Checkpoint 6 Maths · Number · Fractions, decimals and percentages

Fractions, Decimals and Percentages: Equivalence and Ordering

Change confidently between fractions, decimals and percentages, find percentages of shapes and amounts, and compare and order mixed numbers with =, > and <.

British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course

In the curriculum

Know that a fraction, a decimal (with one or two decimal places) and a percentage can have the same value, e.g. 3/4 = 0.75 = 75% (6Nf.04)
Recognise percentages of shapes and of whole numbers: 1% and multiples of 5% up to 100%, e.g. 35% of a shape, 15% of 60 (6Nf.07)
Compare and order decimals (one or two decimal places), proper fractions with different denominators and percentages, using =, > and < (6Nf.08)

Fractions, decimals and percentages are three different ways of writing the same amount. Half a pizza is 1/2, 0.5 or 50% of it. If you can move quickly between the three forms, you can compare any two numbers, find percentages of amounts in your head and answer a lot of Checkpoint questions in seconds.

Key words

Word / phrase	Meaning / use
per cent (%)	في المئة — 'out of 100'; 35% means 35 out of 100
percentage	النسبة المئوية — an amount written out of 100 with the % sign
decimal	عدد عشري — a number with a decimal point, e.g. 0.35
tenth / hundredth	جزء من عشرة / جزء من مئة — 0.1 = 1/10, 0.01 = 1/100
equivalent	مكافئ / متساوي في القيمة — worth the same, e.g. 1/4 and 25%
simplest form	أبسط صورة — e.g. 35/100 simplified to 7/20
common denominator	مقام مشترك — the same denominator for two fractions so you can compare them
greater than >, less than <	أكبر من، أصغر من — the open side of the sign faces the bigger number
ascending order	ترتيب تصاعدي — from smallest to largest

1. What does 'per cent' mean?

Per cent means 'out of 100'. If a square is cut into 100 equal small squares and 35 are shaded, then 35/100 of it is shaded. That is 35 hundredths, which as a decimal is 0.35, and as a percentage is 35%.

One picture, three ways to write it: 35/100 (= 7/20 in simplest form) = 0.35 = 35%.

The whole shape is always 100%. So 100% = 1 = 100/100. The number line below shows the most important equivalents. Learn them by heart — Checkpoint expects you to know them without working.

Fractions (above), decimals (middle row) and percentages (bottom row) on the same number line from 0 to 1.

Fraction	Decimal	Percentage
1/100	0.01	1%
1/20	0.05	5%
1/10	0.1	10%
1/5	0.2	20%
1/4	0.25	25%
3/10	0.3	30%
2/5	0.4	40%
1/2	0.5	50%
3/5	0.6	60%
7/10	0.7	70%
3/4	0.75	75%
4/5	0.8	80%
9/10	0.9	90%
1 (whole)	1	100%

2. Changing between the three forms

Change …	How	Examples
fraction → percentage	make the denominator 100 with an equivalent fraction	7/20 = 35/100 = 35%; 3/25 = 12/100 = 12%
fraction → decimal	make the denominator 10 or 100, or divide numerator ÷ denominator	3/5 = 6/10 = 0.6; 3 ÷ 4 = 0.75
decimal → percentage	multiply by 100 (digits move 2 places left)	0.6 = 60%; 0.07 = 7%; 0.45 = 45%
percentage → decimal	divide by 100 (digits move 2 places right)	45% = 0.45; 5% = 0.05; 80% = 0.8
percentage → fraction	write it over 100, then simplify	45% = 45/100 = 9/20; 60% = 60/100 = 3/5
decimal → fraction	use place value (tenths or hundredths), then simplify	0.6 = 6/10 = 3/5; 0.24 = 24/100 = 6/25

Making the denominator 100

Ask: 'What do I multiply the denominator by to get 100?' Then multiply the numerator by the same number. Denominators that work easily: 2 (× 50), 4 (× 25), 5 (× 20), 10 (× 10), 20 (× 5), 25 (× 4), 50 (× 2). Example: 9/25 → × 4 → 36/100 = 0.36 = 36%.

الفكرة الأساسية

النسبة المئوية يعني "من 100". علشان تحوّل كسر لنسبة مئوية خلّي المقام 100: مثلاً 7/20 = 35/100 = 35%. ومن عشري لنسبة مئوية اضرب في 100: 0.6 = 60% (مش 6%). ومن نسبة مئوية لعشري اقسم على 100: 5% = 0.05.

3. Percentages of shapes

To find what percentage of a shape is shaded, the shape must be split into equal parts. Work out what percentage one part is (100% ÷ the number of parts), then multiply by the number of shaded parts.

Number of equal parts	Each part is worth	Example
100	1%	23 shaded → 23%
50	2%	18 shaded → 36%
20	5%	9 shaded → 45%
10	10%	7 shaded → 70%
5	20%	3 shaded → 60%
4	25%	3 shaded → 75%

A shape of 20 equal squares: each square is 5% of the shape, so 9 squares are 45%.

You can also go the other way: 'Shade 35% of a shape with 20 squares.' Each square is 5%, and 35% ÷ 5% = 7, so shade 7 squares.

4. Percentages of whole numbers

Build every percentage from a few easy ones. You never need a calculator for 1% or multiples of 5%.

Percentage	How to find it	Of 80	Of 600
100%	the whole amount	80	600
50%	halve	40	300
25%	halve, then halve again (÷ 4)	20	150
10%	÷ 10	8	60
5%	half of 10%	4	30
1%	÷ 100	0.8	6

Then add or subtract the easy pieces: 35% = 10% + 10% + 10% + 5%; 45% = 50% − 5%; 95% = 100% − 5%; 75% = 50% + 25%.

A double number line for 80: 10% steps on top match steps of 8 below. Halfway between 30% (24) and 40% (32) is 35% = 28.

5. Comparing and ordering

To compare numbers written in different forms, change them all into the same form — usually decimals with two decimal places, or percentages. Then compare digit by digit from the left.

Change every number to a decimal with two decimal places (write 0.6 as 0.60).
Compare the tenths first, then the hundredths.
Put them in order, then write the original numbers (not your decimals) in the answer.
For two fractions, you can also use a common denominator: 3/4 and 7/10 → 15/20 and 14/20, so 3/4 > 7/10.

5/8 = 15/24 and 7/12 = 14/24. Bars of the same length show 5/8 is a little longer, so 5/8 > 7/12.

The signs =, > and <

The wide open side of the sign faces the bigger number: 0.75 > 0.7 (0.75 is greater than 0.7), 0.09 < 0.1 (0.09 is less than 0.1). Use = only when the values are exactly the same, e.g. 2/5 = 40%.

6. Worked examples

Worked example 1 — Write 0.35 (a) as a fraction in its simplest form (b) as a percentage. [2]

(a) 0.35 = 35 hundredths = 35/100. The HCF of 35 and 100 is 5: 35/100 = 7/20. (b) 0.35 × 100 = 35%. Common mistake: writing 35/10 or 3.5%.

Worked example 2 — A shape is made of 20 equal squares. 9 are shaded. What percentage is shaded? [1]

Each square = 100% ÷ 20 = 5%. 9 × 5% = 45%. (Check: 9/20 = 45/100.)

Worked example 3 — Find 35% of 80. [1]

10% of 80 = 8, so 30% = 24. 5% = half of 8 = 4. 35% = 24 + 4 = 28.

Worked example 4 — Find (a) 1% of 600 (b) 95% of 600. [2]

(a) 600 ÷ 100 = 6. (b) 95% = 100% − 5%. 10% of 600 = 60, so 5% = 30. 600 − 30 = 570. Common mistake: finding 95% as 9 × 10% + 5% and adding wrongly — subtracting from 100% is quicker and safer.

Worked example 5 — Write >, < or = in each box. [2] (a) 3/4 ☐ 0.7 (b) 2/5 ☐ 40% (c) 0.09 ☐ 0.1

(a) 3/4 = 0.75 and 0.7 = 0.70. 0.75 > 0.70, so 3/4 > 0.7.
(b) 2/5 = 4/10 = 40/100 = 40%, so 2/5 = 40%.
(c) 0.09 has 0 tenths; 0.1 = 0.10 has 1 tenth. So 0.09 < 0.1. Common mistake: thinking 0.09 is bigger because 9 > 1.
Mark scheme style: 2 marks for all three correct, 1 mark for two correct.

Worked example 6 — Which is bigger, 5/8 or 7/12? Show how you know. [1]

Find a common denominator: 24 is a multiple of both 8 and 12. 5/8 = 15/24 (× 3) and 7/12 = 14/24 (× 2). 15/24 > 14/24, so 5/8 is bigger (see the fraction bars above).

Worked example 7 — Write these numbers in order, starting with the smallest: 0.62 3/5 66% 0.58 16/25 [2]

Number	0.62	3/5	66%	0.58	16/25
As a decimal (2 d.p.)	0.62	0.60	0.66	0.58	0.64

Order of the decimals: 0.58, 0.60, 0.62, 0.64, 0.66. Answer: 0.58, 3/5, 0.62, 16/25, 66%. Write the numbers as they were given, not the decimals.

Common mistakes

'Longer decimals are bigger': 0.56 is NOT bigger than 0.6. Write 0.6 as 0.60 and compare tenths first.
0.6 = 6% ✗. Multiply by 100: 0.6 = 60%. And 0.06 = 6%.
5% = 0.5 ✗. Divide by 100: 5% = 0.05.
Comparing fractions by the denominator only: 1/8 looks 'bigger' because 8 > 4, but 1/8 < 1/4.
Counting shaded parts of a shape that is not split into equal parts, or forgetting how many parts the whole has.
In ordering questions, writing the converted decimals instead of the original numbers, or ordering largest first when the question says smallest first.

How Checkpoint asks this

Matching lines: join each fraction to the equal decimal or percentage.
Complete a table of equivalent fractions, decimals and percentages.
'What percentage of the shape is shaded?' or 'Shade 45% of this shape' on a grid.
Find a percentage of an amount (often money or pupils), e.g. 15% of $60 — show 10% and 5% in your working.
Write >, < or = in a box, ring the largest/smallest, or order a mixed list. Change everything into decimals or percentages first and write your conversions next to the numbers.
'Explain how you know' — show the common denominator or the conversions; a bare answer may not get the mark.

Practice

Remember

Per cent means out of 100: 35% = 35/100 = 0.35.
Know the key equivalents: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/5 = 0.2 = 20%, 1/10 = 0.1 = 10%, 1/100 = 0.01 = 1%.
Percentages of amounts: build from 10% (÷ 10), 5% (half of 10%), 1% (÷ 100), 50% and 25%.
Percentage of a shape: 100% ÷ number of equal parts = value of one part.
To compare or order, change everything to the same form (two-decimal-place decimals or percentages), then compare tenths first.

All pages in this course

Start here

Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

Number: place value, decimals, fractions, percentages and ratio

Geometry and measure

Statistics and probability

Practice tests