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Percent - a special type of fraction | Percent
Models | Ratios |
| Relationships:decimal fractions, common fractions,
percent and ratio | Rates | Quick
quiz |
Percent - a special type of fraction
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0.25, 1/4, 25%
These
expressions tell us what portion of the square is coloured orange.
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The
word percent come from the expression 'per cent' and literally means
'a part of one hundred'. A percent is a part, or fraction, out of
100. For example:
| 100% |
=100/100 |
=1 |
|
=
1.0 (decimal) |
| 50%
|
=
50/100 |
=
5/10 |
=
1/2 |
=
0.5 = 0.50 (decimal) |
| 25%
|
=
25/100 |
=
5/20 |
=
1/4 |
=
0.25 (decimal) |
| 40%
|
=
40/100 |
=
4/10 |
= 2/5 |
=
0.4 (decimal) |
| 5%
|
=
5/100 |
=
1/20 |
|
=
0.05 (decimal) |
| 0.5%
|
=
5/1000 |
=
1/200 |
|
=
0.005 (decimal) |
We
can see that to write a percent as a fraction we express the percent
as a fraction with a denominator of 100. We may then be able to simplify
the fraction further.
For
example, 75%
= 75/100 = 3/4
To
express a fraction as a percent we must first convert the fraction
into hundredths (in simple cases we can do this by using equivalent
fractions) and then replace '/100' by the percent '%' sign.
For
example, 4/5
= 80/100 = 80%
We
can see that we express a percent as a decimal by dividing by 100.
For
example,
| 25%
= 25/100 = 0.25 (twenty-five hundredths) |
| 47.3
% = 47.3/100 = 0.473 (forty seven hundredths and 3 thousandths) |
| 200%
= 200/100 = 2 |
To
express a decimal as a percent we multiply the decimal number by 100.
For
example,
| 0.108
= 0.108 x 100 = 10.8% |
| 0.75
= .75 x 100 = 75% |
| 1.2
= 1.2 x 100 = 120% |
Some
percents expressed as fractions and decimals
Example
1: 30 out of 50 apples in a box are too bruised to sell. What percent
of apples cannot be sold?
| Working
Out |
Thinking |
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30
out of 50 apples are bruised. To represent 30/50 as a percent
we need to find out how many apples out of 100 are bruised.
By equivalent fractions we know that 30 out of 50 equals 60
out of 100, so 60% of the apples are bruised.
We could also say that,
3/5 of the apples are bruised
0.6 of the apples are bruised.
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Example
2: Ryan spent 25 minutes in the bank, 11 minutes of which was spent
waiting in a queue. What percent of time did he spend waiting in the
queue?
| Working
Out |
Thinking |
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Ryan
spent 11 minutes out of 25 minutes waiting in a queue. To turn
this into a percent we are asking, 11 out of 25 minutes equals
how many minutes out of 100 minutes?
We
can see that 11 mins out of 25 mins equals 44 mins out of 100
mins by equivalent fractions (because we know 25 x 4 = 100)
.
We
can say that Ryan spent 44%, 0.44 or 11/25 of his time in the
bank waiting in a queue.
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Example
3: What percent is 7 cm of 20 cm?
| Working
Out |
Thinking |
|
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To
find out what percent 7 out of 20 is, we need to ask: 7 out
of 20 is how many out of 100?
5
groups of 20 make 100, so 7 out of 20 is 35 out of 100 (5 x
7 out of 5 x 20).
Therefore
7/20 equals 35%, or 0.35 if we represent it as a decimal.
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Percent
models
Dual-scale
number line model
We
can use the dual-scale number line, also called the proportional number
line, to illustrate example 1 from above.
| Recall
example 1: 30 out of 50 apples in a box are too bruised to sell.
What percent of apples cannot be sold? |
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Thinking |
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The
left side of the number line below has a percent scale. The
right side of the number line has a number scale. We can label
each scale using the information we are given in the problem.
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We
know that there are 50 apples in total, ie. 50 apples equals
100% of the apples. We know that 30 out of the 50 apples are
bruised and we need to find what percent this is.
In
more complicated problems this dual-scale number line is a good
way of organising the information we are given and to work out
what information we need to find.
Once
we have represented the problem in this way we can write a proportion
equation directly from the number line.
30/50
= ?/100
By
equivalent fractions we know that 30/50 = 60/100.
(Or
we might have just noticed that it is a 'multiply by 2' relationship,
so 30 x 2 = 60)
Therefore
60% of apples are too bruised to sell.
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The
dual-scale number line model is discussed further in the other pages
of the Percent, Ratio and Rates topic.
Elastic
tape measure model
The
tape measure model is a good linear model of percent. Teachers can
easily make these models using a ruler, such as a 1 metre ruler, and
elastic. The elastic needs to be marked with a percent scale. It can
then be stretched to the desired length.
For
example, what is 60% of 50?
To
find the answer we line up the zeros of the ruler and the elastic.
We then stretch the elastic so that 100% lines up with the whole amount,
which in this case is 50. We then look for 60% on the elastic and
read the corresponding amount on the ruler. We can see below that
60% of 50 is 30.
The
intention is NOT to use this model accurately. It is a good way of
showing that percent always involves a proportional comparison of
something to 100.
1
metre ruler
Elastic
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By
manipulating the tape measure, this model can be used for the 3 types
of percent problems, discussed in Percent
Examples. Examples of which are,
What
is 20% of 50?
What
percent
is 10 of 50?
30% of what number is 15?
(Note:
for a lesson, a teacher will need elastics tape measures of various
lengths, because the elastic can only be stretched - it cannot be
shrunk).
(This
elastic tape measure model was developed by J. H. Weibe)
Ratios
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The
ratio of 1 : 3 tells us the ratio of shaded : unshaded
The
ratio of 3 : 1 tells us the ratio of unshaded : shaded
The
ratio of 1 : 4 tells us the ratio of shaded : whole
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A
ratio is another way of comparing quantities. Each quantity must be
measured in the same units. An advantage of ratios is that we can
compare several things at once.
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scale
on a map
1:10000
(every 1 cm on the map represents 10000 cm on the ground, every
inch on the map represents 10000 inches on the ground)
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ratio
of blue to white paint is 1 : 4
(for
every 1 litre of blue paint there are 4 litres of white paint;
for every cup of blue paint there are 4 cups of white paint,
i.e. 4 times as much white)
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ratio
of gears on a bicycle 8 : 16 : 24
8:16:24
teeth on each cog
(cog size increases in the proportion of 1:2:3:4 etc)
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ratio
of number of girls to number of boys in class is 5 : 2
(for
every 5 girls in class there are 2 boys)
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Although
ratios must have each quantity measured in the same units, the
units are not fixed. 1 litre of blue paint to 4 litres of the
white paint represents the same ratio as 1 tin of blue paint to 4
tins of white paint, or 1 bucket of blue paint to 4 buckets of white
paint. This fact makes ratios very versatile to use in everyday situations.
The
order in which a ratio is written is very important. If we say the
ratio of the number of girls to the number of boys is 5:2 this is
very different to saying the ratio of the number of girls to the number
of boys is 2:5.
A
ratio can be written in different ways;
-
in words
- the ratio of the number of girls to the number of boys is 5 to 2,
and this is the way we say it
-
using a colon
- number of girls : number of boys = 5 : 2
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We
must always talk about ratios in context. To say or write
5 : 2, for example, has no meaning on its own.
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Example
4: Let's
say I want to make the paint colour 'sky blue' and I know that the
way to do this is to mix 1 part blue with 4 parts white. This means
there is a ratio of blue to white of 1:4. In this case 1 litre of
blue to 4 litres of white, making 5 litres of sky blue paint.
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1
blue : 4 white
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2
blue : 8 white
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To
make double the quantity of paint I can mix the blue to white as a
ratio of 2:8. This will make the same colour. The ratios of 1:4 and
2:8 are equivalent, and worked out in the same way as equivalent fractions.
We multiply each part of the original ratio by the same number and
we can find equivalent ratios.
| 1
: 4 |
|
1
: 4 |
|
1
: 4 |
|
|
x
2 |
|
x
3 |
|
x4 |
| 2
: 8 |
|
3
: 12 |
|
4
: 20 |
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Sharing
quantities in a given ratio
Example
5: We
have a small inheritance of $15000 to be shared among 3 people in
the ratio of 2 : 2 : 1, how much does each person receive?
| Working
Out |
Thinking |
| Person
1 |
Person
2 |
Person
3 |
|
2
:
|
2
:
|
1
|
| (2
x $3000) |
(2
x $3000) |
(1
x $3000) |
| $6000 |
$6000 |
$3000 |
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The
ratio of 2 : 2 : 1 means that the inheritance is divided into
5 portions - two people each receive 2 portions and one person
receives 1 portion.
$15000
divided by 5 - each portion is worth $3000
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Relationships
- decimal fractions, common fractions, percent and ratio
We
can use examples to illustrate the relationships between decimal fractions,
common fractions, percent and ratio.
Example
6: An inheritance of $15000 is to be distributed among 3 people in
the ratio of 2 : 2 : 1. ($15000 will be divided into 5 portions)
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Person
1
|
Person
2
|
Person
3
|
| ratio |
2
:
(2
x $3000)
|
2
:
(2
x $3000)
|
1
(1 x $3000)
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common
fraction |
(of $15000)
|
(of $15000)
|
(of $15000) |
decimal
fraction |
0.4
(of $15000)
|
0.4
(of $15000)
|
0.2
(of $15000)
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| percent |
40
% (of $15000)
|
40
% (of $15000)
|
20
% (of $15000)
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Example
7: Ratios and fractional parts.
A litre of mixed cordial requires 250 mls of cordial and 750 mls of
water. How can we represent this as a ratio and a fraction?
| Thinking |
|
The
ratio of cordial to water is 250 : 750 or 1 : 3. One part cordial
to 3 parts water.
In fraction terms, the 1000 mls of mixed cordial is 250/1000
(1/4) cordial and 750/1000 (3/4) water.
We talk about the ratio of cordial to water in many different
ways: For every cup of cordial there are 3 cups of water. There
is 3 times as much water as cordial. 1 out of every 4 parts
of the mixed cordial is cordial. Can you think of any more ways?
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Example
8: Ratios and fractional parts.
A group of 100 people is made up of 60 males and 40 females. How can
we represent this as a ratio and a fraction?
| Thinking |
|
The
ratio of males to females is 60 : 40 or 6 : 4. This means that
overall there is a higher proportion of males in the group,
and for every 6 males there are 4 females.
In fraction terms, where we are talking about the group of 100
people, 60/100 (6/10) are male and 40/100 (4/10) are female.
So
we can also say that 6 out of every 10 people are males and
4 out of every 10 people are females.
Whereas
fractions only enable us to represent the part to whole relationship
(in this case, males/people and females/people), different aspects
of the relationships between quantities (people) can be shown
using ratios. For example,
The
3 ratios that represent the relationships of males and females
in this group of people are:
-
the ratio of 6 males to 10 people can be represented as 6 : 10
- the ratio of 4 females to 10 people can be represented as
4 : 10
- for every 6 males there are 4 females can be represented as
6 : 4
Which
ratio we choose depends on we want to say.
The number of males to the number of females is 6 : 4
The number of males to the number of people is 6 : 10
The number of females to the number of people is 4 : 10
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In
the early stages of introducing ratio at a primary level we generally
discuss ratios in terms of a part to part or, quantity to quantity,
comparison. At this stage part to whole relationships are often better
represented by fractions or percents with which students already have
some experience. When part to whole ratios are introduced care must
be taken to ensure students clearly understand what is being represented.
Rates
We
use rates when we are measuring one quantity or amount in relation
to another quantity or amount. We use them to compare how quantities
change, usually over a period of time. A significant difference between
rates and ratios is that when we are forming rates, each quantity
is measured in different units
to form new composite units.
For
example, let's say you are travelling at a rate of 60 kilometres per
hour (km/h). Here we are measuring kilometres in relation to hours
and the rate unit becomes 'kilometres
per hour', often written as km/h.
Other
examples of rates are, an athlete running at 10 metres per second
(m/s), and a factory using water at a rate of 450 litres per hour
(l/h).
Some
rates we commonly use are:
| km/h |
kilometres
per hour |
| c/L |
cents
per litre |
| $/m |
dollars
per minute |
| c/m |
cents
per minute |
Example
9: A hose is running water at the constant rate of 100 litres an hour.
1. How many litres will run in 2 hours?
2. How many hours will it take to run 350 litres?
| Working
Out |
Thinking |
|
100
litres/1 hour = ? litres /2 hours
200
litres will run in 2 hours.
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We
know that 100 litres runs in 1 hour and we need to find out
how many litres runs in 2 hours. We can write this as litres/hour
because this is what we are trying to find out.
By equivalent fractions we know that 100/1 = 200/2.
Therefore 200 litres will run in 2 hours.
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| Working
Out |
Thinking |
| 100
litres/1 hour = 350 litres /? hours
350
litres/ ? hours = 100 litres /1 hour
350
litres/3.5 hours = 100 litres/1 hour
|
We
know that 100 litres runs in 1 hour and we need to know how
long it takes to run 350 litres.
350 litres is 3.5 times 100 litres, so it takes 3.5 times as
long i.e. 3.5 hours.
Therefore 350 litres will run in 3.5 hours.
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Quick quiz
| 1. |
Express
the following percents as fractions and decimals: |
| a) |
95% |
| b) |
13.5% |
| c) |
42% |
| d) |
1% |
| e) |
0.1% |
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| 2. |
Express
the following fractions as percents: |
| a) |
37/100 |
| b) |
164/100 |
| c) |
25/50 |
| d) |
14/20 |
| e) |
16/25 |
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| 3. |
Express
the following decimals as percents: |
| a) |
0.01 |
| b) |
0.83 |
| c) |
0.005 |
| d) |
1.10 |
| e) |
0.2 |
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| 4. |
Express
the following quantities as ratios: |
| a) |
There
were 3 boys for every 5 girls at school assembly. |
| b) |
Nine
people out of every ten watch television every night. |
| c) |
In
a class of 25 students 3 are left handed and 22 are right handed. |
| d) |
To
cook rice you need 1 cup of rice to 2 cups of water. |
| e) |
To
make ANZAC biscuits, you add the same amount of flour to sugar. |
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| 5. |
The
following ratios of pets owned have been obtained from surveys
of 5 local neighbourhoods. Find at least one equivalent ratio
for each of the following: |
| a) |
dogs : cats = 10 : 20 |
| b) |
cats
: dogs = 30 : 50 |
| c) |
dogs : cats = 100 : 100 |
| d) |
dogs : cats : guinea pigs = 6 : 4 : 2 |
| e) |
fish : turtles = 75 : 25 |
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| 6. |
Express
each part-to-part ratio below as a common fraction, decimal fraction
and a percent of the whole: |
| a) |
men : women : children = 3 : 3 : 4 |
| b) |
men : women : children = 11 : 4 : 5 |
| c) |
adults : children = 4 : 1 |
| d) |
men
: women : children : pets = 5 : 3 : 6 : 1 |
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| 7. |
Write
each of these sentences as a rate: |
| a) |
She
ran 100 metres in 15 seconds. |
| b) |
The
bus travelled 850 kilometres in 10 hours. |
| c) |
Ilana
can type 160 words in 2 minutes. |
| d) |
The
factory packaged 600 packets of biscuits in 5 minutes. |
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Extra
questions:
Shade
20% of this diagram
(a)
What percentage of the following shape is shaded?
(b)
What percentage is not shaded?
3.
What is the ratio of pink to white in the following figure?
4.
A biro is 23 cm long and a ruler is 30 cm long.
(a)
What is the ratio of the biro to the ruler?
(b)
What is the ratio of the ruler to the biro?
5.
Complete the following table.
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Percentage
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Common
Fraction
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Decimal
Fraction
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32%
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0.06
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6.
A dog is three times as long as a bird. The dog is 90 cm long.
(a)
How long is the bird?
(b)
Write a ratio to show the relationship of the dog to the bird.
7.
I have a 300g bag of sugar and a half a kilo of butter.
(a)
What is the ratio of the sugar to the butter?
(b)
What is the ratio of the butter to the sugar?
8.
What two quantities are being measured when the units are km/hr?
To
view the quiz answers, click
here.
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