Multiplying and dividing by numbers between 0 and 1

Multiplying by a number between 0 and 1

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

For example:

2 \times 3 means 2 lots of 3 ( {=6})

6 \times 8 means 6 lots of 8 ( {=48})

So, {10}\times\frac{1}{2} means {10} lots of \frac{1}{2} ( {=5})

And {12}\times\frac{1}{3} means {12} lots of \frac{1}{3} ( {=4})

curriculum-key-fact
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: {10}\times\frac{1}{2}={5}

We also know that {10}\div{2}={5}

Therefore: {10}\times\frac{1}{2}={10}\div{2}

This shows that ' \times~\frac{1}{2}' is the same as ' \div~{2}'.

In general:

m \times \frac{1}{n} = m \div n

Examples

8 \times \frac{1}{4} = 8 \div 4 = 2 (as ' \times~\frac{1}{4}' is the same as ' \div~{4}')

20 \times \frac{1}{5} = 20 \div 5 = 4 (as ' \times~\frac{1}{5}' is the same as ' \div~{5}')

Dividing by a number between 0 and 1

Imagine that you had 10 bars of chocolate that you wanted to share amongst some children.

10 bars of chocolate

If you gave the children 2 bars each, you would have enough for 5 children.

10 \div 2 = 5

10 bars of chocolate divided by 2

If you gave the children \frac{1}{2} bar each, you would have enough for 20 children.

10 \div \frac{1}{2} = 20

10 bars of chocolate divided by 1/2

The pattern

Can you see what's happening?

10 \div 2 = 5

10 \div \frac{1}{2} = 20

When you divide by a whole number the answer is smaller than the original number. When you divide by \frac{1}{2} the answer ( {20}) is greater than the original number ( {10}).

curriculum-key-fact
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: 10 \div \frac{1}{2} = 20

Similarly: 10 \div \frac{1}{3} = 30 and 10 \div \frac{1}{4} = 40

In general: m \div \frac{1}{n} = mn

Question

Q1. What is 10 \div \frac{1}{7}?

Q2. Find the value of 4 \div \frac{1}{3}

A1. {10}\div\frac{1}{7}={10}\times{7}={70}

A2. 4 \div \frac{1}{3} = 4 \times 3 = 12

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