Unlike the other punctuation marks, the semi-colon is very rarely necessary. Its role in language is to link two related sentences together. Generally speaking, it is used to make a text more varied or assist with the flow, but it isn’t necessary. Many accomplished writers will never use semi-colons because it isn’t part of their style, while others may use them frequently.

The first main use of a semi-colon is to join two related sentences together:

These two parts of the sentence could have formed separate sentences:

By joining them with a semi-colon, it’s clear that they relate to one another.
The second use of a semi-colon is to mark the components of a list:

Excluding the final point, each item on the bulleted list is suffixed with a semi-colon.

Question marks are used to signify that the sentence asks a question. Apart from this detail, it functions in the same way as a full stop, showing the end of a sentence:

Sometimes a sentence will imply a question, but will not contain a question mark. This happens when the subject of the sentence comes before the verb:

An exclamation mark is used to show commands or other statements which are either forceful or surprising. It is mostly used in dialogue to express a raised voice, but it can also appear in ordinary text as well. Like the question mark, it operates in the same way as a full stop: it signifies the end of a sentence.

Brackets are used to denote an aside or other additional information without disrupting the flow of the sentence. Usually these details aren’t absolutely vital, but are added for extra clarification. The contents inside the brackets are usually afterthoughts.

Brackets always occur in pairs. If they are opened, they need to be shut:

One of the most important skills that you’ll need is the ability to work with fractions, and understand how to simplify them. To simplify a fraction, you need to find the lowest possible denominator. In other words, you need to find a number which both the top number (numerator) and bottom number (denominator) will divide into whilst still being whole numbers. For example:

This fraction can be simplified because both 5 and 20 are divisible by 5:

However, not all fractions can be simplified so neatly. For example:

This cannot be simplified any further because 7 is a prime number. This means that it is only divisible by 1 and itself (7). 15 is divisible by 1, 3, 5 and itself (15). Since dividing by 1 won’t simplify the fraction any further, and there are no other numbers which both 7 and 15 are divisible by, this fraction is as simple as it can be. Simplification of fractions might not appear on its own as a question topic, but it will be vital for most questions which include fractions.

Many people find fractions to be rather tricky, especially when multiplying or dividing them. The easiest way to multiply fractions is as follows:

1. Multiply the top two numbers: 3 x 2=6

2. Multiply the bottom two numbers: 5 x 6=3

3. Insert the two new figures into the fraction:6/30

4. Simplify the fraction:6/30=1/5

= Simplify the fraction as much as possible: 630=15

Dividing fractions is slightly more complicated. The best way to divide two fractions is to swap the two digits of the second fraction around:

Then, turn the division operation into a multiplication, then carry on as though it were an ordinary fraction multiplication:

Currently, this is an improper fraction. It needs to be simplified:

For addition of fractions, you need to make sure that the denominator is the same. This may involve simplification. In this example, the first fraction has been multiplied by 6, and the second by 5. After that, the top two numbers are added together:

Then, all that’s needed is to simplify the final fraction:

For subtraction, the exact same process is followed, but by subtracting the top two numbers rather than adding them together.