CHAPTER 2 Arithmetic

Learning outcomes

• Understand the broad principles of arithmetic

• Understand what is meant by fractions and decimals

• Be able to simplify fractions; change fractions to decimals; change decimals to fractions

• Understand and be able to use proportion

• Understand the concept of percentages

Numerical skills

Numbers are an essential part of everyday life. When shopping at the supermarket for milk or fruit juice, for example, how do we know the best value for money between containers of different volumes? In order to accurately calculate drug doses and interpret data in healthcare situations, it is necessary to have certain arithmetical skills. On most occasions, the skills required are not complex; in fact, they are relatively straightforward. The skills routinely needed, either singly or in combination, are as follows:

• Adding*

• Subtracting*

• Multiplying*

• Dividing*

• Simplifying fractions

• Changing fractions to decimals

• Changing decimals to fractions

• Using proportion

• Using percentages

Basic arithmetical principles

Numbers

• All numbers are either even numbers or odd numbers

• Even numbers (2, 4, 6, 8, etc.) can all be divided by 2 to leave a whole number

• Odd numbers (1, 3, 5, 7, etc.) cannot be divided by 2 to leave a whole number

• If the individual digits of a number add up to 3 or a multiple of 3, it can be divided by 3, e.g. 12, 651, 2121

• Any number ending in either a 5 or a zero can be divided by 5

• If the individual digits of a number add up to 9 or a multiple of 9, it can be divided by 9, e.g. 81, 7281

Expressing figures

When undertaking any type of calculation, there may be times when you need to express a number in a way that will assist you. Take any number, let’s say 6.

The figure 6 can be expressed as without altering the fact that it is 6.

The figure 6 can also be expressed as 6.000 without altering the fact that it is 6. Any number of zeros can be placed after the decimal point; the number will still be valued as 6.

Great care is needed when working with very large numbers (e.g. 100 000; 250 000). It is no longer common to see in print the use of commas to signify thousands (e.g. 100,000; 250,000). It is advisable, however, to use commas when working on paper to ensure you have the right number of zeros.

Explaining fractions and decimals

The word fraction means a ‘fragment of something’.

The word decimal means ‘based on the number ten’.

Fractions may be expressed in two ways:

• as a vulgar fraction (e.g. , often simply referred to as a ‘fraction’)

• as a decimal fraction (e.g. 0.6, often simply referred to as a ‘decimal’)

‘Fractions’ and ‘decimals’ are two ways of expressing a part of something. There will of course be times when the fraction or the decimal will be added to a whole number, e.g. , 1.25.

Fractions

‘Fractions’ are commonly used in everyday life, e.g.,

• the class

• of an apple

• working days (whole number plus a fraction)

but are less often used for precise measurement.

A fraction for our purposes is formed when a number is placed above another number and the two numbers are separated by a dividing line, e.g.,

The number above the line is known as the numerator; the number below the line is the denominator. If the two numbers are the same, they will cancel each other out to become one. So,

If the number above the line is less than the number below the line, the fraction is always less than one. So,

If the number above the line is more than the number below the line, the fraction must be more than one. So,

Decimals

‘Decimals’ are used in the metric system (see p. 10) and are measured in tenths, e.g.,

• 28.3 g jar of marmalade ( g)

• 304.8 mm length of wood ( mm)

• 3.6 kg baby ( kg)

The decimal point is placed before the decimal fraction.

Quantities that are less than one should normally be written as a whole number of the next unit down, e.g.,

• 750 mg rather than 0.75 g

• 125 micrograms and not 0.125 mg

Never start a figure with a decimal point. For example, always write 0.6 and not .6. In this case, no doubt you can detect the decimal point but there may be times when the ‘decimal point’ is a blob of ink on the page!

When a fraction is expressed in decimal form:

• Any number starting with ‘0.’ must be less than one (e.g. 0.6)

• The quantity one is expressed as 1

• Any number more than one is either a whole number or a whole number plus a decimal fraction (e.g. 1.6)

These figures can be illustrated on a scale, as in Fig. 2.1.

Figure 2.1 Decimal scale

Simplifying fractions (cancelling down)

When using fractions, it is always best to simplify them as far as possible, since it is easier (and safer) to work with smaller figures than with bigger ones. For example, could be immediately simplified by removing the zeros (dividing by 10) to give , making sure of course to delete the same number of zeros above and below the line. The fraction could then be simplified by dividing the numbers above and below the line by 25 (or in two stages by 5) to give . Whatever you do above the line, you must do the same below the line. Also, simplifying a fraction does not change its value. (fifty over a hundred) has exactly the same value as ½ (one over two).

Some exercises are given throughout the next piece of text. Answers are on pp. 193–194.

Exercise 2.1

Here are some fractions for you to simplify:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Taking another example of a calculation which involves a fraction,

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