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Percentages and parts of a whole

Name the reference whole before converting or finding a percentage part.

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Count out of a hundred

Per cent means out of 100. First find the whole.

18 of 60 pupils walk to school. The part is the 18 walkers; the whole is all 60 pupils. A percentage compares that part with its whole, expressed per hundred. The symbol % means “per hundred”.

One class, divided into ten equal groups

Each full strip is one equal-sized whole.

The full strip represents all 60 pupils. Each of its ten equal parts represents 6 pupils, or 10% of the class. Three parts are marked for the 18 walkers.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
Whole: 60 pupilsmarked; marked; marked; empty; empty; empty; empty; empty; empty; empty
Whole used when finding the percentage who walk

Use % in a percentage answer and / in a fraction. Write an ordinary number when a field asks for a count or a volume.

Show workingHide answer
18/60 = 3/10; = 30/100; = 30%.

Three of the ten equal groups are marked. Each group is 10% of the class, so the marked part is 30%. “Per hundred” describes a proportion; the actual class does not need to contain 100 pupils.

How can you write the same share in three ways?

Connect fractions, decimals and percentages

All three forms below describe the same proportion:

1/4 = 25%; = 0.25.

To turn a percentage into a fraction, write its number over 100, then simplify. To express a fraction or decimal as a percentage, find how many hundredths it represents.

Write the same amount in different forms

Try the conversions below. Give a fraction in simplest form when asked.

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35% = 35/100; = 7/20; = 0.35.

The % sign already means “divided by 100”, so 35% is not 35.

3/8 = 0.375; = 37.5/100; = 37.5%.

Expressing the proportion per hundred multiplies its numerical value by 100 and appends %.

0.07 = 7/100; = 7%.

Writing 0.07% would divide by 100 again and make a different proportion.

Does your new form still name the same share?

Practise: Primary 5 Mathematics Percentage parts, representations and contexts practice · Primary 5 MathematicsPractise: Primary 6 Foundation Mathematics Percentage parts, representations and contexts practice · Primary 6 Foundation Mathematics

Check your understanding 1

Keep the same proportion

Write 12% as a simplest fraction and as a decimal. Then write 3/5 and 0.045 as percentages. Keep the % sign on each percentage answer.

Which calculation writes 12% as a decimal?
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12% = 12/100; = 3/25; = 0.12.

Also:

3/5 = 60/100; = 60%.
0.045 = 4.5/100; = 4.5%.

Since 0.045 is smaller than 0.05, its percentage must be smaller than 5%. You write the same share in different forms.

How many books does one equal part stand for?

Find a part from its percentage

A library has 80 books, of which 15% are borrowed. Here the whole is known and the part is needed. The picture groups the books into equal 5% parts.

Count equal parts of the known whole

Each full strip is one equal-sized whole.

The full strip represents 80 books. Each of its twenty equal parts is 5% of the books. The three marked parts together represent the borrowed 15%.
Figure data

This table is included in print while it is open.

Equal parts in the picture
WholeParts in order
Whole: 80 booksmarked; marked; marked; empty; empty; empty; empty; empty; empty; empty; empty; empty; empty; empty; empty; empty; empty; empty; empty; empty
Show workingHide answer

The picture has 20 equal parts, so each part contains 80÷20 = 4 books. The three marked parts contain 3 × 4 = 12 books.

Another route uses 10% + 5%. Ten percent of 80 is 8; five percent is half of 8, or 4. Together, 15% of 80 is 12.

Both routes agree with 15/100 × 80 = 12. The percentage is the scale factor; the result keeps the whole’s unit: books.

Take the same percentage of a bigger whole. What happens?

Practise: Primary 5 Mathematics Percentage parts, representations and contexts practice · Primary 5 MathematicsPractise: Primary 6 Foundation Mathematics Percentage parts, representations and contexts practice · Primary 6 Foundation Mathematics

Check your understanding 2

Same percentage, different part

Find 20% of 35 and 20% of 60. Predict which part will be larger, then calculate both amounts. Explain why the answers differ even though the percentages match.

Which calculation gives the larger part?
Why is 20% of 60 larger than 20% of 35?
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20% = 20/100; = 1/5.

One fifth of 35 is 7; one fifth of 60 is 12. The same fraction of a larger whole gives a larger amount.

The percentage describes the proportion. It needs a reference whole before it can determine the amount.

Can the same number of pupils be different percentages of two classes?

Practise: Primary 5 Mathematics Percentage parts, representations and contexts practice · Primary 5 MathematicsPractise: Primary 6 Foundation Mathematics Percentage parts, representations and contexts practice · Primary 6 Foundation Mathematics

Check your understanding 3

Compare the two class proportions

Twelve pupils travel by bus in a class of 40. In another class of 60, twelve pupils also travel by bus. Which class has the larger percentage travelling by bus? Name the whole in each comparison before calculating.

Why is 12 pupils a larger proportion of 40 than of 60?
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The first whole is 40 pupils:

12/40 = 30/100; = 30%.

The second whole is 60 pupils:

12/60 = 20/100; = 20%.

The 40-pupil class has the larger percentage travelling by bus. The same part is a larger fraction of the smaller whole. Dividing by 12 would compare the whole with the part instead.

What percentage means none, and what percentage means all?

Include none and the whole

A tray contains 20 eggs. What percentage of the eggs is represented by none of them? What percentage is represented by all 20?

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0/20 = 0; = 0%.

For the whole tray:

20/20 = 1; = 100%.

The reference whole is 20 eggs in both cases.

You can find zero percent of a whole greater than zero. You cannot divide to find a percentage for 0 out of 0. That would mean dividing by zero.

Which amount is the whole for this percentage?

Practise: Primary 5 Mathematics Percentage parts, representations and contexts practice · Primary 5 MathematicsPractise: Primary 6 Foundation Mathematics Percentage parts, representations and contexts practice · Primary 6 Foundation Mathematics

Check your understanding 4

Connect fullness with water volume

A tank is 0.6 full. Its capacity is 45 ℓ. Write its fullness as a percentage and find how many litres of water are in it. Show how the percentage, the capacity and the water volume are related.

For a 45 ℓ tank that is 60% full, what does 60% refer to?
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0.6 = 60/100; = 60%.

The water volume is 0.6 × 45 = 27 ℓ.

Equivalently, 0.6 = 3/5, so the water occupies three of five equal capacity parts: 45÷5 × 3 = 27 ℓ. Sixty percent refers to the 45 ℓ capacity, rather than to 60 ℓ.

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