1. Percentages II (Tutor Version)

2. Overview Percentages B
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.
Inclusive means inside or within so GST inclusive means the GST is included in the price.
Exclusive means outside or without so GST exlcusive means the shop price before GST is added on.
A proportion is a part to whole fraction. So 4 out of 5 is a proportion. Four is the part and five is the whole.
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

3. 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.
When we say that fractions are equivalent we mean they refer to the same number. This requires us to think about them operating on the same whole. The tricky part is that in real life the wholes are often different. For example, 32 out of 40 involves a different whole to 42 out of 50. The percentages of 80% and 84% for these proportions is like mapping them to the same whole of 100.

4. 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.
Frequencies are the easiest percentage problems to solve since the part and the whole are clearly defined.

5. 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.
It is important to note that for most ratios the parts must be combined to get the whole. So for the ratio 5:8 the whole is = 13 parts.

6. 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.

7. 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).
Ratios where two wholes are compared are common in finance, e.g. Comparing prices over time, and construction, e.g. Trigonometric ratios are relationships between side lengths.

8. What percentage of the herd are Jerseys?
Section 2: Activity
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?
400 cows
150 Jerseys
This is a 150:250 ratio so the whole is 400 cows.
150/400 = 3/8 (dividing by 50)
3/8 = 37.5/100 = 37.5% (multiplying by 12.5)

9. Page 2: Cow of a problem solved
Section 2: Activity
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%
400 cows
150 Jerseys
200 cows
75 Jerseys
Note that even though the wholes are changing 400 to 200 to 100 we are treating them as the same whole.
100 cows
37½ Jerseys

10. Suppose a basketball player sinks 45 out of 60 shots at goal.
Section 2: Activity
Page 3: Sporty Spice
Suppose a basketball player sinks 45 out of 60 shots at goal.
What is her shooting percentage?
60 Shots
45 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%
Breaking up ratios like this involves looking for common factors, that is numbers that divide into both the parts and wholes of the ratio. In this problem 45 and 60 divided by 3,5,and 15. These are the common factors of 45 and 60.
60 Shots
15 Shots
15 Shots
15 Shots
15 Shots

11. Page 4: Sporty Spice Harder
Section 2: Activity
Page 4: Sporty Spice Harder
Suppose a basketball player sinks 36 out of 45 shots at goal.
What is her shooting percentage?
45 Shots
36 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%
Both 36 and 45 divide by 9. So 9 is a common factor of 36 and 45.
45 Shots
36 Shots

12. Page 5: Smoking Statistics A group of 200 young adults are surveyed.
Section 2: Activity
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
60 smokers
There are two methods on this page:
Common factors: 60/200 (Both have 20 as a common factor), so 60/200 = 3/10 = 30/100
Scaling: 60/200 = 30/100 (Scaling both numbers by one half)
200 adults
60 smokers

13. Section 2: Activity
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

14. The letters GST stand for Goods and Services Tax.
Section 2: Activity
Page 6 GST
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%
The choice of a percentage for GST always considers the size of the percentage and the difficulty of calculating GST on goods and services.
10% was a good first choice because 10% = 1/10 so calculations were relatively easy.

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% %
GST inclusive
Article cost $ GST $10
GST inclusive price $110
Article costs $ GST $5
GST inclusive price $55
10%

16. Page 8: Finding the GST Component when it was 10%
Section 2: Activity
Page 8: Finding the GST Component when it was 10%
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)
10%
$10
Drawing diagrams, especially strips, is always helpful to find out the parts and the whole in a problem. This is used in Singapore with great success.
The hard aspect of taking off GST is realising that the whole is now greater than the exclusive price. So when 10% was added the whole became 110% of the original shop price. So 110% is the new whole.
10%
105 $5
GST

17. Dividing by 1.1 is like taking off on one eleventh.
Section 2: Activity
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.
GST exclusive price
GST inclusive price
x 1.1
÷ 1.1
This idea will work for some learners and not others. It relies on their belief that opposites undo one another, e.g. Turning off a light undoes turning it on. Operations that undo one another are called inverse operations.
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.

18. Page 10: An historical view
Section 2: Activity
Page 10: An historical view
At 10% GST...
$660 was 110% of the original price.
So 660 ÷ 11 = $66 must have been 10% of the original price. That is the GST amount that was added.
The plasma television cost $660 (GST inclusive).
How much of the price was GST?

19. Page 11: Finding the GST component when it was 12%
Section 2: Activity
Page 11: Finding the GST component when it was 12%
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 $ The total price of the article being sold was $ (GST inclusive) To find the GST component you divided by 9, because there are nine lots of 12% in the GST inclusive price. $ ÷ 9 = $12.50 Find the GST component on something that costs $36.
12% +
$12.50
GST
12.5% seemed an untidy percentage but the fact that it was one-eighth of 100% made calculation quite easy. So people quickly learned that 1/8 = 12.5%.
At GST of 12.5% the component for a $36.00 item was $36 ÷ 9 = $4

20. Page 12: Doing and undoing again
Section 2: Activity
Page 12: Doing and undoing again
You could see this as doing and undoing. Multiplying by has the same effect as adding on one eighth.
GST exclusive price
GST inclusive price
x 1.125
÷ 1.125
It is a good idea to check the doing and undoing method with the answers you get from other methods.
Dividing by is like taking off on one ninth.
So dividing the GST inclusive price by returned you to the GST exclusive price.

21. Page 13: Another historical view
Section 2: Activity
Page 13: Another historical view
At 12.5% GST...
The solution is:
$1,200 ÷ 9 = $133.33
That means the GST exclusive price was $1,200 - $ = $1,
By doing and undoing:
$1, x = $1,200 (it works!)
$1,200 ÷ = $1, (it works again!)
The plasma television cost $1,200 (GST inclusive).
How much of the price was GST?

22. GST exclusive price Section 2: Activity 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%
Finding 1% is called a unit rate approach. This is a common method for solving rate problems and always works, no matter what the problem is.

23. GST exclusive price Section 2: Activity 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% $ $ = $48.00
Add this on to the exclusive price $320 + $48 = $368
GST exclusive price
10%
100%
GST
15%
115%
In this method we also treat the percentage as a rate, like this:
Percentage 100% 10% 5% 15% 115%
Dollars $320 $32 $16 $48 $368

24. Section 2: Activity
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

25. GST inclusive price GST exclusive price Section 2: Activity
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 1%
100%
115%
GST exclusive price
GST
The important thing is to treat 115% of the exclusive price as the new whole.
Finding 1% is the unit rate approach again.

26. You are working out an expense claim for your boss.
Section 2: Activity
Page 18: An example
You are working out an expense claim for your boss.
You bought a digital camera for $ (GST included).
How much of the price was GST?
One solution strategy is:
$230 ÷ 115 = $2 so two dollars is 1% of the GST exclusive price.
15 x $2 = $30 so 30 dollars is 15% of the GST exclusive price.
You paid $30 GST.

27. You put $69 of petrol in you car.
Section 2: Activity
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?
One solution strategy is:
$69 ÷ 115 = $0.60 so 60 cents is 1% of the GST exclusive price.
15 x $0.60 = $9.00 so nine dollars is 15% of the GST exclusive price. You paid $9.00 GST.
100 x $0.60 = $60.00 so the garage that much.

28. 15% of GST exclusive price
Section 2: Activity
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.
Shop price (100%)
15% of GST exclusive price
GST
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.
This is a harder way to look at the GST as it involves common factors.
Both 15%,100% and 115% all have 5% as a common factor.
In this way you can think of 15/115 = 3/23 (dividing by 5).

29. You put $69 of petrol in you car.
Section 2: Activity
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.
The solution strategy is:
$69 ÷ 23 = $3 so $3 is 1/23 of the GST inclusive price.
3 x $3 = $9 so nine dollars is 3/23 of the GST inclusive price. You paid $9.00 GST.
20 x $3 = $60 so 60 dollars is 20/23 of the GST inclusive price. The garage got $60 from the sale.

30. Section 2: Activity
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%.
GST exclusive price
GST inclusive price
x 1.15
÷ 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.

31. Page 23: Petrol Again and Again You put $69 of petrol in you car.
Section 2: Activity
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.
The solution strategy is:
$69 ÷ 1.15 = $60 so the garage got $60 from the sale.

32. Section 2: Activity
Page 24: Try some problems
Go to slides 5-8 of the examples section to find some GST problems.
60 smokers
60 smokers

33. Of the 1200 baby turtles that hatch out only 384 make it to the sea.
Section 3: Examples
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?
Strategies to solve this problem might be:
1. 384/1200 = 32/100 (dividing both numerator and denominator by 12)
This could be done in steps by looking for common factors, 384/1200 = 192/600 = 96/300 = 32/100
÷ 1200 = So each turtle is worth %. 384 x = %

34. Last year only 25% of the crayfish you caught were under sized.
Section 3: Examples
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?
Strategies to solve this problem include:
16/48 = 8/24 = 1/3 so the percentage of undersized crayfish was %
16/48 is about 32/100 (doubling both numerator and denominator)
Both answers show the percentage of undersized crayfish is increasing.

35. Karl got 60 of his 75 arrows on target. What percentage was that?
Section 3: Examples
Page 3: Sharp Shooting
Karl got 60 of his 75 arrows on target.
What percentage was that?
Strategies to solve this problem include:
60/75 = 20/25 (dividing by 3), 20/25 = 80/100 (multiplying by four)
100 ÷ 75 = So each arrow is worth %. 60 x = %

36. After one month it measures 6.0 metres long and weighs 4.0 tonnes.
Section 3: Examples
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?
Possible strategies include:
Length – 6.0 ÷ 4.2 = which is % (2dp.) so the whale has grown by about 43% of its birth weight or 6.0/4.2 is about 150/100 (multiplying by 25) so about 50%
Weight – 4.0/2.5 = 40/25 (multiplying by 10), 40/25 = 160/100 (multiplying by four), so the baby whale has added 60% to its birth weight.

37. Section 3: Examples 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?
Some solution strategies are:
10% of $72.00 is $7.20 so 5% is $3.60. $ $3.60 = $10.80 which is 15% (GST)
So you will pay $ $10.80 = $82.80
2. 1% of $72.00 is $0.72. $0.72 x 15 = $10.80 (GST) and $0.72 x 115 = $82.80 (GST inclusive price)

38. GST is often charged on the sale of land.
Section 3: Examples
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?
Some solution strategies are:
$483,000 ÷ 100 = $4,830 so that is 1% of the price.
$4,830 x 15 = $72,450 is the GST
2. 10% of $483,000 = $48,300 so 5% is $24,150.
$48,300 + $24,150 = $72,450 (15% of the price).

39. The Hoata family bought a new Laptop for $2,990 (incl. GST).
Section 3: Examples
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?
Some solution strategies are:
1. $2,990 ÷ 115 = $26 so $26 is 1% of the GST exclusive price.
$26 x 15 = $390 is the GST (15% of the GST exclusive price).
$2,990 ÷ 23 = $130, $130 x 3 = $390 is 3/23 of the GST inclusive price.
$2,990 ÷ 1.15 = $2,600 (GST exclusive price), $2,990 - $2,600 = $300 (GST).

40. How much of the cost is GST and how much does the taxi driver get?
Section 3: Examples
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?
Some solution strategies are:
1. $57.50 ÷ 115 = $0.50 so $0.50 is 1% of the GST exclusive price.
$0.50 x 15 = $7.50 is the GST (15% of the GST exclusive price).
So the driver gets $50.00 of the fare.
$57.50 ÷ 23 = $2.50, 3 x $2.50 = $7.50 is 3/23 of the GST inclusive price.
$57.50 ÷ 1.15 = $50.00 (GST exclusive price), $ $50.00 = $7.50 (GST).

41. There are about 2.5 million cars for about 4 million people.
Section 4: Assessment
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?
Some solution strategies are:
2.5/4 = 62.5/100 (multiplying by 25) so 62.5%
2.5/4 = 5/8 = 62.5%

42. Page 2: GST Burger Please
Section 4: Assessment
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?
Possible solution strategies are:
$5.80 ÷ 115 = 5.04c, 5.04 x 15 = 76 cents
$5.80 ÷ 23 = 25.2c, 25.2 x 3 = 75.6 cents