Year 8NumberNumber structures and operationsYear 8 Maths

Fractions, decimals and percentages

What this lesson covers

The product of two fractions can be found by multiplying the numerators and multiplying the denominators.
Percentages can be used to proportionally increase or decrease a quantity.

Key idea

One amount, written three ways

Three quarters. Nought point seven five. Seventy-five percent.
Those are not three related quantities.
They are one quantity in three notations.

Key idea

The ones to know on sight

A half, a quarter, three quarters, a fifth, three fifths.
These come up constantly, in all three forms.
Knowing them saves you converting the same five all year.

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
3/5	0.6	60%

Key idea

Fraction to decimal: just divide

A fraction bar is a division sign. That is all it ever was.
7/8 means 7 ÷ 8.
Which is 0.875. No rule to remember, just the division.

Going from	Do this	Example
Fraction to decimal	divide top by bottom	7 ÷ 8 = 0.875

Key idea

Decimal to percentage: times 100

Percent means "per hundred". Out of a hundred.
So 0.875 becomes 87.5 percent.
Going back the other way, divide by 100.

Going from	Do this	Example
Fraction to decimal	divide top by bottom	7 ÷ 8 = 0.875
Decimal to percentage	multiply by 100	0.875 = 87.5%
Percentage to decimal	divide by 100	87.5% = 0.875
Decimal to fraction	read the place value	0.875 = 875/1000

Key idea

Decimal to fraction: read the columns

0.875 is 875 thousandths, because the last digit is thousandths.
So it is 875/1000.
Then simplify. Both divide by 125, giving 7/8.

Key idea

When it does not come out evenly

2/3 means 2 ÷ 3, which never stops. 0.666 and on.
So the decimal and the percentage are rounded.
Only the fraction is exact. Sometimes it is the right form.

Form	Note	Value
2 ÷ 3	never stops	0.666...
As a decimal	rounded to 1 dp	0.7
As a percentage	rounded to 1 dp	66.7%
Exactly	only the fraction is exact	2/3

Worked example

Comparing across the three

Which is largest: 3/5, 0.56, or 58%?
You cannot compare them as they stand. Pick one form.
As decimals: 0.6, 0.56, 0.58. So 3/5 wins.

Value	Converted	As a decimal
3/5	as a decimal	0.6
0.56	already one	0.56
58%	as a decimal	0.58
Largest		3/5

Key idea

Multiplying by 10, 100, 1000

3.7 × 10 is 37. Every digit moves one column left.
× 100 moves them two columns. 3.7 becomes 370.
The number of zeros tells you how many columns.

Calculation	What happens	Answer
3.7 × 10	digits move 1 left	37
3.7 × 100	digits move 2 left	370
3.7 × 1000	digits move 3 left	3700

Key idea

And dividing goes the other way

45 ÷ 10 is 4.5. Digits move one column right.
45 ÷ 1000 moves them three. That gives 0.045.
Zeros appear to fill the columns nothing landed in.

Calculation	What happens	Answer
45 ÷ 10	digits move 1 right	4.5
45 ÷ 100	digits move 2 right	0.45
45 ÷ 1000	digits move 3 right	0.045

Key idea

The digits move, not the point

You may have been told to move the decimal point.
Same answers, but the columns are a fixed ladder.
The digits slide along it. The point marks the ones, always.

Key idea

Multiplying two fractions

Tops times tops. Bottoms times bottoms. That is the whole rule.
2/3 × 3/4: the tops give 6, the bottoms give 12.
So 6/12. No common denominator needed anywhere.

Part	Working	Gives
Numerators	2 × 3	6
Denominators	3 × 4	12
So		6/12

Key idea

But 6/12 is not the answer yet

6/12 is correct and unfinished.
Both numbers divide by 6, giving 1/2.
Simplest form is part of the question, not an extra.

Key idea

Why the answer got smaller

Multiplying by 3/4 means taking three quarters of something.
Three quarters of anything is less than you started with.
Multiplying only makes things bigger when you multiply by more than 1.

Worked example

A fraction of a whole number

2/5 of 20. Write the 20 as 20/1.
Then it is an ordinary fraction multiplication: 40/5.
Which is 8. "Of" has always meant multiply.

Step	Working	Gives
2/5 of 20	write 20 as 20/1	2/5 × 20/1
Multiply	2 × 20 over 5 × 1	40/5
Simplify	40 ÷ 5	8

Worked example

Mixed numbers: convert first

1 1/2 × 8. Do not multiply the 1 and the half separately.
Turn 1 1/2 into halves first: that is 3/2.
Then 3/2 × 8/1 is 24/2, which is 12.

Step	Working	Gives
1 1/2	two halves in the 1, plus 1	3/2
3/2 × 8	as 3/2 × 8/1	24/2
Simplify	24 ÷ 2	12

Key idea

Percentages that increase or decrease

Increase 40 by 25%: you end up with 125% of it.
So 40 × 1.25 = 50. One multiplication, not two steps.
Decrease 80 by 15%: you keep 85%. 80 × 0.85 = 68.

Change	What you end with	Working
Increase 40 by 25%	you end with 125%	40 × 1.25 = 50
Decrease 80 by 15%	you end with 85%	80 × 0.85 = 68
Either way	one multiplication, not two steps

Your turn

Your turn: four questions

Pause here and try all four.
Questions 1 and 2 both want simplest form. Check before you stop.
Question 4 has two good methods. Either is fine.

	Question
1	Work out 2/3 × 6/7, in simplest form.
2	Write 0.35 as a fraction in simplest form.
3	What is 2.4 × 1000?
4	What is 3/4 of 24?

Show answer

1. 4/7. Multiplying gives 12/21, and both divide by 3. Stopping at 12/21 is the commonest lost mark in the topic. 2. 7/20. 0.35 is 35 hundredths, so 35/100, and both divide by 5. 3. 2400. The digits move three places left. 4. 18. Either 3/4 × 24/1 = 72/4 = 18, or find a quarter of 24 first, which is 6, and take three of them.

Worked example

The answers

2/3 × 6/7 gives 12/21, which simplifies to 4/7.
0.35 is 35/100, which is 7/20. And 2.4 × 1000 is 2400.
3/4 of 24 is 18.

	Answer
1	4/7
2	7/20
3	2400
4	18

Summary

What to take away

One amount, three notations. Swap to whichever is easiest.
Multiplying fractions: tops times tops, bottoms times bottoms.
Then simplify, every time, before you call it done.

	Remember
One amount	three notations, swap freely
Fraction to decimal	divide the top by the bottom
Times 100	digits move two places left
Multiplying fractions	tops times tops, bottoms times bottoms
Always	simplify before you call it finished

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