Year 7NumberNumber structures and operationsYear 7 Maths

Multiplying and dividing fractions and decimals

What this lesson covers

In the simplest form of a fraction, the numerator and denominator do not share a common factor.
Scaling by powers of 10 and using number facts supports multiplication with decimals.

Key idea

12 ÷ 1/4 = 48

Dividing just made the number bigger.
That is not a typo and it is not a mistake.
By the end of this it will be obvious.

Calculation	Answer
12 ÷ 4	3
12 ÷ 1/4	48

Key idea

Multiplying by ten shifts the digits

Every digit moves one column to the left.
Dividing by ten moves them one to the right.
The decimal point never actually moves.

Calculation	What happens	Result
3.5 × 10	each digit moves one column left	35
3.5 × 100	two columns left	350
3.5 ÷ 10	one column right	0.35
3.5 ÷ 100	two columns right	0.035

Key idea

Watch it in the columns

3.5 has a 3 in ones and a 5 in tenths.
× 10 and the 3 becomes tens, the 5 becomes ones.
Same digits, one column left. 35.

Key idea

One fact, four answers

If you know 7 × 5 = 35, you know all of these.
Work out the easy one, then fix the size.
0.7 × 50 is 35; the shifts cancel out.

Calculation	Reasoning	Answer
7 × 5	the fact you know	35
0.7 × 5	7 is ten times too big	3.5
0.7 × 50	7 down ten, 5 up ten	35
0.07 × 5	7 is a hundred times too big	0.35

Worked example

0.35 × 6

Ignore the point: 35 × 6 = 210.
0.35 has two decimal places.
Put two back: 2.10, which is 2.1.

Step	Working	Result
Ignore the point	35 × 6	210
Count the places	0.35 has two
Put them back	210 with two places	2.10
Tidy up	drop the trailing zero	2.1

Key idea

Check the size is right

0.35 is about a third. Six thirds is about two.
The answer was 2.1. That fits.
21 or 0.21 would not have.

Answer	Against "about 2"
2.1	right size
21	ten times too big
0.21	ten times too small

Key idea

A fraction of an amount

Three quarters of 20.
Divide by the bottom to get one quarter: 5.
Multiply by the top to get three of them: 15.

Step	Working	Result
3/4 of 20	divide by the bottom	20 ÷ 4 = 5
	multiply by the top	5 × 3 = 15
Answer		15

Key idea

The same thing as a picture

Twenty, cut into four equal parts of 5.
Take three of those parts.
15 out of 20.

Key idea

Times a fraction is the same thing

3/4 of 20 and 20 × 3/4 are the same question.
Both give 15.
And 15 is smaller than 20: multiplying shrank it.

Written	Answer
3/4 of 20	15
20 × 3/4	15

Key idea

Why it shrank

× 2 means take two of them.
× 3/4 means take three quarters of one.
Less than a whole one, so less than you started with.

Key idea

Always leave it in simplest form

An answer of 6/21 is correct but not finished.
6 and 21 both divide by 3.
So write 2/7.

Answer	Shared factor	Simplest
6/21	3	2/7
2/12	2	1/6
15/20	5	3/4

Worked example

3/4 ÷ 3

Three quarters, shared between three people.
One quarter each.
Dividing made it smaller, as usual.

Worked example

2/3 ÷ 4

Two thirds shared four ways.
Cut each third into four: twelfths. 2/12.
Simplify: 1/6.

Calculation	Working	Answer
3/4 ÷ 3	three quarters, shared three ways
Each share	one quarter	1/4
2/3 ÷ 4	2/12, then simplify	1/6

Key idea

Now the other kind

12 ÷ 1/4 is a different question entirely.
Not "share 12 between a quarter of a person".
It asks: how many quarters fit into 12?

Worked example

How many quarters in 12?

Each whole one holds four quarters.
There are twelve wholes.
12 × 4 = 48.

Step	Working	Answer
12 ÷ 1/4	how many quarters in 12?
Each whole	holds 4 quarters
12 wholes	12 × 4	48

Key idea

The pattern behind both

Multiplying by less than 1 shrinks.
Dividing by less than 1 grows.
The old rules were only true for numbers above 1.

Operation	Effect
× a number above 1	grows
× a number below 1	shrinks
÷ a number above 1	shrinks
÷ a number below 1	grows

Your turn

Your turn: four questions

Pause here and try all four.
On question 1, start from 6 × 4.
On question 4, ask how many thirds fit.

	Question
1	Work out 0.6 × 40.
2	Find 2/5 of 35.
3	Work out 6/7 ÷ 3 in simplest form.
4	Work out 5 ÷ 1/3.

Show answer

1. 24. Use 6 × 4 = 24: the 6 went down ten and the 4 went up ten, so they cancel. 2. 14. 35 ÷ 5 = 7, then 7 × 2 = 14. 3. 2/7. Six sevenths shared three ways is 6/21, and 6 and 21 share a 3. 4. 15. There are three thirds in every whole, and five wholes, so 5 × 3.

Worked example

The answers

0.6 × 40 is 24; the shifts cancel.
2/5 of 35 is 14. And 6/7 ÷ 3 is 2/7.
5 ÷ 1/3 is 15: three thirds in each of five wholes.

	Answer
1	24
2	14
3	2/7
4	15

Summary

What to take away

Do the easy whole-number fact, then fix the size.
A fraction of an amount: divide by the bottom, times the top.
Dividing by a fraction asks how many fit, so it grows.

	Remember
× 10, ÷ 10	the digits shift columns
Decimal × whole	do the easy fact, then place the point
Fraction of an amount	divide by the bottom, times the top
Fraction ÷ whole	the pieces get smaller
Whole ÷ unit fraction	how many fit; the answer grows

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