Ratio and Proportion
Demonstration: Science in Math ?!?!
Reflect on what you have just seen…
If the Montreal Canadiens scored 52 goals in the first 18 games of the season,
how many goals would they be expected to score in an 80 game season?
In order to answer this question, we have to take a closer look at Ratio and Proportion.
I. Ratio
A ratio is…
Examples of ratios:
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The Golden Ratio
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Coloured Squares
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Students in the class
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Ratios can be written in different ways:
§
Using a colon
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As a fraction
§
Using the word “to”
When writing ratios, ___________ is important!
Example.
1. In a bowl, there is a mixture of 13 blue M&M’s and 16 red M&M’s.
What is the ratio of (a) blue: red (b) red to blue (c) red : all?
2. In another bowl, there is a mixture of peanuts and almonds in a ratio
of 2:3. Should we necessarily conclude that there are 5 nuts in the
bowl?
Example #2. What is the ratio of inches to feet? (Hint: There are 12 inches in 1 foot)
Ø Web Activity: http://math.rice.edu/~lanius/proportions/rate.html
Ø Activity:http://mathforum.org/escotpow/print_puzzler.ehtml?puzzle=
40#Open%20Java
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Exercises.
1. Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange.
What is the ratio of books to marbles? Write your answer in 3 different
ways.
2. In the diagram below, what is the ratio of (a) squares to circles? (b) circles:
squares?
3. Write each ratio in another form:
a. 21 to 3
d. 7:12
b. 3:5
e. ¾
c. ½
4. Write as a ratio:
a. an ounce to a pound (Hint: There are 16 ounces in 1 pound)
b. If the length of a rectangle is twice as long as the width, what is the ratio
of width:length?
Activity: Colour & Corn
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II. Ratio & Like Quantities: When comparing like quantities, we use the same units.
What are ‘like quantities’?
What are some examples of units?
Example. What is the ratio suggested by the following?
1. Spring compared to the other seasons of the year
2. One hour compared to a day
3. One second compared to one hour
4. 5 cents compared to $10
Example. The fuel used in two-stroke engines is a mixture of gasoline and motor oil.
As a general rule, 150 mL of oil are mixed with 10 L of gasoline. What is the ratio of
oil/gasoline?7
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Exercises. What is the ratio suggested by the following?
1. A minute compared to an hour
2. A cm compared to a m
3. A week compared to a year
Ø Quiz yourself: http://www.aaamath.com/B/g62a_rx1.htm
Ø Quiz yourself again (this one is challenging!):
http://regentsprep.org/Regents/math/ratio/PracRatio.htm
III.
Ratio & Parts: If we change one part of the ratio, the other part will be
affected.
Example. One cup of coffee is served with 2 cubes of sugar in it. Will the coffee be
sweeter or not as sweet if the following are added?
a. another sugar cube
b. more coffee
c. two more sugar cubes and one cup of coffee
d. more sugar cubes and more coffee
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Example. A teacher wrote up an exam in which every group of 10 questions contained
2 difficult questions, 6 questions of average difficulty, and 2 easy questions. What
should he do if he wants to:
e. decrease the level of difficulty of the exam?
f. increase the level of difficulty of the exam?
Exercises.
1. To make orange juice from concentrate, 1 part concentrate must be mixed with
3 parts water. What would happen to the mixture if:
a. more orange juice concentrate was added to the same amount of water?
b. more water was added to the same amount of concentrate?
c. more orange juice concentrate was added to less water?
d. 2 times more orange juice concentrate was added to three times more
water?
e. one part orange juice concentrate and one part water were added?
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2. In his last basketball game, Marco scored 8 baskets out of 14 shots. Which of
the following people performed better than Marco?
a. John scored as many baskets in fewer shots
b. Isabel scored more baskets with the same number of shots
c. Charlotte scored one basket less with one additional shot
d. Tommy scored fewer baskets with one additional shot
e. Paula made one more basket with one additional shot
IV.
Equivalent Ratios/Reducing Ratios
Cherry Kool-Aid is to be mixed with water in a ratio of 1:6; this means that
for every unit of Kool-Aid, ____ units of water will be used. What if I
wanted to make more Kool-Aid but keep the same concentration?
Unit of Kool-Aid
1
Units of Water
6
2
4
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The ratios above are __________________.
So, ____ :_____ is EQUIVALENT to ____: ____.
A statement indicating that two ratios are equivalent is called a PROPORTION.
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How can we find equivalent ratios (or proportions)?
Have you ever seen this kind of situation before?
Example. At Cote St Luc summer day camps, there must be 1 counselor for every 8
campers. How many counselors are needed if there are 24 campers?
In the above example, ____ :_____ is in SIMPLEST FORM.
Simplest Form:
Ratios can be simplified by _______________________ both sides by the same number
(note: this is similar to Lowest Terms with Fractions). A ratio in simplest form cannot
be ________________________.
Example. Write each ratio in simplest form:
a. 7: 14
d. 80: 100
b. 15:25
e. 22:77
c. 10:4
f. 18:24
g. Jennifer mixes 600 mL of orange juice with 900 mL of apple juice to make a fruit
drink. Write the ratio of orange juice to apple juice in its simplest form.
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Try This! Bryan wants to make a tossed green salad for a barbeque his family is
planning. There will be 12 guests attending the party. Bryan looks in several
cookbooks for a salad-dressing recipe. Here is the one he found:
Summer Vinaigrette
125 mL light olive oil
75 mL balsamic vinegar
15 mL Dijon mustard
10 mL brown sugar
5 mL chopped rosemary
Place all ingredients into a glass jar.
Put the lid on the jar and shake vigorously. (Serves 6)
1. How would Bryan have to adapt this recipe to use on his salad? (Remember,
he needs enough for 12 people.)
2. Write a ratio Bryan could use to adapt the recipe.
3. Use a
a.
b.
c.
d.
ratio to rewrite the recipe to serve the following:
18 people
3 people
15 people
2 people
Exercises.
1. Write each ratio in simplest form:
a. 14:21
d. 24:4
b. 3:15
e. 6:9
c. 20:100
f. 8:20
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2. Write 3 equivalent ratios for each of the following:
a. 11:14
d. 4 to 18
b. 39:15
e. 7 to 8
c. 3:17
3. A builder mixes 10 shovels of cements with 25 shovels of sand. Write the ratio
of cement to sand in simplest form.
4. If 6 copies of a book cost $20.00, how much would 18 copies cost?
V.
Comparing Ratios. In some situations, we need to compare ratios.
Can you think of some situations where you would want to compare ratios?
Example. Louis mixed 0.5 L of oil with 5 L of gasoline. Lucy mixed 2 L of oil in 25 L
of gasoline. Which of the two mixtures has the higher concentration of oil?
There are 2 methods to solve this problem.
Method #1. Find a common denominator (or numerator)
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Method #2. Change to a decimal equivalent & compare
Exercises.
1. Which is the greater ratio?
a. 6:15 or 7:15
c. 1:9 or 1:10
b. 2:5 or 7:20
d. 4:22 or 5:21
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PACKAGE #2
Challenge: 3 way ratios
VI.
Properties of Proportions
Initial q: But, we STILL don’t know… If the Montreal Canadiens scored 52 goals in the
first 18 games of the season, how many goals would they be expected to score in an 80
game season?
VII. Proportional Situations
VIII. Rate & Unit Rate
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