In several different parts of the world, footprints from prehistoric human civilisations have been found preserved in either sand or volcanic ash. From these tracks it is possible to measure the foot length and the length of the stride. These measurements can be used to estimate both the height of the person who made the footprint and also whether the person was walking or running by using the following three formulas:

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Now try the quiz, and see if there are any areas you need to work on.

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Discount can be calculated in the same way as an increase by a percentage. For example, Â£8 with 15% discount means you actually pay

Â Â Â£8 less (15% of Â£8)

Â Â 15% of 8 = Ã— 8 = Author(s): The Open University

• (a) How much will this tennis racquet cost if VAT at Author(s): The Open University

Our everyday experience of percentages includes percentage increases (like VAT at %, or a service charge of 15%) and percentage decreases (such as a discount of 15%).

For example, Â£8 plus Author(s): The Open University

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:

Â Â 0.17 Ã— 100% = 17%;

Â Â 0.3 Ã— 100% = 30%;

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1 Express each of the following percentages as fractions:

• (a) 40%

• (b) 8%

• (c) 70%

• (d)
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Percentages are used, particularly in newspaper articles, to indicate fractions (as in â€˜64% of the population votedâ€™) or to indicate changes (as in â€˜an increase of 4%â€™).

Percentages often indicate proportions. For example, labels in clothes indicate the various proportions of different yarns in the fabric. â€˜Per centâ€™ means â€˜per hundredâ€™ and is denoted by the symbol %. 100% is the same as the whole, or one hundred per hundred.

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1 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?

Â Â A 1 year
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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 proporti
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1 A recipe for four people calls for
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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
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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 Author(s): The Open University

1 Which is greater, 1.2 minutes or 70 seconds?

There are at least three ways of answering this:

• (a) 70 seconds is
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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:

Â Â Â 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

• Â
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1 Convert each of the following fraction ratios to decimal ratios.

• (a) Author(s): The Open University

1 A local supermarket sells a popular breakfast cereal in a â€˜Large Packâ€™ and â€˜New Extra Large Packâ€™. They are both being sold at â€˜knock downâ€™ prices. The large pack contains 450 g of cereal priced at Â£1.85. The new extra large pac
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Ratios crop up often in official statistics. The government wants the teacherâ€“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 no
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By the end of this unit you should be able to:

• work with simple ratios;

• convert between fractions, decimals and percentages;

• explain the meaning of ratio, proportion and percentage;

• find percentages of different quantities;

• calculate percentage increases and decreases;

• calculate average speeds in given units and find speeds, distances and times for travel at constant speed;

• convert units;
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• 1. Â Â  The relation â€˜is equal toâ€™ on is reflexive, symmetric and transitive.

• It is reflexive since, for all x
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