1.4 Rounding decimals Using a calculator often gives a long string of digits. For example, 1 ÷ 3 might give .333333333. But very often, for practical purposes, this level o
1.3.1 Try some yourself 1 Round the numbers below: (a) to the nearest 10. (b) to the nearest 100. (c) to the nearest 1000. 325 089, 45 982, 11 985 1.3 Rounding in general Numbers are often approximated to make them easier to handle, but sometimes it doesn’t help very much to round to the nearest 10 or the nearest 100 if the number is very large. For example, suppose the monthly balance of payments deficit was actually £24 695 481. Rounded to the nearest 10, it's £24 695 480; and to the nearest 100, it's £24 695 500. But £24 695 500 is still a complicated number to deal with in your head. That's why it was rounded to £25 000 000 in the newspaper 1.2.1 Rounding to the nearest hundred You will probably think to yourself that the coat shown costs about £300. £290 is considerably closer to £300 than it is to £200, so £300 is a reasonable approximation. In this case, 290 has been rounded up to 300. Similarly, 208 would be rounded down to 200 because it is closer to 200 than it is to 300. Both numbers have been rounded to the nearest hundred pounds. When rounding to the nearest hundred, anything below fifty rounds down. So 248 rounds to 200. Anything o 1.1 Rounding in daily life The English mathematician Charles Babbage, father of modern computing, once wrote to Tennyson regarding one of his poems: ‘In your otherwise beautiful poem,’ Babbage wrote, ‘one verse reads, Every moment dies a man, Every moment one is born. ‘If this were true, the population of the world would be at a standstill. In truth, the rate of birth is slightly in excess of that of death. I would suggest: Learning outcomes By the end of this unit you should be able to: round a given whole number to the nearest 10, 100, 1000 and so on; round a decimal number to a given number of decimal places or significant figures; use rounded numbers to find rough estimates for calculations; use a calculator for decimal calculations involving +, −, × and ÷, giving your answer to a specified accuracy (e.g. decimal places or significant figures) and checking your an Introduction For many calculations you use a calculator. The main aim of this unit is to help you to do this in a sensible and fruitful way. Using a calculation to solve a problem involves four main stages: Stage 1: working out what calculation you want to do; Stage 2: working out roughly what size of answer to expect from your calculation; Stage 3: carrying out the calculation; Stage 4: interpreting the answer – Doe Acknowledgements The content acknowledged below is Proprietary (see terms and conditions). This content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence Grateful acknowledgement is made to the following sources for permission to reproduce material in this Unit: Ficure 2: Crown copyright 1.8: Summary This Unit has looked at a variety of ways of comparing prices, and the construction of a price index. Important statistical ideas that contributed to this included mean, weighted mean and median, as well as the general notion of an index. You now know quite a lot about the CPI, the RPI, and price indices in general, and so you should be able to explain what politicians and journalists really mean when they make sweeping statements about inflation and the cost of living. In the course of 1.7.4: Is a picture worth a thousand words? This final subsection is an overview of the various modes of mathematical communication used so far, like words, tables and graphs, and diagrams. You may have a preference for one over the others as a way of presenting ideas and of receiving information. However, they can all aid your understanding and communication of different mathematical ideas. So you need to develop your skills in using and interpreting all of them. Look back at Author(s): 1.7.3 What is proportion? A common criticism of many children's and some adults' drawings is that certain parts are not ‘in proportion’. That means that they are either too big or too small in relation to the rest of the masterpiece. ‘In proportion’ means being in the same ratio. Imagine that you have drawn a picture of the front of your house, reducing it in scale to one twentieth of its size. 1.7.2: Ratio and proportion It is easy to distinguish children from adults. For one thing, children are usually much smaller. But how are we able to tell them apart from a drawing alone? Have a look at the two outline drawings. Which one do you think represents the child and which the adult? 1.7: Some mathematical themes
Aims
The main aim of this section is to review some of the mathematical skills and ideas you have been using, and for you to reflect on some of their more general features and applications. 1.6: Using the price indices
Aims
In this section various uses of the RPI and CPI are discussed. The RPI and CPI are intended to help measure price changes. How they are used to do this is discussed in the audiotape band which follows.
Now listen to the audio clip below, called ‘Using the price indices’. 5.1: What are the CPI and RPI? The Consumer Prices Index (CPI) and the Retail Prices Index (RPI) are published each month by the UK Office for National Statistics. These are the main measures used in the UK to record changes in the level of the prices most people pay for the goods and services they buy. The RPI is intended to reflect the average spending pattern of the great majority of private households. Only two classes of private households are excluded, on the grounds that their spending patterns differ greatly from t 1.5: The UK Government price indices
Aims
The main aim of this section is to discuss what the UK Government price indices (CPI & RPI) measure and how they are calculated. How often have you read statements like these in the newspapers or heard them on the radio? Have you ever wondered how ‘infla 1.4.3 A price index for the shopping basket In the audio session, two methods of constructing a price index for bread were described. They were called the ‘previous year’ method and the ‘base year’ method. In both cases, the value of the index in the base year is 100. So, for the base year method, For the prev 1.4.1: Price ratios In Chapter 1, Section 1.4 of the Calculator Book, you saw that multiplying a price by, say, 1.30 is equivalent to increasing it by 30%. Similarly, multiplying a price by 0.94 is equivalent to decreasing it by 6%. The figures 1.30 and 0.94 are called price ratios. In Table 6, the price of a loaf of bread went up from 50p to 65p. The price 1.4: Price ratios and price indices
Aims
The main aim of this section is to look at some different ways of measuring price increases. In this section you will be looking at measuring price changes using price indices. In order to do this you will need to understand the concept of a price ratio. Price ratios are another way of looking at price increases or decreases, related to the proportional and percentage increases and decreases you have seen before. 1.3.6 Weighted mean The concise formula that you have just used is useful in itself for calculating a mean when you are given data in frequency form. But, even more useful, it can be extended, leading to the idea of a weighted mean, that has many applications, as you will see.
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Example 5: Assignment scores
