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Benchmark percents | Percent
increases and decreases |
Equivalent ratios | Quick
quiz |
Benchmark percents
Some
equivalent percents, common fractions and decimal fractions we should
easily be able to recall are:
 |
 |
=
1.0 = 1 |
 |
 |
=
0.75 |
 |
 |
= 0.5 |
 |
 |
=
0.25 |
 |
 |
= 0.2 |
 |
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=
0.8 |
 |
 |
=
0.15 |
 |
 |
=
0.05 |
 |
 |
=
0.125 |
 |
 |
=
0.33 (rounded to 2 decimal places) |
 |
 |
=
0.67 (rounded to 2 decimal places) |
Knowing
these values which are very commonly used in our daily lives means
that we can quickly and easily find the answer to many percent problems
with a minimum of extended working out.
For
example, if I want to find out what percent of the pizza has been
eaten and I know that there is 1/4 of the pizza left, then I know
75% has been eaten.
For
example, if I know that there is a 33 1/3% increase in wheat production
I also know that wheat production has increased by one third.
Estimation
We
can apply our knowledge of benchmarks to making estimations. When
using percents in every day situations an estimation will usually
suit our needs. Estimations are also useful when we want to have a
fair idea of the answer expected to a problem before working it out
accurately.
For
example,
| 47%
is just under half |
| 70%
is just over two thirds |
| 19%
is about one fifth |
| 69%
of 311 is about two thirds of 311 and is a bit over 200 |
| 48%
of 820 is just under 410 |
Percent increases
and decreases
This
subtopic has been included prior to Percent
Examples because it introduces key ideas about percent in situations
with which most of us are familiar. Please note, many more percent
problems and explanations are included in the Percent
Examples section.
Example
1:
If the price of a jacket was $200 and has now been reduced to $100,
what is the percent decrease?
What
did you answer to this question? Did you answer 100% or 50%?
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Thinking
|
|
If
the jacket was originally $200 and it has been reduced to $100
then it has been marked down by half of its original price ($100
out of $200), it has been reduced by 50%.
(If you thought that the price was reduced by 100% then this
would mean that the jacket was actually marked down to nothing.
To check your answer, ask '100% of what?' 100% of the original
price of $200 is $200, so $200 reduced by $200 leaves nothing!
The answer cannot be 100%)
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Percent
increases and decreases are always taken as a fraction of the original
amount, in this case, price.
The
dual-scale number line model, discussed earlier in Meaning
and Models: Percent Models, can be a clear way of illustrating
the problem.
|
We
have labelled the percent scale on the left hand side of the
number line. We have labelled the right hand side with the corresponding
dollar amounts in the problem.
|
We
know that $200 is the original price, ie. the 100% price. We
know that $100 is the reduced price. We need to find out what
percent of the original price is the reduced price. We have
put a '?' next to the information we need to find out.
Now
using this model we can see by equivalent fractions that,
?/100 =$100/$200
and that the discount is 50%.
This
model also clearly shows that the answer could not be 100%,
since 100% is $200.
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Example
2:
If the price of a TV went up from $200 to $300, by what percent did
it increase?
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Thinking
|
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The
TV was originally priced at $200. It was increased to $300,
which means it was increased by half of the original price.
$200 increased to $300 is an increase of $100 out of $200 which
is 50%.
The new price is 150% of the old price.
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People
commonly make errors when working with percent increases and decreases.
The important question to ask is 'the
percent of what?'.
Everyday
usage of percent
In
order to use percent in our everyday lives we need to understand the
common expressions we use to talk about them.
'Gone
up by 20%' means an increase of 20%
of the original quantity.
The
movie below shows two different ways to calculate an increased quantity.
Example
3: Movie
- calculating an increased quantity.
Since the recent rains the production of cotton has gone up by 20%.
If a farmer produced 300 tonnes prior to the rain, how much will he
produce now?
We
have used both the methods shown below to find the increased quantity
of 20%.
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new
amount = old amount + (20% of old amount), OR
new
amount = old amount x (1.20)
NOTE:
We have expressed 1.20 as 1.2 because it is easier to read on
the screen. '1.20' is the same as '1.2'.
|
'Gone
up to 110%' means an increase of 10%.
(NOTE:
This expression can easily be confused with 'gone up by 110%'
which means an increase of 110%!)
Example
4: Movie - calculating an increased quantity of 10%.
 |
The
opposition was quoted as saying 'since the introduction of the
GST books have gone up to 110% of their previous price....'. If
the total cost of 250 new books for the library was $3500 pre-GST,
how much would the cost be now? |
As
we are still talking about an increase we have used the same methods
to find the new amount.
|
new
amount = old amount + (10% x old amount), OR
new
amount = old amount x (1.10)
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'Went
down by 20%' or 'discounted by 20%' or 'marked down by 20%' are common
terms used when we are talking about money. All of these terms mean
a decrease of 20%.
Example
5: Movie - calculating a decreased quantity of 20%
A
shoe store is having a sale where everything has been discounted by
20%. At the sale, Tan bought a pair of shoes for $40. What was the
original price of the shoes?
We
also could have used the following methods, but as we knew the 'new
amount' and were looking for the 'old amount' we would have had to
transpose the equations to find the 'old amount'.
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New
amount = old amount - (20% x old amount), OR
new
amount = old amount x 0.80, OR by transposing this equation,
old
amount = new amount / 0.80
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'Went
down to 75%' or 'marked down to 75%' means a decrease of 25%.
(NOTE:
It is easy to confuse 'went down to 75%' with 'went down by
75%' - they do not mean the same thing!)
Example
6: Movie - calculating a decreased quantity of 75%
Since the increase in competition amongst telecommunications companies,
the price of international phone calls has decreased by 25%.
If
the average cost per month of international calls in our household
was $400, what would it be now with the reduced call costs?
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new
amount = old amount - 25% of old amount, OR
new
amount = old amount x (1 - 0.25)
new amount = old amount x 0.75
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Less
common percent increases and decreases
Although
we have covered percent increases and decreases we most commonly come
across, it is important to still be able to work with more complicated
numbers.
Example
7:
Council rates are set to increase by 7.5% this year. If my rates were
$815 last year, what will they be this year?
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Setting
Out
|
| new
amount |
=
old amount x 1.075 |
| |
=
815 x 1.075 |
| |
=
876.125 |
| |
Answer
is $876.13 (rounded to the nearest cent) |
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|
Thinking
|
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I
need to calculate the increased amount of 7.5% on $815. To do
this I multiply the old amount by 1 plus the percent increase.
The percent increase expressed as a decimal is 0.075. Therefore
I will multiply the old amount by 1.075.
NOTE:
A VERY common error is to express 7.5% as 0.75. To avoid this
error, remember that 7.5% is 7.5 divided by 100 (7.5/100). which
is 0.075 (A bit more than 7 hundredths 0.07).
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More
examples are shown in the table below.
| %
change |
Expressed
as a fraction and as a decimal |
Calculation |
| an
increase of 8.3% |
8.3/100,
0.083 |
new
amount = old amount x 1.083 |
| a
decrease of 0.5% |
0.5/100,
0.005 |
new
amount = old amount x (1 - 0.005)
new amount = old amount x 0.995 |
| an
increase of 27.5% |
27.5/100,
0.275 |
new
amount = old amount x 1.275 |
Example
8:
It has been forecasted that property values are set to rise 3% per
year over the next 3 years. If Lionel's property is worth $250 000
now, what will it be worth at the end of the next 3 years?
| Setting
Out |
Thinking
|
| Year
1 |
|
| new
amount |
=
$250 000 x 1.03 |
| |
=
$257 500 |
| |
|
| Year
2 |
|
| new
amount |
=
$257 500 x 1.03 |
| |
=
$265 225 |
| |
|
| Year
3 |
|
| new
amount |
=
$265 225 x 1.03 |
| |
=
$273 181.75 |
| |
|
|
We
know that to calculate an increase of 3% we use the following
formula,
new
amount = old amount x 1.03
(we
multiply by 1.03 because we have an increase of 3/100 or 0.03)
This
will calculate our increase for the first year.
As
there is the same percent increase in the second year we can
also multiply the amount at the end of the first year by 1.03
As
there is also the same percent increase in the third year we
can multiply the amount at the end of the second year by 1.03.
The
property has increased in value to $273 181.75 by the end of
the third year.
|
| Alternative
method |
Thinking |
| new
amount |
=
$250 000 x 1.03 x 1.03 x 1.03 |
| |
=
$273 181.75 |
| |
|
|
We
can see from above that for each year of increase we multiplied
the new amount by 1.03. We could have worked out this problem
in one step by multiplying the original amount by 1.03, 3 times.
Try it on your calculator!
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Value
for Money
A
common application of percent is comparing value for money.
Example
9:
Which video recorder is the best buy? Find the percent discount of
each video and determine which has the biggest percent discount?
|
$190, save $15
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$320
save $40 |
$225,
save $30 |
|
$375,
save $40
|
$495,
save $60
|
$550,
save $75
|
The
percent discount for each item is,
| 15/205x100
= 7.32% |
40/360x100=11.11% |
30/255x100=11.76% |
| 40/415x100=9.64% |
60/555x100=10.81% |
75/625x100=12% |
The
video recorder with a discounted price of $550 has the largest percent
discount (although not by much) of 12%.
Equivalent
ratios
Example
10:
Jay
loves white chocolate mud cake. He is eating a piece of chocolate
mud cake which contains 1 part white chocolate to 2 parts milk chocolate
when he sees another chocolate mud cake in a shop window which is
advertised as containing 6 parts white chocolate to 12 parts milk
chocolate. Does the cake in the window contain a higher ratio of white
to milk chocolate?
|
Thinking
|
|
Jay's
piece of cake has a ratio of white to milk chocolate of 1 :
2. The cake in the window has a ratio of white to milk chocolate
of 6 : 12.
We
can see that 6 : 12 is the same ratio as 1 : 2 because if there
are 6 parts of white to 12 parts of milk, then there must be
1 (smaller) part of white to 2 parts of milk chocolate. 6:12
can be simplified to 1 : 2 (by dividing both terms by 6). Therefore
the cakes contain the same ratio of white to milk chocolate.
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In
example 4 in Meaning
and Models, Ratio we saw that the ratio of 1 : 4 is equivalent
to 2 : 8. If we mixed blue to white paint in the ratio of 1 : 4 and
then mixed some more paint later in the ratio 2 : 8 we would get exactly
the same colour.
Realising
that ratios have many equivalent ratios is crucial to using ratios
efficiently.
Simplifying
Ratios
Example
11: Simplify the following:
| Example |
Thinking |
| boys
: girls = 6 : 12 |
Both
numbers can be divided by 6, so 6 : 12 = 1 : 2 |
| hot
chocolate : coffee : tea = 3 : 27 : 18 |
All
numbers can be divided by 3, so 3 : 27 : 18 = 1 : 9 : 6 |
| blue
: red : yellow = 16 : 12 : 20 |
All
numbers can be divided by 4, so 16 : 12 : 20 = 4 : 3 : 5 |
| students
: computers = 28 : 9 |
There
is no whole number which will divide both numbers exactly, however
we can divide both numbers by 2 to give, 14 : 4.5 |
The
total number of parts
Another
important aspect of ratios is understanding how many parts there are
in total. We
often want to compare the ratio of parts but we may also want to know
the ratio of parts compared to all the parts in total.
Example
12:
If
red to yellow paint has been mixed in the ratio of 1 : 3, this means
there is 1 part red to 3 parts yellow and that there are 4 parts in
total.
One
ratio can give us a lot of information. If I know that the ratio of
red to yellow paint is 1 : 3 I also know:
| the
ratio of yellow to red is 3 : 1 |
| the
ratio of red to the total number of parts is 1 : 4 |
| the
ratio of yellow to the total number of parts is 3 : 4 |
| We
could also say we have 1/4 red and 3/4 yellow. |
| We
could also say that we have 25% red and 75% yellow. |
We
can see that when we are comparing parts to the whole we can do so
using ratios, fractions or percents.
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Example
13: Equivalent ratios..
Concrete is mixed in the ratio of 3 : 2 : 1 = screenings : sand
: cement. Work out the quantity of each amount required to make
a concrete mix.
(a) If I add 6 shovels of sand, how much screenings and cement
will I need?
(b) If I add 12 shovels of screenings,how much sand and cement
will I need to add?
(c) If I add 4 bags of cement, how much sand and screenings
will I need to add?
(d) If I add 3 wheelbarrows of sand, how many barrows of screenings
and cement will I need?
|
|
Thinking
|
|
(a)
If I need to mix screenings : sand : cement in the ratio of
3 : 2 : 1 and I have 6 shovels of sand then this represents
2 parts. So 1 part would be 3 shovels. So the amount of screenings
would be 3 x 1 part = 3 x 3 shovels = 9 shovels. The amount
of cement would be 1 x 1 part = 1 x 3 shovels = 3 shovels. So
the amounts are 9, 6 and 3 respectively.
(b)
Now I have 12 shovels of screenings. This represents 3 parts
in the ratio of screenings 3 : sand 2 : concrete 1. Therefore
1 part is four shovels. So the amount of sand is 2 x 4 shovels
= 8 shovels. The amount of concrete required is only 1 part
so this is 4 shovels. 3 : 2 : 1 = 12 : 8 : 4
(c)
I now have 4 bags of cement. There is only 1 part cement to
3 parts screenings and 2 parts sand required, so 4 bags of cement
is equivalent to 1 part. To make the concrete mix in this case,
I need 12 bags of screenings and 8 bags of sand. All the bags
must be the same size.
3 : 2 : 1 = 12 : 8 : 4
(d)
Now I have 3 wheelbarrows of sand. There are 2 parts sand in
the concrete mix ratio, so 3 wheelbarrows represents 2 parts,
and 1.5 wheelbarrows represents 1 part. Therefore 3 x 1.5 wheelbarrows
of screenings are required and 1 x 1.5 wheelbarrows of cement
is required.
3 : 2 : 1 = 4.5 : 3 : 1.5
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| NOTE:
This ratio of 3
: 2 : 1 = screenings : sand : cement will
apply for any measure of volume, such as a bucket, bag,
shovel, spade etc. It is not a ratio of a measure of weight. We
can see that this makes sense since the weights and volumes of
sand, screenings and cement would vary considerably. Volume is
an easier measure to use when concreting. |
Quick quiz
| 1. |
Express
the following percents as fractions and decimals: |
| a) |
33 1/3% |
| b) |
231% |
| c) |
0.03% |
| d) |
75% |
| e) |
12.5%
|
| |
|
| 2. |
Calculate
the new price if the following items were marked down by 20%: |
| a) |
a
bottle of juice at $2.30 |
| b) |
a
computer at $1 900 |
| c) |
a
litre of oil at $6.50 |
| d) |
a
movie ticket at $11 |
| e) |
a
new car at $40 000
|
| f) |
a CD at $30 |
|
|
| 3. |
Calculate
the new price if the following items went up by 15%: |
| a) |
a bottle of juice at $2.30 |
| b) |
a computer at $1 900 |
| c) |
a litre of oil at $6.50 |
| d) |
a movie ticket at $11 |
| e) |
a new car at $40 000 |
| f) |
a CD at $30 |
|
|
| 4. |
Estimate
the following percentages (DO NOT WORK THEM OUT ON YOUR CALCULATOR): |
| a) |
Estimate 9% of 70
|
| b) |
Estimate 48.5% of 120 |
| c) |
Estimate 24% of 200 |
| d) |
Estimate 0.5% of 100 |
| e) |
Estimate 98% of 163 |
| f) |
Estimate 19% of 5000 |
| g) |
Estimate 47% of 820 |
| h) |
Estimate 69% of 8960 |
|
|
| 5. |
Find
at least one equivalent ratio for the following: |
| a) |
pens : pencils = 6 : 12 |
| b) |
blue eyes: brown eyes = 1 : 3 |
| c) |
teachers : students = 2 : 7 |
| d) |
water : orange juice : softdrink = 7 : 22 : 35 |
| e) |
A : B : C : D = 3 : 9 : 24 : 36 |
| f) |
width
: length : height = 5 : 10 : 26 |
|
|
| 6. |
Simplify
the following ratios into lowest terms: |
| a) |
X : Y : Z = 14 : 6 : 28 |
| b) |
cm on map: cm on ground = 2 : 50 000 |
| c) |
L
: W : H = 12 : 4 : 48 |
| d) |
inches on ground : inches on map = 30 000 : 3 |
To
view the quiz answers, click
here.
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