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7 Subtracting decimals by lining them up

Subtracting whole numbers such as 52 from 375 is fairly straightforward. Subtracting decimal numbers such as 6.892 from 223.6 uses the same process but with one extra step – you have to line the decimal points up first.

Rather than arranging your two numbers so that they line up on the right-hand side, you need to line up the decimal points, regardless of how many numbers there are after the decimal point. In the example below, the top number has one number after the decimal point. It
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11 Example of long division

The example of 25546 divided by 53 is suitable for long division. First write the calculation down on paper in the same way you did before.

Described image

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7 Dividing when you have to carry

If the number you are dividing by does not go exactly (with no remainder) into the digit you are dividing into, you need to do something called carrying.

Say you want to divide 952 by 7. The process is basically the same as in the previous section. First write it down on paper. Then, to do the calculation, you take each digit from the number being divided in turn, starting with the one on the far left, and see how many times the dividing number, 7 in this case, goes into it. The calcula
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4.5: The mode

The USA workforce data in Table 2 were usefully summarised in Figure 6, w
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Acknowledgements

Course image: Jörg Reuter in Flickr made available under Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence.

Grateful acknowledgement is made to the following sources for permission to reproduce material in this course:

The content acknowled
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5 Approaches to problem solving

You should not expect always to be able to read a problem and then just write down the answer. When you are faced with a written mathematical question or problem to solve, read it carefully. It is important that you get to grips with the question in two ways: first, that you absorb the information given; and second, that you find out what the question is really asking. Your solution will link the two. This method can be summarised by the following questions.

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2.2.1 Try some yourself

Activity 2

Here is a poor example of mathematical writing, although the final answer is correct. Rewrite it, correcting the layout and the mathematical punctuation.

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3.6.1 Try some yourself

Activity 33

The population of a village is 5481. Round this:

  • (a) to the nearest thousand people;

  • (b) to the nearest hundred people.


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3.4.1 Try some yourself

Activity 24

For each of the following calculations make suitable rough estimates before doing the calculation on your calculator and check the result.

  • (a) 22.12 ÷ 4.12


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Try some yourself

Activity 21

Without using your calculator solve the following calculations.

  • (a) 3 + 5 × 2 = ?

  • (b) 12 − 6 + 6 = ?

  • (c) 6 + (5 +
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2.1.1 Try some yourself

Activity 14

Measurement of a ceiling gives a length of 6.28 m and a width of 3.91 m.

  • (a) Make a rough estimate of the area of the ceiling (the length times the width).


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1.5 Significant figures

Sometimes it doesn’t make sense to round to a specific number of decimal places. If, say, you were calculating the cost of fencing at £10.65 per metre, for a garden boundary, the length of which had been given to you as 185 feet, then you would want to multiply 10.65 × 185 × 0.3048. (Conversion of feet to metres was given in Author(s): The Open University

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
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5.1 Identifying edges of a polygon

In this section, we revisit the construction of surfaces by identifying edges of polygons, as described in Section 2. Recall that, if we take any polygon in the plane and identify some of its edges in pairs, then we obtain a surface. When specifying how a given pair of edges is to be identified, we choose one of the two possible re
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2.1 Surfaces in space

In Section 2 we start by introducing surfaces informally, considering several familiar examples such as the sphere, cube and Möbius band. We also illustrate how surfaces can be constructed from a polygon by identifying edges. A more formal approach to surfaces is presented at the end of the section.

Figure 3 shows
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Modelling heat transfer

The main teaching text of this course is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click the link below to open the workbook (PDF, 0.4 MB).

workbook

Click the link below to open the answerbook (PDF, 0.2 MB).<
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1 Analysing skid marks

The main teaching text of this course is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click the link below to open the workbook (PDF, 0.2 MB).

workbook

Click the link below to open the answerbook (PDF, 0.1 MB).<
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Learning outcomes

After studying this course, you should be able to:

  • create simple models, given a clear statement of the problem

  • identify the simplifying assumptions that underpin a model

  • identify the key variables and the parameters of a model

  • apply the input–output principle to obtain a mathematical model, where appropriate.


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Try some yourself

1 Calculate the area of a carpet in a model house if the real carpet has an area of 22 m2. On the scale used, 1 cm represents 0.25 m.

Answer

(a) Since 1
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3.3 Volumes

What is a volume? The word usually refers to the amount of three-dimensional space that an object occupies. It is commonly measured in cubic centimetres (cm3) or cubic metres (m3).

A closely related idea is capacity; this is used to specify the volume of liquid or gas that a container can actually hold. You might refer to the volume of a brick and the capacity of a jug – but not vice versa. Note that a container with a particular volume will not nec
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