be able to calculate the percentage discount. To do this we need to return to the idea of expressing one thing as a percentage of another, sometimes finding the percentage represented by the fraction made from putting the smaller over the larger, sometimes by putting the larger over the smaller. In this context we use the expedient of always putting the new amount over the old amount. In the case of an increase this will give a fraction greater than 1 from which the percentage increase can be read; in the case of a decrease it will give a fraction less than 1. We look at examples of both. Increasing Example 1 Find the percentage increase when a population of a town increases from 150,000 to 200,000.

200,000  20  4  1.333.... 150,000 15 3

so the percentage increase is 33.3% to one

decimal place. Example 2 Find the percentage increase when the price of a game of squash is increased from 70p per session to 80p per session. 80  8 . Here the figures do not work out so 70

7

easily so it’s best to use a calculator for this one: 87 on a calculator gives 1.1428571 so the percentage increase is 14.3% correct to one decimal place. Example 3 If the policy at the Sports Centre is always to increase by units of 10p the next increase is likely to be to 90p. 90  9  1.125 (remembering that 12 1 % is an eighth) 80

8

2

so the percentage increase that time will be 12 1 %. 2

page 7

Decreasing Example 1 Find the percentage decrease when the population of a village falls from 1200 to 900. Here, following the given procedure, 900  9  3 so the population has been 1200

12

4

reduced by a quarter or 25%. Example 2 Find the percentage discount when the price of an article is reduced from £450 to £400. 400  8 so the reduction is 1  0.1111... (either with a calculator or by dividing) 450

9

9

so the reduction is 11.1% to one decimal place. Some people prefer to look at the amount of the change over the original rather than as presented here. There is nothing wrong with this in practice, particularly in the case of a decrease, because this gives the percentage decrease directly instead of requiring a further small calculation but, for the sake of consistency with the earlier section on expressing one thing as a percentage of another, we use the expedient of making a fraction of the given numbers in all cases. The eventual choice is up to you. Exercise 5. Find the % increase or decrease represented by changes from one to the other in each of the following pairs of numbers: 250, 275 275, 250 1251, 2139 2139, 1251 1256, 3016 3016, 1256 8. Some comments on commercial practice While in mathematics it is customary to make the starting amount 100% when dealing with increases in percentage, in the commercial world this is not the custom. There, since takings are the crucial amount rather than costs to the retailer, it is customary to make the selling price 100%. Otherwise the methods of calculation are the same. There are also some other different conventions in commerce and other real world situations so it is advisable always to check on the practice in particular circumstances before proceeding. Sustained practice of percentage examples and constant noting of percentages quoted in the media should bring you to a confidence about their use which you did not have before.

page 8

ANSWERS Exercise 1 37% 168.72 £207.20 867.65

456 £560 2345 Exercise 2 1 33.33% 300%

2 61.54% 162.5%

Exercise 3 1 (to 2 dec places) 2 (to nearest penny) Exercise 4 1 2 3 4

240 1188 2686 434

56% 255.36 £313.60 1313.2

3 71.43% 140%

5653.44 £423

Exercise 5 1 10% increase 2 9.09% decrease 3 70.98% increase

4 47.64% 209.92%

4964.88 £2791.80

120 594 1343 217

320 1584 3581.33 576.67

4 5 6

62% 282.72 £347.20 1453.9

76% 346.56 £425.60 1782.2

5 32.39% 308.76%

6233.28 £27.61

6668.16 £523.76

6595.68

244.8 1211.76 2739.72 442.68

41.51% decrease 140.13% increase 58.36% decrease

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 email at [email protected] The information in this leaflet can be made available in an alternative format on request using the email above. page 9