Year 8NumberNumber structures and operationsYear 8 Maths

Decimals, fractions of amounts, and working backwards

What this lesson covers

Ratios can be used to describe proportional relationships and unequal division of a whole.

Key idea

Before you multiply, trap it

2.3 × 45. Do not reach for the method yet.
2 × 45 is 90. 3 × 45 is 135.
2.3 sits between 2 and 3, so the answer sits between those.

Benchmark	Compared	Value
2 × 45	too small	90
3 × 45	too big	135
So 2.3 × 45	lands between	90 to 135

Worked example

Now the actual multiplication

2.3 is just 23 divided by 10.
So do 23 × 45 first, which is 1035.
Then divide by 10, giving 103.5. Inside our bracket.

Step	Working	Gives
2.3 is	23 ÷ 10
So 2.3 × 45 is	23 × 45, then ÷ 10
23 × 45		1035
÷ 10		103.5

Key idea

Which is where the shortcut comes from

Count the decimal places in the question.
Put the same number into the answer.
It is not a separate rule. It is that division, tidied up.

Calculation	Places	Answer
2.3 × 45	1 decimal place in the question	103.5
1.5 × 2.4	2 decimal places	3.6

Worked example

The one that looks wrong

0.4 × 0.2. The digits give 4 × 2 = 8.
Two decimal places in the question, so two in the answer.
0.08, not 0.8. Two numbers under 1 multiply to something smaller.

Calculation	Places	Answer
2.3 × 45	1 decimal place in the question	103.5
1.5 × 2.4	2 decimal places	3.6
0.4 × 0.2	2 decimal places	0.08

Worked example

A fraction of a whole number

2/5 of 42. Write the 42 as 42/1 and multiply.
2 × 42 over 5 gives 84/5.
That is an improper fraction, and it is a correct answer.

Step	Working	Gives
2/5 of 42	write 42 as 42/1	2/5 × 42/1
Multiply	2 × 42 over 5	84/5

Key idea

Tidying it into a mixed number

84/5 means 84 divided by 5.
5 goes into 84 sixteen times, with 4 left over.
So 16 4/5. Or 16.8, if the question wants a decimal.

Step	Working	Gives
2/5 of 42	write 42 as 42/1	2/5 × 42/1
Multiply	2 × 42 over 5	84/5
As a mixed number	84 ÷ 5 is 16, r 4	16 4/5

Worked example

Percentages, built from ten percent

35% of 180. Start with 10%, which is 18.
Three lots gives 30%, which is 54.
5% is half of 10%, so 9. Together, 63.

Step	Working	Gives
10% of 180	180 ÷ 10	18
30% is	three lots	54
5% is	half of 10%	9
35% is	54 + 9	63

Key idea

Or in a single multiplication

35% is 0.35 as a decimal.
180 × 0.35 is 63. Same answer, one step.
Use this with a calculator. Use the other to check it.

Step	Working	Gives
35% as a decimal	35 ÷ 100	0.35
180 × 0.35		63
Same answer	one step instead of three

Worked example

A quick one, no calculator

15% of 60. A tenth of 60 is 6.
Half of that is 3, which is the 5%.
6 + 3 = 9. Faster than typing it.

Key idea

Everything so far went one way

You were given the whole and asked for a part.
2/5 of 42. 35% of 180. 15% of 60.
Now we turn it round, and this is the hard bit.

Given	Asked for	Direction
the whole	a part of it	forwards
a part	the whole	backwards

Key idea

One question before any method

8 is three fifths of a set. How big is the set?
Do not work it out. Just answer this: bigger or smaller than 8?
Bigger. Three fifths is not the whole thing.

Worked example

Multiply by the fraction, upside down

Going forwards you multiply by 3/5.
Going backwards you multiply by 5/3.
8 × 5/3 is 40/3, which is 13 1/3.

Step	Working	Gives
8 is 3/5 of it	so multiply by the flip	8 × 5/3
Multiply	8 × 5 over 3	40/3
As a mixed number	40 ÷ 3 is 13, r 1	13 1/3

Key idea

And check it, in one line

13 1/3 is bigger than 8, as we said it had to be.
Take three fifths of 40/3 and you get 8 back.
Reverse problems always check themselves. Always do it.

Step	Working	Gives
8 is 3/5 of it	so multiply by the flip	8 × 5/3
Multiply	8 × 5 over 3	40/3
As a mixed number	40 ÷ 3 is 13, r 1	13 1/3
Check	3/5 of 40/3	8

Key idea

Why turning it over works

8 is three fifths, so one fifth is 8 ÷ 3.
The whole is five of those: 8 ÷ 3 × 5.
Which is exactly 8 × 5/3. Same thing, written shorter.

Step	Working	Gives
Three fifths is		8
So one fifth is	8 ÷ 3	8/3
Five fifths is	8/3 × 5	40/3

Key idea

And a word you will need next

A ratio describes a split that is not equal.
3 : 2 means three for every two. Part against part.
3 out of 5 is the same split as a fraction. Part against whole.

Written	Compares	Means
3 : 2	part to part	3 for every 2
3 out of 5	part to whole	a fraction
Same split	described two ways

Your turn

Your turn: four questions

Pause here and try all four.
Question 4 is the backwards one. Decide the direction first.
Then check your answer by going forwards again.

	Question
1	Work out 1.4 × 25.
2	What is 3/8 of 20?
3	What is 20% of 85?
4	If 9 is 3/4 of a number, what is the number?

Show answer

1. 35. 14 × 25 = 350, and one decimal place in the question puts it at 35.0. 2. 7 1/2. 3/8 × 20/1 = 60/8, which is 7.5, or 7 1/2. 3. 17. A tenth of 85 is 8.5, so two tenths is 17. 4. 12. Multiply by the flip: 9 × 4/3 = 36/3 = 12. Check it, three quarters of 12 is 9. And 12 is bigger than 9, which it had to be.

Worked example

The answers

1.4 × 25 is 35. And 3/8 of 20 is 60/8, which is 7 1/2.
20% of 85 is 17, since a tenth is 8.5.
9 × 4/3 is 12. And three quarters of 12 is 9, so it checks.

	Answer
1	35
2	7 1/2
3	17
4	12

Summary

What to take away

Estimate a decimal product before you place the point.
Going backwards, multiply by the fraction turned over.
And always ask whether the answer should grow or shrink.

	Remember
Decimal products	estimate first, then count the places
Fraction of an amount	multiply, then tidy to a mixed number
Percentages	build from 10%, or multiply once
Going backwards	multiply by the fraction turned upside down
Always ask	should the answer be bigger or smaller?

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