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Ratios and equal shares

Use equal-sized ratio units to compare and share quantities with whole-number terms.

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Check your understanding 1

Remember what a ratio names

Count 2 red units for every 3 blue units. How many units in all?

A collection has red : blue = 2 : 3. What fraction of all its counters is red?

Red as a fraction of the whole
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There are 2 + 3 = 5 equal ratio units altogether. Red is 2/5 of the whole. The ratio 2 : 3 compares red with blue, whereas 2/5 compares red with all counters.

Which colour comes first in the ratio?

Keep the named order

A box has 6 red counters and 9 blue counters. A ratio compares the named quantities in order. Use whole-number ratio terms separated by colons, fractions with /, and ordinary numbers for amounts.

Five equal ratio units

Each full strip is one equal-sized whole.

The full strip represents all fifteen counters. Two hatched units represent red counters; three plain units represent blue counters. Each unit stands for three counters, not one counter.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
All fifteen countersmarked; marked; empty; empty; empty
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6:9 = 2:3 after dividing both terms by 3. Reversing the order gives 9:6 = 3:2. The five equal units make the whole collection: two red units and three blue units. Red is 2/5 of the whole; 2/3 compares red with blue instead.

Compare volumes in the same unit

The first jug holds 750 ml and the second holds 1 ℓ. Write first volume : second volume in simplest whole-number form.

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The second volume is 1 × 1000 = 1000 ml. With both volumes in millilitres, 750:1000 = 3:4. Writing 750:1 would compare numbers measured in different units.

Scale all parts of the ratio together

For 6 red, 9 blue and 15 white counters, the ratio is 6:9:15 = 2:3:5. The same scale applies to every term, including the third.

Simplify 20 : 30 : 45. Then choose a different whole-number ratio equivalent to 2 : 3 : 5.

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Dividing every term of 20:30:45 by 5 gives 4:6:9. A simplest ratio has no common whole-number factor greater than 1. For a different equivalent ratio, one possible choice is 4:6:10, obtained by doubling every term. Changing only one or two terms changes the comparison.

Check your understanding 2

One scale for all three terms

Complete 2:3:5 = 8:12: ?. Then decide whether any third term could make 2:3:5 = 8:11: ? true.

Can any third term make 2:3:5 = 8:11:? true?
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The scale factor from 2 to 8 is 4. It also changes 3 to 12, so the missing term is 5 × 4 = 20. In the second statement, the first two terms already disagree: 3 × 4 = 12, not 11. No third term can repair that disagreement.

Does the known amount name one part, the total or the difference?

Share a total

Share 84 dollars between two people in the ratio 2 : 5. This time the known amount is the whole total.

The total covers every ratio unit

Each full strip is one equal-sized whole.

All seven equal units together represent 84 dollars. The first person's share is the two hatched units; the second person's share is the five plain units.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
All 84 dollarsmarked; marked; empty; empty; empty; empty; empty
Show workingHide answer
all units = 2 + 5 = 7; one unit = 84÷7 = 12; first share = 2 × 12 = 24; second share = 5 × 12 = 60.

You can also find 2/7 and 5/7 of 84. Both routes use the same seven-unit whole. Check 24 + 60 = 84 and 24:60 = 2:5. Checking the sum alone is not enough. 30 and 54 also total 84, but their ratio is different.

Share between three groups

Three groups share 72 counters in the ratio 2 : 3 : 4. Find all three counts in the named order.

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There are 2 + 3 + 4 = 9 units, each worth 72÷9 = 8 counters. The counts are 2 × 8 = 16, 3 × 8 = 24 and 4 × 8 = 32. Check that they total 72 and simplify to 2:3:4. The second group is 3/9 = 1/3 of the total, not 3/4.

Sometimes a known amount is one part

Red : blue is 5 : 2, and there are 14 blue counters. The 14 belongs to the named blue part. It is not the total of all the counters.

Attach the known count to its named part

Each full strip is one equal-sized whole.

The whole collection has five hatched red units and two plain blue units. The two blue units together contain 14 counters.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
All red and bluemarked; marked; marked; marked; marked; empty; empty
Show workingHide answer

One unit is 14÷2 = 7 counters, so red has 5 × 7 = 35 counters. Check 35:14 = 5:2. Dividing 14 by seven would treat the blue count as if it were the whole collection.

Use the first group’s count

Three groups have counters in the ratio 3 : 4 : 5. The first group has 24 counters. Find the other two counts.

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The named first group fixes one unit: 24÷3 = 8 counters. The second group has 4 × 8 = 32 and the third has 5 × 8 = 40. Check 24:32:40 = 3:4:5. Use the ratio part for the group you know. It may come anywhere in the ratio.

Choose a view that helps you

Use red : blue = 5 : 2 with 30 red counters. Choose a view that helps you find one equal unit.

Use objects

Build equal units from the known objects

Use red : blue = 5 : 2 with 30 red counters. These five plates show the five red ratio units. They show only red counters, not the whole collection.

Still to share

Empty

Plate 1

Plate 2

Plate 3

Plate 4

Plate 5

All counters shared. Do the plates have the same number?

One plate has 6 counters, so each ratio unit is worth six. The blue collection needs two more units of that same size, or 12 blue counters. The original thirty still counts only red counters.

Use a bar

Match the known amount to the picture

One named group, not the whole

Each full strip is one equal-sized whole.

The whole contains seven equal units. Five hatched units are red; two plain units are blue. The known red count labels only the five red units.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
All red and bluemarked; marked; marked; marked; marked; empty; empty

Use red : blue = 5 : 2 with 30 red counters. Put 30 above the five red units. It does not label all seven units. The two blue units have the same unit size.

Show the calculation

Match the picture to the calculation

For red : blue = 5 : 2 with 30 red counters:

5 units = 30 → 1 unit = 30 ÷ 5 = 6 → 2 units = 2 × 6 = 12 blue counters.

Check both the known amount and the ratio: 30 : 12 = 5 : 2. Dividing 30 by seven would treat the known red count as the whole collection.

Check your understanding 3

What does the given amount represent?

A class has boys : girls = 4 : 5. There are 20 girls. Find the boys and the fraction of the class who are boys.

In boys:girls = 4:5 with 20 girls, what does 20 label?
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Five units are 20 pupils, so one unit is 4 and four units are 16 boys. There are 16 + 20 = 36 pupils, giving 16/36 = 4/9 boys. The known 20 represents the girls, not the whole class.

Sometimes the known amount is a difference

Two collections of counters are in the ratio 3 : 7. The second collection has 28 more counters than the first.

Compare the two groups of equal units

Each full strip is one equal-sized whole.

Both full strips represent the same total of ten ratio units. The first row marks three units for the first collection; the second marks seven for the second collection. Compare the marked amounts.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
First collectionmarked; marked; marked; empty; empty; empty; empty; empty; empty; empty
Second collectionmarked; marked; marked; marked; marked; marked; marked; empty; empty; empty
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The difference is 7-3 = 4 units, so one unit is 28÷4 = 7 counters. The collections have 3 × 7 = 21 and 7 × 7 = 49 counters. Check both 49-21 = 28 and 21:49 = 3:7. Dividing 28 by ten would wrongly treat a difference as a total.

Check your understanding 4

A difference in a three-part ratio

Three groups have counters in the ratio 2:3:5. The third group has 18 more counters than the first. Find every group’s count in the named order.

Which ratio units represent the 18-counter difference?
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The difference is 5-2 = 3 units, so one unit is 18÷3 = 6 counters. The counts are 12, 18 and 30. Check 30-12 = 18 and 12:18:30 = 2:3:5. The known difference is not the total of the three groups.

How many ratio units make the whole?

Turn a fraction into a part-to-rest ratio

3/5 of a collection is red; the rest is blue. Find red : blue in simplest whole-number form. Then give the blue fraction of the whole in simplest form.

A fraction's denominator names the whole

Each full strip is one equal-sized whole.

Five equal units make the whole collection. Three hatched units are red and the other two plain units are blue. A part-to-rest ratio compares these two groups.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
The whole collectionmarked; marked; marked; empty; empty
Show workingHide answer

Five units form the whole. Three are red and 5-3 = 2 are blue. Red : blue is 3:2, while red : whole is 3:5. From the ratio 3:2, the blue fraction is 2/(3 + 2) = 2/5. Name the comparison before choosing a denominator.

Check your understanding 5

Name the denominator

Two fifths of the beads in a bag are red; the rest are blue. Write red : blue in simplest whole-number form. Separately, a mix has red : blue : white = 2:3:4. What fraction of that mix is blue?

For red:blue:white = 2:3:4, what whole gives blue’s fraction of the mix?
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The bag has two red units and three blue units, giving red : blue = 2:3. In the mix, blue occupies three of nine units: 3/9 = 1/3. Blue : all other colours would be 3:(2 + 4) = 1:2, a different comparison. One part’s fraction does not tell you how the other colours split. You need more information to find all three ratio terms.

Compare adding with multiplying

Check your understanding 6

Compare the two changes

A ratio is 2:3. Doubling both actual quantities gives 4:6. Starting again from 2 and 3, adding two to each quantity gives 4:5. Compare the first quantity’s fraction of the total after each change. Write each fraction in simplest form.

Which change preserves the ratio 2:3?
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The original first fraction is 2/5. After doubling, it is 4/10 = 2/5. After adding two to each original quantity, it is 4/9 instead. A common multiplier preserves a ratio; adding the same fixed amount need not preserve it.

For a grown-up

Pip’s synthetic voice uses the University of Edinburgh’s Alba speech dataset, via Piper’s Alba voice, under CC BY 4.0. Audio is optional.

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