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Overview: Learning about percentages 1Key words: Percentage,
discount, mark up, tax, GST, increase, decrease, difference,
wastageIncrease, decrease Purpose: This unit is designed to help
tutors who teach courses that require calculations with
percentages, e.g. GST, discounts, wastageTutor Outcomes:By the end
of the unit tutors should be able to: 1. Recognise contexts and
problems that involve percentages 2. Develop lessons in their
teaching context that help learners to solve problems with
percentages
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Section 1: Mathematical BackgroundPage 1: What does % mean?
The symbol % is a combination of the two zeros from 100 and the
sign / which means out of. So % means out of one hundred.
This can be quite misleading for learners because in most
contexts the percentage operates on a quantity that is not 100,
e.g. Find 35% of $86 means you are actually working with $86 not
$100.
Another way to look at it is through the word percent. Per means
for every and cent is the prefix for 100, like a century is 100
years or 100 runs. So percent means for every hundred.
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Section 1: Mathematical BackgroundPage 2: Percentage as a
rate
One way to think about a percentage is as a special rate.At 35%
off you pay 65% or $65 in every $100.At the same rate how much do
you pay for something that normally costs $86?
Normal Price Discount Price1006586?
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Section 1: Mathematical BackgroundPage 3: Percentage as a
rate
All of the things you can do with other rates, like kilometres
per hour, you can do with percentages. Both numbers in the rate can
be multiplied or divided by the same number. 100 100x 86x 86
Normal Price ($)Discount Price ($)1006510.658655.90
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Section 1: Mathematical BackgroundPage 4: Why do we have
percentages?Percentages are used in two main ways in everyday
life:
As operatorsIn many real life situations you find a percentage
of an amount. For example, if you buy something at 30% discount you
pay 70% of the usual price.70% operates on the usual price, e.g.
70% of $60 is $42
2. As proportionsPercentages are often used to compare two or
more proportions. For example, to compare two shooters in a netball
game you might convert the statistics into percentages. Selma gets
32 out of 40 shots so her shooting percentage is 80%Niki gets 33
out of 44 shots so her percentage is 75%
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Section 1: Mathematical BackgroundPage 5: Are percentages always
less or equal to 100%?
Most situations involve percentages less than 100.
In a sale a percentage is taken off the full price so you pay
less than the full price, less than 100%.When you toss a coin at
the start of a sporting match your chances of winning the toss are
one-half or 50%.
Comparison situations can involve percentages greater than
100%.
For example the price of a house was $200,000 in 2000 and
$280,000 in 2010.Compared to the $200,000 the house is now worth
140% of what it was in 2000.
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Section 2: ActivityPage 1: What is a percentage?
Write 35% on the board.What does this mean?Discuss this in small
groups of 3-4 learners.
Record the ideas from each group as they report back.Discuss
things like:% means out of one hundred (/ means divide by, 00 comes
from 100)Per means for every, Cent means one hundred, e.g. Century
is 100 years or 100 runs35% is less than one half but bigger than
one quarter because 50% is one half and 25% is one quarter35% is
about one third because one third is 33.3%35% of something, what is
the something? (Whole needs to be given, e.g. 120 kg)
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Section 2: ActivityPage 2: When do we use percentages
(examples)?
Provide each group of learners with a copy of copymaster 1.This
provides possible real life situations in which percentages may be
involved.
Ask the learners:How might percentages occur in each of these
situations?Can you think of other situations in which percentages
are used?
Share the ideas from each group.Important points are:Percentages
are used in situations where the whole varies, e.g. Goalkickers
take different numbers of shots, people borrow different amounts of
money.Percentages can be more than 100% in comparison situations,
e.g. Lambing percentages are usually between 150-200% where the
number of lambs is compared to the number of ewesPercentages must
be no more than 100% in out of situations, e.g. Jenny goals 35 out
of 60 shots in netball.Percentages are special types of fractions
with denominators (bottom numbers) of 100, e.g. One quarter is 25
hundredths ( ).
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Section 2: Activity
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Section 2: ActivityPage 4: Percentage to Fraction Snap
Play a game of snap with cards made from copymaster 3.This game
is designed to practise simple percentage to fraction
knowledge.
Points that may arise:Nine tenths is one tenth less than the
whole. This is because the whole is ten tenths. So nine tenths is
90% (100% - 10%)Four fifths is one fifth less than the whole. This
is because the whole is five fifths. So four fifths is 80% (100% -
20%)33.3% is another name for one third. This is because 100 3 =
33.3 (recurring).
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Section 2: ActivityPage 5Finding a percentage using place value
knowledge.
To find 10% is the same as dividing by 10.When we divide be 10
the number gets 10 times smaller. The digits move one place to the
right, e.g. 46 10 = 4.6
Use this method to find 10% of:Find 10% of:8075136589Ask
learners to find 5% of 24Record students methods.Look for methods
such as finding 10% then halving to find 5%100%1%
hundredstensonestenthshundredths4646046
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Section 2: ActivityPage 5Finding a percentage using place value
knowledge.
To find 1% is the same as dividing 10% by 10.When we divide be
10 the number gets 10 times smaller. The digits move one place to
the right, e.g. 46 10 = 4.6
Use this method to find 10% of:Find 1% of:8075136589Ask learners
to find 3% of 24Record students methods.Look for methods such as
finding 10% then dividing by ten.
hundredstensonestenthshundredths4646046
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Section 2: ActivityPage 6: Finding percentages of something
Present this problem to your learners or pose a problem with the
same numbers but a different story.Kegs hold 50 litres of
beer.There is 10% allowance for wastage. What a shame!How much beer
is wasted out of each keg?Note: Wastage is loss of beer through
pouring overflow, clearing the hose lines when kegs are changed and
the beer left behind in the keg.Ask the learners to solve the
problem and share their strategies.For example, I know that 10% is
one tenth and one tenth of 50 is 5 litres or 10% is ten out of 100
so it must be 5 out of 50 litres.Present the problem using the
strip diagram (Copymaster 4).
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Section 2: ActivityPage 7: Practice Examples
Refer to Section Three, problem examples 1 - 3, for your
students to practise the ideas introduced so far.
You will need to run off copies of Copymaster 4 for your
students to use.
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Section 2: ActivityPage 8: Adding on GSTAsk your learners what
they understand by GST (Goods and Service Tax).The total price you
pay for any item includes net price, mark up and GST.
Net price is how much the shop pays for the item and the mark up
is the profit the shop makes. These two parts add up to the shop
price. GST is charged on top of the shop price at a rate of
15%.
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Adding on GSTGST is 15%To add on GST we can mentally workout 10%
plus 5%.Look at the following example:
100% 15%115%We can also calculate the GST inclusive price by
multiplying the 200 by 1.15. 200 x 1.15 = $230Section 2:
ActivityItem costs $200 GST = $30
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Pose the following problems:Before GST is added the bottle of
milk costs $4.00.How much do you pay for the milk after GST is
added on?
Section 2: Activity10%5%
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Section 2: ActivityPractice Examples
Refer to Section Three, problem examples 4-5, for your students
to practise the ideas introduced so far.
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Section 3: ExamplesPage 1: Shopping Spree
Mareea wants to buy a top that usually costs $60The shop has a
20% off sale.How much will Mareea save?How much will she pay for
the top?
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Section 3: ExamplesPage 2: Horsing Around
A horse eats about 60% of its own body weight each month.
This horse weighs 550 kilograms.
How much does it need to eat this month?
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Section 3: ExamplesPage 3: Credit Crunch
Warren has $1760 owing on his credit card.He pays 18% interest
per month on what he owes.
How much will Warren pay in interest this month if he does not
pay anything off his card.
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Section 3: ExamplesPage 4: Credit Crunch
The shop price of a pair of jeans is $120.
Add the GST and find out how much you pay for these jeans.
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Section 3: ExamplesPage 5: Honest Phils Car Dealership
The shop price of a car you want is $13,500Honest Phil forgot to
tell you about the GST.
How much GST needs to be added?
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Section 4: AssessmentPage 1: Shoes
At Shoes 4 Less there is a 25% off sale. This pair of shoes
normally costs $160. How much will the shoes cost on sale?
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Section 4: AssessmentPage 2: Weed SprayingThe instructions say
that the spray should be 80% water and 20% concentrate. Your
sprayer takes 5 litres of liquid. How much water should you put in
before topping it up with concentrate?
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Section 4: AssessmentPage 3: Brakes
Ralph has fixed your car brakes. The bill is $280 but GST has to
be added. What will the total bill be?
Be aware that out of describes a part-whole (part of a whole)
relationship. Most percentage problems involve either finding the
part, e.g. 25% of $36 means the part which is one quarter of the
whole ($36). Rarely do we find the whole given the part, e.g. 25%
of an amount is $9. What is the amount?*Rate thinking is very easy
in that in comes from the most basic type of multiplication
problem. It is natural to look for a unit rate, e.g. What is 1% of
$86? Or What is 65% of $1? In the next slide we find 65% of $1.
*Here the unit rate $1 (price): $0.65 is scaled up by 86. This
is like multiplying 86 x 0.65 = $55.90. Alternatively you can work
out 1% of $86 = $0.86 then calculate 65 x 0.86 = $55.90.Either way
it works.*When we use a percentage as an operator it is easy to
think of it like a whole number even when it isnt. Its actually a
fraction, e.g. 25% means 25/100.Treating it like a whole number
lets you use rate thinking to make finding the answer easier. For
example, 35% of something can be found using 10% + 20% + 5% of the
something.Thinking about percentages as proportions, part-whole
relationships, is usually harder to do.*Percentages equal of less
that 100% usually, but not always, involve a part-whole
relationship that can be described as out of.Percentages greater
than 100% always involve a comparison of one whole with another
whole, e.g. Comparing the value of something with its value 10
years ago.*Be aware that there are many legitimate ways to think
about percentages that can prove obstacles later on. For example
out of 100 can restrict the person in terms of comparisons where
the percentages are greater than 100. Ask the learners to provide a
situation that matches their idea about percentages and ask
connecting questions, e.g. What is the same about the Kathmandu
discount situation and the 95% fat free label on the cereal box?
*You may find that drawing strip diagrams of the situations in
copymaster 1 may help students to see the structural similarity
between different problems.Refer to the webinar of percentages to
see how to do this.*Benchmark fractions such as 25%, 50%, 10% are
very useful for calculation and estimation of percentage problems.
These common benchmarks need to be remembered so they are
accessible.*You may find other ways to help learners remember the
benchmark percentages. Bingo is another game that may
work.*Learners often have rules for multiplying by ten, e.g. Add a
zero. Trusting a rule that works is useful but bear in mind that
such rules can also cause confusion unless supported by ideas about
why it works. For example, mistakes with adding zero are common,
e.g. 4.3 x 10 = 4.30. Connect multiplying be ten and dividing by
ten as inverse operations that undo one another. Practising with
anticipation using a calculator helps learners considerably.Enter
is 3.7. If you divide by ten what do you get? (think ahead then
check) What multiplication operation returns 0.37 to 3.7? *In
general:Dividing 100% by 10 gives 10% so dividing an amount by 10
gives 10% of the amount, e.g. $56 10 = $5.60 so 10% of $56 is
$5.60Dividing 100% by 100 gives 1% so dividing an amount by 100
gives 1% of the amount, e.g. $56 100 = $0.56 so 1% of $56 is
$0.56*It is normal to call the shop price the GST exclusive price
to show that GST is not included.Naturally the total price is
called the GST inclusive price to indicate GST has been added.*10%
of $4.00 is $0.40 so 5% of $4.00 is $0.20.So 15% of $4.00 is
$0.60.The GST inclusive price is $4.00 + $0.60 = $4.60*20% is one
fifth so Mareea pays one fifth of $60 which is $12.80% is one fifth
off so Mareea pays $60 - $12 = $48.Pose another question such
as:Mareea wants to buy a something that usually costs $60The shop
has a 35% off sale.How much will Mareea save?How much will she pay
for the something?
*60% of 550 is 0.6 x 550 = 330 kilograms.Another way to do it
is:10 % of 550 = 55kg6 x 55 = 330*18% of $1760 can be solved as
0.18 x 1760 = $316.80It could be solved as 10% of $1760 = $1765% of
$1760 = $881% of $1760 = $17.60 so 3% of $1760 = $52.80176 + 88 +
52.80 = $316.80*15% of $120 is 0.15 x 120 = $18This could be worked
out as 10% of $120 = $12 so 5% of $120 = $6, $12 + $6 = $18*15% of
$13,500 is 0.15 x $13,500 = $2,025*Shoe sale solutionThere are two
main ways to work this out: Method One 25% is one-quarter. 1/4 of
$160 is $40. So you pay $160 - $40 = $120 Method Two 10% of $160 is
$16 5% of $160 is $8 20% of $160 is $32 $32 + $8 = $40 (25% of
$160) So you pay $160 - $40 = $120 *Weed Spray SolutionThere are
two main ways to solve this problem. Method One 10% of 5 litres is
0.5 litres So 80% is 8 x 0.5 = 4 litres Method Two 20% is one-fifth
1/5 of 5 litres is 1 litre 5 litres - 1 litre = 4 litres
*Car Bill SolutionYou can find the total bill this way: 10% of
$280 is $28 5% of $280 is $14 $28 + $14 = $42 So the bill will be
$280 + $42 = $322 On a calculator you could go: $280 + 15% = or
$280 x 1.15 = *