The topics in this free course, Ratio, proportion and percentages, are concerned with dividing something into parts. For example, if there are 200 people living in a small village, and 50 of these are children, this could be expressed as a percentage: 25% of the village population are children; or as a ratio: one in every four people is a child or there is 1 child for every three adults; or a proportion: the proportion of children in the village population is a quarter. This OpenLearn course provides a sample of level 1 study in Mathematics. After studying this course, you should be able to:
Ratios crop up often in official statistics. The government wants the teacher–pupil ratio in schools to be increased to one teacher to thirty pupils or less. The birth rate has fallen: the ratio of children to women of child bearing age has gone down. It used to be 2.4 to 1, and now it is 1.9 to 1. Predictions for the ratio of working adults to retired adults is disturbing. Predictions are, that by 2030 the ratio will be two working adults to every retired person, instead of three to one now, and four to one ten years ago. Often ratios are implicit in the language rather than explicitly referred to: one teacher for 30 pupils; 2.4 children per woman of child bearing age; one retired person per two working adults. The word ‘per’ often indicates that the concept of ratio is being used. To make short crust pastry, one recipe book says ‘use one part of fat to two parts of flour’; another recipe says ‘use fat and flour in the ratio of one to two’; and yet another says ‘use half as much fat as flour’. These are different ways of expressing the same ratio. Ratios are often expressed as fractions. So in this case:  Since you can multiply top and bottom of a fraction by the same number and get an equivalent fraction, you can use the ratio in a number of ways. If you have 100 grams of fat then So you need 200 grams of flour to 100 grams of fat. There are many ways to arrive at this answer. The important point is that a ratio of 100 to 200 is equivalent to 1 to 2. To make concrete, the instructions are ‘use sand and cement in the ratio three to one’. This means If you have 30 kg of cement, then you need 90 kg of sand. The conversion rates between currencies or different units are often easier to remember as ratios. Many people remember that the ratio of distance in miles to the same distance in kilometres is five to eight. At the time of writing the ratio of prices in pounds sterling to prices in euros is two to three (2 : 3). What is the equivalent price in pounds for a coat costing 150 euros? 3 euros are equivalent to 2 pounds. This means that So price in pounds = 2/3 × price in euros = 2/3 × 150 = 100 The equivalent price of the coat is £100. A friend is painting the inside walls of a garage. So far she has used a 2 litre tin of emulsion paint and covered an area of 9 m2. She needs some more paint. How much more would you advise her to purchase if she intends to paint all the walls and ceiling, which is a total area of 75 m2. An area of 9 m2 requires 2 litres of paint. The ratio of paint to area is 2 to 9 or . So an area of 1 m2 would require of a litre of paint. She needs to paint an area of (75 − 9) m2 or 66 m2. This will require If the ratio of distance measured in miles to the same distance measured in kilometres is five to eight, which is the higher speed limit, 70 miles per hour or 110 kilometres per hour? 70 is 5 × 14, so 70 miles is about the same as 112 kilometres. Alternatively 1km = miles. So 110 km = 110 × miles = 68.75 miles. So the two speed limits are very close but with 70 miles per hour being the higher. Time conversions are also ratios. The ratio of time measured in minutes to time measured in seconds is one to sixty (1:60), as there are sixty seconds in a minute. Adam's grandfather ran a mile in
So When shopping for a bargain, the ratio of price to quantity is often a useful way of comparing prices of different sized packets. A local shop sells ready-made custard at £1.45 for a special offer pack of three 425 g tins. It also sells the same brand of custard in 1 kg cartons costing £1.29 each. Which is the better bargain? To compare the prices it would be best to compare the ratio of prices to amounts (measuring amounts in the same units) i.e. prices per kg. Three 425 g tins will contain 3 × 425 g = 1275 g or 1.275 kg. £1.45 for 1.275 kg is
The 1 kg carton costs £1.29 per kg. The three tins are 15 p cheaper per kg. So the three-tin pack is the better bargain. A local supermarket sells a popular breakfast cereal in a ‘Large Pack’ and ‘New Extra Large Pack’. They are both being sold at ‘knock down’ prices. The large pack contains 450 g of cereal priced at £1.85. The new extra large pack contains 625 g and is priced at £2.35. Which is the better bargain? In order to compare the prices of the two cereal packs it is best to work out the price per gram for each. The large packet will cost £1.85 for 450 g. This is The New extra large packet will cost £2.35 for 625 g. This is (But the extra large package is 37 cm tall (about 13˝). Unluckily it might not fit in the kitchen cupboard.) Baking potatoes are priced at 75 p for a pack of three (very similar) potatoes or at £2.70 for a 5 kg bag. How heavy does each of the three potatoes in the pack have to be in order for the pack to be a better bargain than the 5 kg bag? Does your answer seem reasonable? (Try to imagine a potato of this size.) 5 kg costs £2.70. You want to know, at this rate, how much you get for 75 p. First work out how much you get for £1: if £2.70 buys 5 kg then £1 buys So 75 p should buy 0.75 × If three potatoes weigh 1.39 kg each, one weighs about 0.46 kg. So each potato in the pack would need to weigh around half a kilogram for the pack to be a better bargain. Baking potatoes are big, but it's unlikely that a pack of three baking potatoes would weigh Although ratios are often given as fractions, they can also be expressed as decimals. You need to deal with a mixture of fractions and decimals, and to compare ratios given in either form, so you need to be able to convert between the two forms. The ratio of the circumference of a circle to its diameter is a constant denoted by (the Greek letter p) pronounced pi, it has been approximated by a number of different fractions. One such fraction is If you have a calculator handy then you could key in 22 ÷ 7 to convert to a decimal. However, if not, you might use long division or an informal method of division. Either way you should get 3.1428 …. So You will probably find it easier to use a calculator for dividing 355 by 113. The result is 3.14159292, which agrees to 7 significant figures. Sometimes you are given a ratio as a decimal, but might find it easier to use and/or remember as a fraction. Suppose you had been told that the ratio between a distance measured in miles and the same distance measured in kilometres is about 0.625. Convert this to a fraction. This fraction is an adequate answer, or you can divide top and bottom by common factors to reach a simpler equivalent fraction. Divide top and bottom by 5 to get Divide top and bottom by 5 again to get Divide top and bottom by 5 again to get . So the fraction is , which is the fraction used earlier. An alternative method is do this in one line cancelling Convert each of the following fraction ratios to decimal ratios.
Convert each of the following decimal ratios to fraction ratios.
A recipe for a casserole involves soaking dried beans. The beans require You need to multiply Follow-up in Section 3.1.2 Alternatively 2.5 × 1.75 = 4.375. So Speed is the ratio of distance travelled to time taken. A runner's speed may be quoted in metres per second, miles per hour or kilometres per hour. The units are given as: unit of distance per unit of time. When you have a distance covered (such as a mile) and a time taken (such as four minutes) the average speed is defined as The formula for average speed applies even over a journey made up of several stages. In 1999, Hicham El Guerrouj held the record for running a mile. He covered the distance in just over 3 minutes 43 seconds. In the same year he also held the world record for the 1500 m race. He completed this distance in 3 minutes 26 seconds. Work out:
Which is greater, 1.2 minutes or 70 seconds? There are at least three ways of answering this:
Answer the following questions:
Which is longer, 11 minutes or 0.17 hours? As a fraction of an hour, 11 minutes is 11/60 hours. To convert this to a decimal, divide 11 by 60 to get 0.18333…. This is greater than 0.17, so 11 minutes is longer. (Alternatively, 0.17 hours is 60 × 0.17 = 10.2 minutes, so 11 minutes is longer.) A van driver averages 50 km per hour travelling on ordinary roads and 70 km per hour on motorways. Estimate:
when travelling on (a) ordinary roads and (b) motorways.
Proportion is another way of expressing notions of part and whole. You might say that the proportion of village inhabitants who are children is a quarter, or that the proportion of fruit juice in the punch is two thirds, or that the proportion of sand in the concrete is three quarters. All these examples involve the fractions , , . Problems involving proportions are best handled by manipulating fractions, generally, by division or multiplication. The task is to decide which fraction to manipulate, in what way, and at what stage! In a recipe the quantity of each ingredient needed depends upon the number of portions. As the number of portions increases, the quantity required increases. The quantity per portion is the same. This is called direct proportion. The quantity is said to be directly proportional to the number of portions. If 2 potatoes are required for one portion, 4 will be required for two portions etc. A useful method for direct proportion problems is to find the quantity for one and multiply by the number you want. John lives with three cats. His daughter asks him to look after her cat for a week while she goes away. John normally buys two tins of cat food a day for the three cats. How many tins should he buy for the four cats for a week? There are several ways to do this. Here is one. 3 cats eat 2 tins a day 1 cat eats tin a day 4 cats eat 4 cats eat So John should buy 19 tins for the week. Note it helped to simplify the problems by considering 1 cat, rather than going straight to 4 cats. Debbie is checking her phone bill. Her mobile phone calls have all been charged at the same rate, 30 pence per minute. (Call charges are rounded to the nearest penny and charged to the nearest second.)
(a) 1 minute cost 30 p. (i.e. 60 seconds for 30 p).
(b) 0.5 p will allow her to talk for 1 second.
A recipe for four people calls for teaspoon of mustard powder. How much should you use for ten people? For one person you need For ten people you need So almost 2 teaspoons of mustard are needed. The length of time it takes to cook a Christmas pudding in a pressure cooker depends on the weight of the pudding. Kim has forgotten the time per pound but remembers that her How long will it take to cook a 2 kg pudding? If it takes 5 hours to cook So a 2 kg pudding takes In Section 2.2 you saw that direct proportion described relationships between two quantities, where as one increased, so did the other. Sometimes as one quantity increases the other decreases instead of increasing. This is called indirect proportion. Team tasks are often an example of this. The time taken to do a job is indirectly proportional to the number of people in the team. A difficulty with the real-life context of such problems is that, in many cases, it is hard to believe that people working in a team will work at the same rate regardless of the size of the team, unless the team work independently, i.e. ‘in parallel’. The main idea behind this type of problem is that increasing the number of people working decreases the time taken to complete the task. (An obvious exception to this is decision-making in a committee: if two people can reach a decision in an hour, four people are liable to take twice as long!) Such problems can be compared with certain problems involving speed: doubling the number of people working is the same as doubling the speed at which the team work. In either case the time is halved. It is useful to find out how long it would take one person to do the whole job, then divide by the number of people sharing the work. This is a good approach to most indirect proportion problems. A team of five people can deliver leaflets to every house on a housing estate in three hours. How long will it take a team of just two people? Take the same approach as in Examples 7 and 8, and first work out how long it would take one person to deliver leaflets to the estate. It will take one person five times as long as a team of five people. (If you find this hard to accept, imagine that the estate consists of five streets, and that each person delivers leaflets to one of these streets in the three hours.) So each street takes 3 hours to leaflet. It would take one person 5 × 3 hours to leaflet all five. So it takes one person 15 hours to deliver leaflets to the whole estate. Two people will take half this time, so two people take (As a check: you would expect two people to take longer than five.) A piece of computer software is to be developed by a team of programmers. It is estimated that a team of four people would take a year. Which of the following times is the length of time taken by three programmers? A 1 year 3 months B 9 months C 1 year 4 months C: 4 programmers take 1 year. So 1 programmer would take 4 years. 3 programmers would take years = A 10 kg bag of potatoes lasts for a week when used in catering for 7 people.
Two workers in the Open University warehouse take 20 minutes to stick labels on 500 packages for an MU120 mailing. There are still 4000 more packages. How many workers are required, if the job is to be done in about a further hour? One worker should take twice as long as two. So 1 worker takes 40 mins for 500 labels. 4000 is 8 times 500 so 1 worker takes 40 × 8 min = 320 mins. 1 hour is 60 mins. So you need Practically, you could have 5 workers for Percentages are used, particularly in newspaper articles, to indicate fractions (as in ‘64% of the population voted’) or to indicate changes (as in ‘an increase of 4%’). Percentages often indicate proportions. For example, labels in clothes indicate the various proportions of different yarns in the fabric. ‘Per cent’ means ‘per hundred’ and is denoted by the symbol %. 100% is the same as the whole, or one hundred per hundred. 100% cotton indicates that the fabric is made entirely from cotton. (100 parts out of 100 parts). 60% cotton means that 40% polyester means that The percentages on the label should total 100%, just as the corresponding fractions add up to 1, because the total (100%) refers to the whole garment. 60% + 40% = 100%, 0.6 + 0.4 = 1. Percentages can also be manipulated as either fractions or decimals. A building society offers 90% mortgages to first-time buyers. How much would the Smiths get on a house valued at £150 000? They want to find 90% of £150 000. 90% = 0.9 × 150 000 = 135 000 So the Smiths would get £135 000. Express each of the following percentages as fractions:
Express each of the following percentages as decimals:
A survey was carried out on 840 couples to investigate family income. 75% of the women were in paid work; 90% of the men were in paid work; 18% of the couples had a joint income of less than £160 per week.
Fractions and decimals can also be converted to percentages, by multiplying by 100%. So, for example, 0.17, 0.3 and can be expressed as percentages as follows: 0.17 × 100% = 17%; 0.3 × 100% = 30%; Decimals or fractions bigger than 1 correspond to percentages greater than 100%. For example, Care needs to be taken when talking in percentages. A percentage is a percentage of a given quantity: 50% of the voters, 25% of the budget, 10% of the population. In newspapers, it is not always clear what quoted percentages are percentages of. Politicians too can give misleading statements. ‘We are giving an increase in funding of 10% and 5% of this has no strings attached’. In this statement it is not clear whether the 5% is 5% of the original funding or 5% of the 10% increase in funding (which would be To avoid this confusion some people use the term percentage points, when they are comparing the percentages of different quantities. In elections the percentage of the vote at one election for a given party is often compared with the percentage of the vote at the previous election. The difference is referred to as the ‘swing’ and expressed in percentage points. In a local election 2540 votes are cast for the Purple party out of a total of 5000 votes. At the previous election 42.1% of votes had been for the Purple party. What is the swing to the Purple party in percentage points? 2540 out of 5000 as a percentage is Swing = 50.8 − 42.1 = 8.7 percentage points. Convert each of the following to percentages. Round off the percentages to whole numbers.
In the election in Example 11 the Average party gets 2100 votes. In the previous election they had 10%. Find the swing. 2100 out of 5000 is
The swing is 42 − 10 = 32 percentage points. Our everyday experience of percentages includes percentage increases (like VAT at For example, £8 plus £8 + (
so the VAT is £1.40, and the sum you pay is £8 + £1.40 = £9.40. There is another way of doing this calculation: 100% of £8 plus that is 117.5% of £8. 117.5% of £8 = £(1.175 × 8) = £9.40 (as before). A restaurant bill comes to £76 before VAT and the service charge are added. VAT is added at Adding VAT first gives 100% + Adding the service charge on to £89.30 gives 100% + 15% = 115% 115% of £89.30 = 1.15 × £89.30 = £102.70 (correct to the nearest penny). So the total bill is £102.70. Adding these extras the other way round: 115% of £76 = 1.15 × £76 = £87.40; = £102.70 (correct to the nearest penny). In the first case 1.15 × 1.175 × £76, and in the second 1.175 × 1.15 × £76. The order doesn’t matter in multiplication: 1.175 × 1.15 = 1.15 × 1.175. The total bill is the same. The order does not matter from the customer's point of view. However, it does from the VAT collector's point of view (and indeed from that of the restaurant management, who receive a bigger service charge if it is calculated last). If VAT is added first, VAT is Answer the following questions
A commuter pays £1260 for a season ticket. The train company announces an increase of 7.4% on all its fares. How much will the season ticket cost after the increase? If increase on fares is 7.4% then you would need to find (100% + 7.4%) i.e. 107.4% of £1260. 1.074 × 1260 = 1353.24. So the season ticket will cost £1353.24 after the increase. Discount can be calculated in the same way as an increase by a percentage. For example, £8 with 15% discount means you actually pay £8 less (15% of £8) 15% of 8 = £8 − £1.20 = £6.80 Alternatively the actual amount can be calculated in another way, as follows: (100% − 15%) of £8 is 85% of £8. 85% of £8 = £(0.85 × 8) = £6.80 (as before).
A new train operator boasts ‘Train times reduced by 12%’. Decrease 90 minutes by 12%. Give your answer as minutes and seconds. A 12% decrease would reduce it to 88% of the original time, i.e. 0.88 × 90 = 79.2 minutes. 0.2 of a minute = 0.2 × 60 = 12 secs. So 79.2 minutes = 79 minutes and 12 secs. The population of a small town is 4650. It is predicted that the population will decrease by approximately 4% each year. What is the population likely to be after (a) one year, (b) two years?
In lots of everyday situations percentages are used to make predictions and comparisons. The number of casualties handled by the outpatients department of a hospital increases by approximately 8% per year. The number of casualties this year was 1920. Make a prediction for the number of casualties handled (a) next year, (b) in two years' time.
Answer the following questions
In an election 1248 votes out of 3000 are cast for the Gold party. What is this as a percentage? If in the previous election 35.1% voted Gold, what is the swing to Gold in percentage points?
The swing is 41.6 − 35.1 = 6.5 percentage points. A quote from a builder for some home improvements is £2200 plus VAT.
An airline decides to increase its fares from London to Europe by 17% and its fares to North America by 15%. Before the increase a special offer European fare from London to Geneva was £165 and the North American fare from London to New York was £376. What would the new fares be for these trips after the increases? The London to Geneva fare is increased by 17%. So it will now cost 117% of £165, i.e. 1.17 × £165 = £193.05. The London to New York fare is increased by 15%. So it will now cost 115% of £376, i.e. 1.15 × £376 = £432.40. Now try the quiz, and see if there are any areas you need to work on. This free course provided an introduction to studying Mathematics. It took you through a series of exercises designed to develop your approach to study and learning at a distance and helped to improve your confidence as an independent learner. There are more than 800 courses on OpenLearn for you to choose from on a range of subjects. Find out more about all our free courses. Find out more about studying with The Open University by visiting our online prospectus. If you are new to university study, you may be interested in our Access Courses or Certificates. Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence Course image: Tom Sandars in Flickr made available under Creative Commons Attribution 2.0 Licence. All written material contained within this course originated at The Open University. Don't miss out: If reading this text has inspired you to learn more, you may be interested in joining the millions of people who discover our free learning resources and qualifications by visiting The Open University - www.open.edu/ openlearn/ free-courses |
The ratios of the time taken by three runners to cover a particular distance is 3 6 9
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