Year 6NumberRational numbersYear 6 Maths

Comparing and adding fractions

What this lesson covers

Percentages can be used to compare quantities to a value or whole.
Fractions should have the same denominator before using them in addition or subtraction.

Key idea

What is 1/2 + 1/4?

A very common answer is 2/6.
Add the tops: 1 + 1 = 2. Add the bottoms: 2 + 4 = 6.
It looks tidy. It is completely wrong.

Key idea

You can prove it is wrong without any rules

2/6 is the same as 1/3, which is about 0.33.
But we started with 1/2, which is 0.5.
So adding a quarter supposedly made it smaller.
That cannot happen. The method is broken.

Stage	Fraction	Value
We started with	1/2	0.5
We added	1/4	0.25
Tops-and-bottoms says	2/6	0.33

Key idea

You can only add pieces of the same size

Halves and quarters are different-sized pieces.
Adding 1 half to 1 quarter is like adding 1 cat to 1 dog.
You get 2 animals, not 2 of either one.
First make the pieces match.

Key idea

One half is also two quarters

Cut each half in two and you have quarters.
The shading has not moved: 1/2 = 2/4.
Same amount, smaller pieces, more of them.

Worked example

Now the addition is easy

Rewrite 1/2 as 2/4. Leave the 1/4 alone.
Now both are quarters, so just add the tops.
2/4 + 1/4 = 3/4.
And 3/4 is bigger than 1/2, as it has to be.

Key idea

The rule, now that you can see why

Fractions must share a denominator before you add them.
When one bottom is a multiple of the other, use the bigger.
4 is a multiple of 2, so quarters were the right choice.
Rewrite only the one that needs it.

Sum	Rewritten	Answer
1/2 + 1/4	2/4 + 1/4	3/4
1/3 + 1/6	2/6 + 1/6	3/6 = 1/2
3/4 − 1/8	6/8 − 1/8	5/8

Worked example

Adding 1/3 and 1/6

6 is a multiple of 3, so work in sixths.
1/3 is 2/6. The 1/6 stays as it is.
2/6 + 1/6 = 3/6.
And 3/6 simplifies to 1/2. Always tidy the answer.

Worked example

Subtracting works the same way

3/4 − 1/8. Eighths are the smaller pieces, so use eighths.
3/4 is 6/8.
6/8 − 1/8 = 5/8.
Same first step every time: make the pieces match.

Worked example

Which is bigger, 2/3 or 5/6?

You cannot tell by looking at the numbers.
6 is a multiple of 3, so put both over sixths.
2/3 is 4/6. And 4/6 is less than 5/6.
So 5/6 is the bigger fraction.

Worked example

Ordering 1/2, 3/8 and 3/4

8 is a multiple of both 2 and 4, so use eighths.
1/2 is 4/8. 3/4 is 6/8. The 3/8 is already eighths.
Now compare the tops: 3, 4, 6.
So the order is 3/8, then 1/2, then 3/4.

Fraction	In eighths
1/2	4/8
3/8	3/8
3/4	6/8

Key idea

When the top is bigger than the bottom

7/4 means seven quarters.
Four quarters make a whole, so there is more than one whole here.
A fraction like that is called improper, which just means top-heavy.

Worked example

Turning 7/4 into a mixed number

Ask how many whole fours fit into 7. One.
That leaves 3 quarters over.
So 7/4 = 1 3/4.
It is a division, and you already know how to do those.

Improper	Divide	Mixed
7/4	7 ÷ 4 = 1 r 3	1 3/4
9/2	9 ÷ 2 = 4 r 1	4 1/2
7/3	7 ÷ 3 = 2 r 1	2 1/3

Key idea

A fraction bar is a division sign

7/4 and 7 ÷ 4 are the same thing written differently.
7 ÷ 4 is 1 remainder 3, which is 1 and 3/4.
The remainder becomes the top of the fraction.
That is why the two methods always agree.

Worked example

And back the other way

Turn 2 1/3 into thirds.
Each whole is 3 thirds, and there are 2 wholes: 6 thirds.
Plus the 1 third already there makes 7.
So 2 1/3 = 7/3.

Step	Working
2 wholes	2 × 3 = 6 thirds
plus the third	6 + 1 = 7
Answer	7/3

Worked example

Adding mixed numbers

1 1/2 + 2 1/4. Deal with the wholes first: 1 + 2 = 3.
Now the fractions: 1/2 + 1/4, which we know is 3/4.
Put them back together: 3 3/4.
Wholes with wholes, fractions with fractions.

Part	Sum
Wholes	1 + 2 = 3
Fractions	2/4 + 1/4 = 3/4
Answer	3 3/4

Key idea

3 3/4 is also 3.75

3/4 is 0.75, so 3 3/4 is 3.75.
Fractions and decimals are the same numbers, differently written.
Which means adding decimals needs the same first step.
Make the pieces match before you add.

Worked example

Adding 2.35 and 1.4

A common answer is 3.39 (adding 4 onto 35).
But 1.4 is 1.40, not 1.04.
Write it as 1.40 so the places line up, then add.
2.35 + 1.40 = 3.75.

	Ones	Tenths	Hundredths
	2	3	5
+	1	4	0
=	3	7	5

Worked example

The same with thousandths

1.256 + 2.4. Line up the decimal points, not the right edges.
Write 2.4 as 2.400 and every column has a digit.
1.256 + 2.400 = 3.656.
The zeros do nothing except hold the places open.

	Ones	Tenths	Hundredths	Thousandths
	1	2	5	6
+	2	4	0	0
=	3	6	5	6

Your turn

Four to try

1. Work out 1/3 + 1/9.
2. Which is bigger, 3/5 or 7/10?
3. Write 11/4 as a mixed number.
4. Work out 4.7 + 3.25.

Show answer

Answers: 1. 3/9 + 1/9 = 4/9: ninths, because 9 is a multiple of 3. 2. 7/10, since 3/5 is 6/10. 3. 11 ÷ 4 = 2 remainder 3, so 2 3/4. 4. 7.95; write 4.7 as 4.70 first; 7.32 or 7.12 means they aligned the right-hand edges. On question 1 a "4/12" answer is the tops-and-bottoms method again, and the beat to revisit is 2.

Summary

Make the pieces match first

Fractions: find a common denominator.
Decimals: line up the points and fill the gaps with zeros.
It is the same instruction twice.

Sum	Rewritten	Answer
1/2 + 1/4	2/4 + 1/4	3/4
1/3 + 1/6	2/6 + 1/6	1/2
2.35 + 1.4	2.35 + 1.40	3.75
1.256 + 2.4	1.256 + 2.400	3.656

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