# Mathematics for the RAE

The basics

The basic mathematical processes are: addition, subtraction, multiplication and division. As long as only 'whole numbers' are involved, such sums are simple.

However, very often we must consider quantities which are which are less than one (unity), for instance , , etc. Here means one‑eighth part of the whole and so on.

is called a vulgar fraction and has two parts: the ‘8’ (the bottom part) is called the 'denominator' and the ‘1' (the top part) is called the 'numerator'. The magnitude of a fraction is not changed if we multiply top and bottom by the same number, ie

As the ‘2’ is on the top and bottom we can 'cancel' it thus:

A fraction should always be cancelled down to its simplest form:

Here top and bottom have been divided by 4.

Fractions can be

(a) Multiplied

(b) Divided

Dividing by 1/2 is the same as multiplying by 2/1, ie

In other words, dividing by a fraction is the same as multiplying by that fraction 'upside down'. Another example is:

Here we divide top and bottom by 4.

(c) Added

If the denominators are different, we must make them the same, ie 'bring them to a common denominator' and normally the lowest common denominator is used. For example

Here we have multiplied top and bottom of 2/3 by 2, making it 4/6. Hence we can add it to 5/6, making 9/6, which is then simplified to 1½ . Another example is

It is generally preferable to divide out fractions greater than one, as we have done above.

(d) Subtracted

Exactly the same rules apply to the subtraction of fractions.

We can also express parts of the whole as 'decimals' or 1/10 parts, written as 0.1, 0.2, 0.3 etc (these are equivalent to 1/10, 2/10, 3/10 etc). The 'full stop' is known as the 'decimal point'. In a decimal, the 'nought' before the decimal point should never be omitted.

The denominator of any fraction can be divided into the numerator to give a decimal, eg

1/8 = 0. 125

3/8 = 0.375

The more common fractions and decimal equivalents should be memorised, eg

1/10 = 0.1 1/8 = 0. 125

2/10 = 1/5 = 0.2 2/8 = 1/4 = 0.25

3/10 = 0.3 3/8 = 0.375

4/10 = 2/5 = 0.4 etc. 4/8 = 1/2 = 0.5 etc.

Numbers can be expressed to 'so many significant figures' or 'so many decimal places'.

Thus 12345 is a number to five significant figures

1234 is a number to four significant figures

123 is a number to three significant figures

Note also that 1.23 is a number to three significant figures (the decimal point is ignored).

12.345 is a number to three decimal places

12.34 is a number to two decimal places

12.3 is a number to one decimal place

Decimals may be 'rounded off'; that means

3.3267 to three decimal places is 3.327 (the 7 is greater than 5, so 6 becomes 7)

3.327 to two decimal places is 3.33 (the 7 is greater than 5, so 2 becomes 3)

3.33 to one decimal place is 3.3 (the 3 is less than 5, so is ignored)

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