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Percentages II (Tutor Version )PowerPoint Presentation

Percentages II (Tutor Version )

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Key words: Percentage, Goods and Services Tax (GST), GST inclusive, GST exclusive, frequency , equivalent

Purpose: To provide tutors with lesson ideas for introducing more difficult percentage calculations.

By the end of the unit tutors should be able to:

1. Solve percentage problems which involve proportions

2. Develop lessons in their teaching context that help learners to solve problems with percentages

Section 1: Mathematical Background

Page 1: Percentages as proportions

Percentages are equivalent fractions. When we say that two fractions are equivalent that means they are different fractions for the same number.

For example: ¾ = 75/100 means that three-quarters and seventy-five hundredths are the same number. So three-quarters of any amount is always 75% of that amount.

Section 1: Mathematical Background

Page 2: Frequencies

The most common situations where percentages occur involve frequencies.

Frequencies are “out of” situations such as “Warren sinks 24 out of 40 first putts” or “12 out of every 100 people are left-handed.”

Frequencies can be expressed as percentages.

For example, 24/40 = 6/10 = 60/100. So Warren sinks 60% of his first putts.

Section 1: Mathematical Background

Page 3: Ratios as percentages

Ratios are combinations of two or more quantities with the same measures.

For example you might mix one shovel of cement to four shovels of sand to make mortar. That is a 1:4 ratio.

You might mix 20 mL of Kahlua to 30mL of Midori to make a cocktail. That is a 20:30 or 2:3 ratio.

Percentages are a bit harder to find in ratios than frequencies since you have to create the “out of” relationships.

Section 1: Mathematical Background

Page 4: Ratios as percentages a bit harder

Ratios can be expressed as part-whole fractions and as percentages.

So our 2:3 cocktail is 2/5 kahlua and 3/5 midori.

2/5 = 40/100 so the cocktail is 40% kahlua

3/5 = 60/100 so the cocktail is 60% midori.

Section 1: Mathematical Background

Page 5: Ratios as percentages harder

Some ratios are not parts of the same whole.

In growth and reduction situations you might compare different wholes.

For example you buy some shares for $400 and sell them two years later for $480.

The ratio 400:480 compares two different amounts.

In this case you sell the shares for 120% of what you paid for them (480/400 = 120/100).

Page 1: Cow of a problem

Suppose a dairy farm has 150 Jersey cows and 250 Fresian cows. That’s 400 cows altogether.

What percentage of the herd are Jerseys?

150 Jerseys

400 cows

Page 2: Cow of a problem solved

150 out of 400 is the same fraction as...

75 out of 200

37.5 out of 100 or 37.5%

150 Jerseys

75 Jerseys

37½ Jerseys

400 cows

200 cows

100 cows

Page 3: Sporty Spice

Suppose a basketball player sinks 45 out of 60 shots at goal.

What is her shooting percentage?

60 Shots

The total of 60 shots can be broken up into quarters. There are 15 shots in each quarter.

45 is three lots of 15 so 45 is three-quarters of 60 or 75%

45 Shots

60 Shots

15 Shots

15 Shots

15 Shots

15 Shots

Page 4: Sporty Spice Harder

Suppose a basketball player sinks 36 out of 45 shots at goal.

What is her shooting percentage?

45 Shots

The total of 45 shots can be broken up into fifths. There are nine shots in each fifth.

36 is four lots of nine shots so 36 is four-fifths of 45 or 80%

36 Shots

36 Shots

45 Shots

Page 5: Smoking Statistics

A group of 200 young adults are surveyed.

60 of them smoke regularly.

What percentage of the group are smokers?

There are ten lots of 20 people in 200 people so the group can be divided into tenths.

So 60 out of 200 is the same as 3 out of 10 or 30 out of 100.

That is 30%.

Of course you could also use equivalent fractions, 60/200 = 30/100.

200 adults

200 adults

60 smokers

60 smokers

Page 6: Try a few problems

At this stage you might like to try a few percentage problems just like these?

Go to slides 1-4 of the examples section to find some problems.

60 smokers

60 smokers

Page 6 GST

Section 2: Activity

- GST is a government tax which is added on to the selling price of all goods we buy.
- The letters GST stand for Goods and Services Tax.
- GST was first applied in New Zealand on October 1, 1986 and was 10%.
On July 1, 1989 it increased to 12.5%.

On October 1, 2010 it increased to 15%

Section 2: Activity Page 7: GST at 10%

When GST was 10% we had to add 10% to the price of goods or any services. We divided the cost by 10 and added the GST on.

Article for purchase + GST

100% + 10%

GST inclusive

Article cost $100 GST $10

GST inclusive price $110

Article costs $50 GST $5

GST inclusive price $55

Page 8: Finding the GST Component when it was 10%

Section 2: Activity

People often had to find the GST component in a price for tax reasons.

Suppose the article cost $110 (GST inclusive)

GST

To find the GST content they divided by 11 because there are 11 lots of 10% in the GST inclusive price. Dividing by 11 gave 1/11 of the GST inclusive price.

$110 ÷ 11 = $10

Article costs $55 (GST inclusive)

$55 ÷ 11 = $5 (GST)

GST

Page 9 Doing and undoing

You could see this as doing and undoing. Multiplying by 1.1 has the same effect as adding on one tenth.

x 1.1

GST exclusive price

GST inclusive price

÷ 1.1

Dividing by 1.1 is like taking off on one eleventh.

So dividing the GST inclusive price by 1.1 returned you to the GST exclusive price.

Page 10: An historical view

At 10% GST...

The plasma television cost $660 (GST inclusive).

How much of the price was GST?

Page 11: Finding the GST component when it was 12%

Section 2: Activity

If you divide 100% by 12% you get 8. So 12.5% is another name for one-eighth.

The price of goods was divided by 8 to find how much GST needed to be added on.

An article cost $100. How much GST has to be added on?

GST

The GST content on $100 was $12.50.

The total price of the article being sold was $112.50 (GST inclusive)

To find the GST component you divided by 9, because there are nine lots of 12% in the GST inclusive price. $112.50 ÷ 9 = $12.50

Find the GST component on something that costs $36.

GST

Page 12: Doing and undoing again

You could see this as doing and undoing. Multiplying by 1.125 has the same effect as adding on one eighth.

x 1.125

GST exclusive price

GST inclusive price

÷ 1.125

Dividing by 1.125 is like taking off on one ninth.

So dividing the GST inclusive price by 1.125 returned you to the GST exclusive price.

Page 13: Another historical view

At 12.5% GST...

The plasma television cost $1,200 (GST inclusive).

How much of the price was GST?

Page 14: Now that GST is 15%

With GST now at 15% adding GST onto an exclusive price is still very easy.

There are a couple of easy ways to do this:

Suppose the GST exclusive price is $320

Method One:

Divide the price by 100 to get 1% $320 ÷ 100 = $3.20

Multiply that result by 15 to get 15% $3.20 x 15 = $48

Add this on to the exclusive price $320 + $48 = $368

GST exclusive price

1%

100%

GST

15%

115%

Page 15: More adding on 15%

Suppose the GST exclusive price is $320

Method Two:

Divide the price by 10 to get 10% $320 ÷ 10 = $32.00

Halve that to get 5% $32.00 ÷ 2 = $16.00

Add 10% and 5% to get 15% $32.00 + $16.00 = $48.00

Add this on to the exclusive price $320 + $48 = $368

10%

100%

15%

GST

GST exclusive price

15%

115%

Page 16: Doing

You could see adding 15% as finding 115% of the exclusive price. Multiplying by 1.15 has the same effect as adding on 15%.

x 1.15

GST exclusive price

GST inclusive price

On our price of $320.00

$320 x 1.15 = $368

Page 17: Taking GST off

If you divide the total price by 115 you will get 1% of the GST exclusive price.

So to work out the GST multiply 1% by 15 to get 15%.

So to work out the shop price multiply by 100 to get 100%

GST inclusive price

GST exclusive price

GST

1%

100%

115%

Page 18: An example

You are working out an expense claim for your boss.

You bought a digital camera for $230.00 (GST included).

How much of the price was GST?

Page 19: Another example

You put $69 of petrol in you car.

How much of the price is GST and how much does the garage get?

Page 20: Another way to take GST off

GST is 15% of the GST exclusive price.

So altogether the GST inclusive price is 115% of the exclusive price.

15% of GST exclusive price

GST

Shop price (100%)

115% of shop price

There are 23 lots of 5% in 115%.

20 lots of 5% are the GST exclusive price and 3 lots of 5% are the GST.

So GST is 3/23 of the GST inclusive price.

Page 21: Petrol Again

You put $69 of petrol in you car.

How much of the price is GST and how much does the garage get?

Use the 3/23 method to work this out.

Page 22: Doing and undoing again

You could see this as doing and undoing. Multiplying by 1.15 has the same effect as adding on 15%.

x 1.15

GST exclusive price

GST inclusive price

÷ 1.15

Dividing by 1.15 is like taking off on three twenty-thirds.

So dividing the GST inclusive price by 1.15 returns you to the GST exclusive price.

Page 23: Petrol Again and Again

You put $69 of petrol in you car.

How much does the garage get from the sale?

Use divide by 1.15 method to work this out.

Page 24: Try some problems

Go to slides 5-8 of the examples section to find some GST problems.

60 smokers

60 smokers

Page 1: Turtle Tragedy

Of the 1200 baby turtles that hatch out only 384 make it to the sea.

The rest are eaten by predators.

What percentage of baby turtles make it?

Page 2: Crayfish Catch

Last year only 25% of the crayfish you caught were under sized.

This year you put back 16 of the 48 crayfish you caught.

Is the percentage of undersized crayfish increasing?

Page 3: Sharp Shooting

Karl got 60 of his 75 arrows on target.

What percentage was that?

Page 4: Whale of a whale

A baby humpback at birth measures 4.2 metres long and weighs 2.5 tonnes.

After one month it measures 6.0 metres long and weighs 4.0 tonnes.

By what percentages has its length and weight grown?

Page 5: Add GST

A naughty shop gives the GST exclusive price of a pair of jeans as $72.00

How much GST will be added on and how much in total will you pay for the jeans?

Page 5: Add GST

GST is often charged on the sale of land.

A small block has a GST exclusive price of $483,000.

How much GST will be charged on top of the price?

Page 7:

The Hoata family bought a new

Laptop for $2,990 (incl. GST).

“I use this for my business,”

said Paora, “So I can claim back

the GST.”

How much GST can Paora claim back?

Page 8: Taxi Fare

You pay $57.50 for a taxi trip.

How much of the cost is GST and how much does the taxi driver get?

Page 1: New Zealand has one of the highest rates of car ownership in the world.

There are about 2.5 million cars for about 4 million people.

What percentage of 4 million is 2.5 million?

Page 2: GST Burger Please

Even a hamburger and fries has GST included in the price.

If the GST inclusive price is $5.80 how much GST are you paying?

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