Faculty of Mathematics
Waterloo, Ontario N2L 3G1
Centre for Education in
Mathematics and Computing
Grade 6 Math Circles
March 3/4, 2015
Proportions Solutions
Powerful Proportions
Today, we’re going to talk about different ways to express and apply proportions. A proportion is a relationship of one thing to another in terms of quantity, size, or number. Everything
we will look at today is related and has a real-world application.
Warm-Up
Try to answer the following 11 questions in 4 minutes. You may only use a calculator for
questions 9 through 11.
1
20
as a fraction in lowest terms.
1. Express 100
5
2. Express
20
100
as a decimal. 0.2
3. Express
20
100
as a percent. 20%
1
4
1
5. Express 0.02 as a fraction in lowest terms.
50
4. Express 25% as a fraction in lowest terms.
6. Express
2
3
as a ratio. 2 : 3
7. If a $100 coat is on sale where the discount is 10%, what is the sale price? $90
8. Mike purchased a bike that cost $120 plus 10% tax. How much did Mike pay? $132
9. Ian makes a 15% commission on all sales. If he sells $300 worth of product, how much
commission did he make? $45
10. Target Canada is closing its doors. Everything must go! During its clearance sale,
Target sells a $30 item for $24. What was the discount rate? 20%
11. If 12 eggs cost $1.20, how much do 8 eggs cost? $0.80
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Terms
A ratio is a comparison of quantities with the same units. They are expressed using the
colon (:) symbol.
A fraction is a part of a whole. They are expressed using a fraction bar or a slash (/) symbol.
The numerator is the number on the top of the fraction. The denominator is the number
on the bottom of the fraction.
A percent is part per 100. Cent means 100 in French.
A decimal is a number based on 10 that may include a decimal point. In this lesson we
consider a decimal to be a number with a decimal point.
Conversions
All four of the quantities mentioned above are closely related. It is important to know how
to go from one representation to another.
Converting Decimals to Percentages:
Multiply the decimal by 100 or move the decimal two places to the right. Then put a percent
sign after the number.
Converting Percentages to Decimals:
Divide the percent by 100 or move the decimal two places to the left. Make sure to remove
the percent sign.
Converting Fractions to Decimals:
Divide the numerator by the denominator.
Converting Percentages to Fractions:
Multiply the percent by 100. This is the numerator. Now write the fraction bar and make
the denominator 100. Now write this fraction in lowest terms. Make sure you’ve removed
the percent sign.
Converting Fractions to Ratios:
Simply replace the fraction bar with a colon, and write the numerator to the left of the
denominator.
Converting Ratios to Fractions:
Simply replace the colon with a fraction bar, and write the number to the left as the numerator and the number to the right as the denominator.
2
Try it out:
1. Convert
3
5
to a ratio, percent, and decimal. 3 : 5, 60%, 0.6
2. Convert 5% to a decimal, fraction, and ratio. 0.05,
1
, 1 : 20
20
Mixing Problems
We can apply percentages to real-life scenarios in determining the concentration of solutions
and determining the quantities of different objects or substances needed.
When solving these problems it is easiest to organize the information in a chart. This allows
you to clearly see what information you have and need.
Percent of
Final Solution
Amount
Final Solution
Amount of
Solution 1
Percent of
Soluti on 1
Amount of
Solution 2
Percent of
Solution 2
Example:
Jolene is creating 2050mL of fruit punch for her friend’s birthday. She adds 1000mL of apple
juice that is 84% concentrated. Jolene also adds 1050mL of orange juice. If the fruit punch
has a concentration of 88% what is the concentration of the orange juice? Round your answer
to the nearest hundredth.
Solution: Let c be the concentration of the orange juice.
(2050)(0.88) = (1000)(0.84) + (1050)c
1804 = 840 + (1050)c
964 = (1050)c
964
c=
1050
.
c = 0.9181 = 91.81%
The concentration of the orange juice is about 91.81%.
3
Try it out:
Gary makes a 100mL solution in chemistry. He adds 25mL of a 63% acidic solution and
75mL of a 43% acidic solution. How acidic is the final solution?
Let a be the acidity of the final solution.
(100)(a) = (25)(0.63) + (75)(0.43)
(100)(a) = 15.75 + 32.25
(100)(a) = 48
48
a=
100
a = 0.48 = 48%
The acidity of the final solution is 48%.
Discounts/Markdowns
When new products come in, stores want to clear the older products. To make the older
products more attractive, stores lower the price. The lowering of the price of products
is called a discount (a percentage), or markdown (a dollar amount). With discounts, a
percentage is taken off.
Example:
Suppose we have a 15% discount on a $463 suit. How much money do you save? What is
the discounted price?
Solution:
Start by converting the percent to a decimal. Well, 15% = 0.15.
Now we can find the discount. This is simply 463 × 0.15. Thus, the discount is $69.45.
∴ you save $69.45.
The discounted price is 463 − 69.45 = 393.55.
∴ the discounted price is $393.55.
Alternatively, you can directly find the discounted price.
Since the discount is 15%, the final price is 100% − 15% = 85% of the original cost.
∴ the discounted price is 463 × 0.85 = $393.55.
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Try it out:
1. Sajeev wants to buy a bike whose price tag is labeled $275. There’s a blowout sale at
the bike store today. All bikes are 8% off. How much will Sajeev have to pay for the
bike today? 275 × 0.92 = $253
2. Jiminy wants to buy a tiny violin to play during sad occasions. Normally, tiny violins
cost $40. But today, there’s a worldwide sale on tiny violins. Jiminy saved $12. What
was the discount percentage? $12 ÷ $40 = 0.3 ∴ he saved 30%
3. Johann wants buy some potatoes. After a 17% discount, his potatoes come to a total
of $4.98. How much were they before the discount?
Johann bought the potatoes for 100 − 17 = 83% of the original cost.
4.98 ÷ 0.83 = $6.00
∴ the potatoes cost $6 before the discount.
Commission
In order to motivate sales, companies pay employees by commission. This means that employees get paid a certain amount (usually a percentage) for each product they sell. The
percentage at which the employee gets the commission is called the commission rate. The
more products you sell, the more you get paid. People who get commission usually get an
hourly rate as well (just in case they aren’t able to sell anything).
To encourage employees to sell even more products, companies can introduce a graduated
commission. If an employee sells more than a certain amount, the commission rate increases,
so that the employee gets more pay per sale.
Example:
Lebron sells basketballs. He gets 1% commission on the first $2000 of his sales, 3% on the
next $3000 of his sales, and 5% for any more sales after that. How much commission will he
receive if he sells $4000 worth of basketballs?
Solution:
Since Lebron sold more than the $2000, the first commission level is “filled up”. That means
there is 4000 − 2000 = $2000 remaining to fill up the second commission level.
(2000)(0.01) + (2000)(0.03) = 20 + 60
= 80
∴ Lebron will receive $80 if he sells $4000 worth of basketballs.
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Try it out:
How much commission will Lebron receive if he sells $6500 worth of basketballs?
Since Lebron sold more than the $5000, the first two commission levels are “filled up”. That
means there is 6500 − 5000 = $1500 remaining at the final commission level.
(2000)(0.01) + (3000)(0.03) + (1500)(0.05) = 20 + 90 + 75
= 185
∴ Lebron will receive $185 if he sells $6500 worth of basketballs.
Taxes
Taxes are a real-life application of percentages. When using percentages, it is important to
remember their relationship with decimals and fractions.
When you buy something from a store in Ontario, you must pay the price on the tag, plus
tax. This is an extra amount stated by the government in terms of a percentage. Currently,
we have a Harmonized Sales Tax (HST) in Ontario of 13% on most items. Certain items,
such as books, only have a 5% tax. The moral of this story is that you should spend more
time reading!
Let’s do an example. Fleur wants to buy some flowers. The price tag says $5.00. The tax
where Fleur lives is 10%. How much will she have to pay?
1
To solve this problem, recall that 10% = 0.1 = . Multiplying by one tenth is the same as
10
dividing by 10. You should understand that multiplying or dividing by 10 results in simply
moving the decimal place. This is thanks to the fact that we work in the decimal system!
1
You may have also heard it being called “base 10” in your math class. So, 5.00 ×
= 0.50.
10
But we are not done yet! We have only found that the tax itself is $0.50. We must add this
to the original price to find the total cost. Thus, $5.00 + $0.50 = $5.50. So, Fleur must pay
$5.50.
This problem was pretty easy to solve without a calculator since the tax was 10%. What
if the tax was 15% or 13% (like in Ontario)? For the sake of simplicity, you may use a
calculator to solve tax problems when the tax is not 10% in this lesson.
There is an alternate approach. For example, let the tax be 13%. Well, this means the final
cost of the item is 100% + 13% = 113% of the pre-tax cost. Thus, you can multiply the
pre-tax cost by 1.13 to find the after-tax price.
6
Try it out:
1. Tyrion wants to purchase bubbles. The price tag says the bubbles cost $15.60. The
tax where Tyrion lives is 13%. How much Tyrion will have to pay?
15.60 × 1.13 = 17.628 ∴ Tyrion must pay $17.63
2. Arya wants to buy a soccer ball. The price tag says the ball costs $160. If the tax
where Arya lives is 27%, how much does she have to pay?
160 × 1.27 = 203.2 ∴ Arya must pay $203.20
3. Jaimie buys a microwave for $48 (after tax). If he pays $8 tax, what is the tax rate?
The pre-tax cost of the microwave is 48 − 8 = $40.
To find what percentage the tax is, we see what proportion of 40 the amount he paid
in tax is:
8 ÷ 40 = 0.2 ∴ the tax rate is 20%
Unit Price
The unit price (also known as the unit cost) tells you the cost per one unit of a quantity or
measure. The technique to find the unit price is straightforward. You divide the cost by the
quantity.
Comparing unit prices can be a good way of finding the best buy.
Example:
Which is the better deal?
(a) 10 chocolates for $40.00
(b) 6 chocolates for $27.00
Solution:
40.00 ÷ 10 = $4.00 per chocolate
27.00 ÷ 6 = $4.50 per chocolate
∴ (a) is a better deal.
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Try it out: Which is the best deal?
(a) 2 litres of milk for $3.80
(b) 1.5 litres of milk for $2.70
(c) 0.75 litres of milk for $1.20
3.80 ÷ 2 = $1.90 per litre
2.70 ÷ 1.5 = $1.80 per litre
1.20 ÷ 0.75 = $1.60 per litre
∴ (c) is a better deal.
Wrap-Up
Today, you looked at the relationships between different ways to express proportions. You
also examined just how useful they can be in real life situations.
Problem Set
Complete the following 18 problems. You may use a calculator.
1. Convert
15
16
into a:
(a) Ratio 15 : 16
(b) Percent 93.75%
(c) Decimal 0.9375
2. Convert 56 : 64 into a:
7
8
(b) Percent 87.5%
(a) Fraction
(c) Decimal 0.875
3. Convert 177% into a:
177
100
(b) Ratio 177 : 100
(a) Fraction
(c) Decimal 1.77
8
4. For every 29 on-time flights between Chicago and San Francisco, there are 3 late flights.
What percentage of flights are on time?
The ratio is 29 : 3. In total, in this format we have 29 + 3 = 32 flights. 29 of these 32
of these flights are on time. ∴ 29 ÷ 32 = 0.90625 = 90.625% of the flights are on time.
5. Ryan is exceptionally good at math. On his last test, he received a score of 214 out of
200 because there were bonus questions. What was his mark as a percentage?
214 ÷ 200 = 1.07 = 107%
6. Charlie’s blood-juice level is 0.06. Unfortunately, Charlie is not allowed to ride a bike
with this much juice in his system. A blood-juice level of 0.06 means that there are
60 mg of juice in every 100 mL of blood in his body. What is this ratio as a percentage?
60 ÷ 100 = 0.6 = 60%
7. With a blood-juice level of 0.06, Charlie is very uncoordinated. For every 5 metres he
walks, he stumbles for 3 metres. If he moves 24 metres in total, what percentage of
the distance did he stumble? What distance did he stumble?
The ratio is 5 : 3. In total, in this format we have 5 + 3 = 8 metres. 3 of these 8
of these metres are stumbles. ∴ 3 ÷ 8 = 0.375 = 37.5% of the metres are stumbles.
Notice that the distance you travel is irrelevant, as you would just adjust the ratio
accordingly. We find that the distance he stumbles is 24 × 0.375 = 9 metres.
8. Julian mixes a solution made of 100mL of pumpkin juice, which is 5% acidic, with
100mL of apple juice. If the resulting solution’s acidity percent is 3 times the pumpkin
juice, how acidic is the apple juice?
The resulting solution is 200mL and has acidity 15%. Plugging into the formula:
(200)(0.15) = (100)(a) + (100)(0.05)
30 = (100)(a) + 5
25 = (100)(a)
25
a=
100
a = 0.25 = 25%
The acidity of the apple juice is 25%.
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9. The sale price of a clock is $15 after a 30% discount. What is the original price and
markdown?
.
The clock is on sale for 70% of the original price. ∴ 15 ÷ 0.7 = $21.43 is the original
price. The markdown was 21.43 − 15 = $6.43.
10. Crazy Cow discounted their toy cars by 60%. The markdown was $42. What is the
original price and sale price of the toy cars?
(orig)(0.6) = 42
42
orig =
0.6
orig = 70
∴ The orginal price was $70. This means the sale price was 70 − 42 = $28.
11. Akshayaa works at a toy car dealership. She is on a graduated commission, making
0% of the first $150, 10% of the next $3000 and 15% of all sales afterwards. How much
must she sell in order to make $1200?
Akshayaa makes ($3000)(0.1) = $300 on her second commission level and $0 on her
first commission level. She needs to make $1200−$300 = $900 on her third commission
= $6000 on her third commission level. In
level. In order to do that, she must sell $900
0.15
total, she needs to sell $150 + $3000 + $6000 = $9150.
12. From question 11, if Akshayaa switches to a single commission rate, what rate should
she have so that she sells the same amount as she did to make $1200?
$1200 .
= 13.11%
$9150
13. Aishwarya used to live in New Zealand where she sold dreamcatchers for $11. She
now lives in Ontario and wants to sell dreamcatchers. If the sales tax is 15% in New
Zealand and 13% in Ontario, what price should she set the dreamcatchers at so that
customers in Ontario pay the same as what the customers in New Zealand paid?
In New Zealand, the customers pay ($11)(1.15) = $12.65. For the customers in Ontario
$12.65
= $11.19.
to pay $12.65, Aishwarya would have to sell the dreamcatchers for
1.13
14. Which is the best deal?
(a) 3 cans of soup for $3.75 Unit price is 3.75 ÷ 3 = $1.25
(b) 53 cans of soup for $33.39 Unit price is 33.39 ÷ 53 = $0.63
(c) 11 cans of soup for $9.90 Unit price is 9.90 ÷ 11 = $0.90
∴ (b) is the best deal.
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15. How much does a discount rate have to be so that the total (after 15% tax) comes to
half the original price?
Let d be the discount rate. We set the selling price at 100%. The customer would pay
(100%)(1 − d)(1.15) = 50%
(1 − d)(1.15) = 0.5
0.5
1−d=
1.15
0.5
=d
1−
1.15
.
d = 56.52%
16. An item is discounted three times: 30%, 15%, then 7%. What single discount rate
would reduce the price by the same amount?
After all the discounts, a product’s price goes from 100% to:
(100%)(0.7)(0.85)(0.93) = 55.34%
So we need to find a d such that:
(100%)(1 − d) = 55.335%
1 − d = 0.55335
1 − 0.55335 = d
d = 44.665%
17. Perry buys a platypus for $78.40 after a 30% discount and 12% tax. If the discount is
applied to the pre-tax price and the tax is applied after the discount is taken off, what
was the pre-tax price?
For this question, we need to reverse the order of changes to the pre-tax price. We
start by reversing the tax:
78.40 ÷ 1.12 = $70.00
This is the discounted price. We must now reverse the discount:
70 ÷ 0.7 = $100
∴ the pre-tax price was $100.
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18. Sapphire earns a 7% commission on all sales on the before-discount, before-tax price.
In her store, the discount is applied to the after-tax price. If the store is having a
clearance event and all items are 20% off and she has made $1630 in sales today, how
much commission did she make if the tax rate where she lives is 10%?
To answer this question, we must find out how much money Sapphire earns commission
on. Just like question 17, we must reverse-engineer this quantity. This time, we begin
by reversing the discount:
1630 ÷ 0.8 = $2037.50
Next, we reverse the tax:
2037.50 ÷ 1.10 = $1852.27
This is the amount of money that Sapphire earns commission from. Lastly, we find
out the amount of commission she makes:
.
1852.27 × 0.07 = 129.6590 = $129.66
∴ Sapphire earns $129.66 in commission.
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