It is useful for you to be able to multiply and divide without using a calculator. Learn how to do this using short division and short multiplication.

The multiplication sign can be replaced by the phrase 'lots of'.

For example:

means lots of ( )

means lots of ( )

So, means lots of ( )

And means lots of ( )

When you multiply by a number greater than 1, the answer is greater than the original number. But when you multiply by a number between 0 and 1, the answer is smaller than the original number.

We have seen that:

We also know that

Therefore:

This shows that ' ' is the same as ' '.

In general:

(as ' ' is the same as ' ')

(as ' ' is the same as ' ')

Imagine that you had bars of chocolate that you wanted to share amongst some children.

If you gave the children bars each, you would have enough for children.

If you gave the children bar each, you would have enough for children.

Can you see what's happening?

When you divide by a whole number the answer is smaller than the original number. When you divide by the answer ( ) is **greater than** the original number ( ).

When you divide by a number greater than 1, the answer is smaller than the original number. But when you divide by a number between 0 and 1, the answer is greater than the original number.

So:

Similarly: and

In general:

- Question
Q1. What is ?

Q2. Find the value of

A1.

A2.