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LESSON
2.1
Page 96
Proportions
When you say, “I got 21 out of 24 questions correct on the last quiz,” you are
comparing two numbers. The ratio of your correct questions to the total number
of questions is 21 to 24. You can write the ratio as 21:24, or as a fraction, 2214 , or as a
decimal, 0.875. The fraction bar means division, so these expressions are equivalent:
Mathematics is not a way of
hanging numbers on things
so that quantitative answers
to ordinary questions can be
obtained. It is a language that
allows one to think of
extraordinary questions.
21
24
21 24
0.875
JAMES BULLOCK
EXAMPLE A
7
8
Write the ratio 210 : 330 in several ways.
7
210
or 11
330
Solution
210 330 or
7 11
To change a common fraction into a decimal fraction, divide the numerator by
the denominator.
When you see the
same difference
that you’ve seen
before, you know
the decimal repeats.
.636363
33021
0.
00
00
00
198 0
12 00
9 90
2 100
1 980
1200
990
...
When you divide 21 by 24, the decimal
form of the quotient ends, or terminates.
The ratio 2214 equals 0.875 exactly. But when
you divide 210 by 330 or 7 by 11, you see a
repeating decimal pattern, 0.636363 . . . .
You can use a bar over the numerals that
repeat to show a repeating decimal pattern,
7
0.6
3. [ See Calculator Note 0A for more
11
about converting fractions to decimals. ]
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CHAPTER 2 Proportional Reasoning and Variation
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A proportion is an equation stating that two ratios are equal. For example, 32 182 is
a proportion. You can use the numbers 2, 3, 8, and 12 to write these true proportions:
12
8
8
3
2
3
12
2
3
2
12
12
3
2
8
8
Do you agree that these are all true equations? One way to check that a proportion
is true is by finding the decimal equivalent of each side. The statement 38 122 is
not true; 0.375 is not equal to 0.16.
In algebra, a variable can stand for an unknown number or for a set of numbers.
In the proportion 23 M
6 , you can replace the letter M with any number, but only
one number will make the proportion true. That number is unknown until the
proportion is solved.
Investigation
Multiply and Conquer
You can easily guess the value of M in the proportion 23 M
6 . In this investigation
you’ll examine ways to solve a proportion for an unknown number when guessing
56
is not easy. It’s hard to guess the value of M in the proportion 1M9 133 .
Step 1
56
Multiply both sides of the proportion 1M9 133 by 19.
Why can you do this? What does M equal?
Step 2
For each equation, choose a number to multiply both
ratios by to solve the proportion for the unknown
number. Then multiply and divide to find the
missing value.
p
132
21
Q
a. 1
b. 3
2 176
5 20
L
30
130
n
c. 3
d. 7
0 200
8 15
Step 3
Check that each proportion in Step 2 is true by replacing the variable with
your answer.
Step 4
In each equation in Step 2, the variables are in the numerator. Write a brief
explanation of one way to solve a proportion when one of the numerators is
a variable.
Step 5
The proportions you solved in Step 2 have been changed by switching the
numerators and denominators. That is, the ratio on each side has been inverted.
p
(You may recall that inverted fractions, like 12 and 1p2 are called reciprocals.) Do
the solutions from Step 2 also make these new proportions true?
12
176
a. p 132
Step 6
35
20
b. 2
1 Q
30
200
c. L 3
0
78
15
d. 130 n
How can you use what you just discovered to help you solve a proportion that
20
12
has the variable in the denominator, such as 135 k ? Why does this work?
Solve the equation.
LESSON 2.1 Proportions
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There are many ways to solve proportions. Here are three student papers each
answering the question “13 is 65% of what number?” What are the steps each
student followed? What other methods can you use to solve proportions?
Step 7
a.
b.
65
13
___
__
100 = x
100
x
___ = __
65 13
13
x 13
__ 100
___ = __
__
1 65 13 1
c.
65
13
___
__
100 = x
13
65 = 13
___
__
100 x
20
20 = x
20 = x
65
13
___
__
100 = x
100
x ___
65 = 13
x
___ __
__ 100
___ __
1 1 100 x 1 1
65x _____
1300
____
65 = 65
x = 20
In the investigation you discovered that you can solve for an unknown numerator
in a proportion by multiplying both sides of the proportion by the denominator
56
under the unknown value. You can also think of a proportion such as 1M9 133
56
.” To find the original
like this: “When a number is divided by 19, the result is 133
number, you need to undo the division. Multiplying by 19 undoes the division.
EXAMPLE B
Solution
Jennifer estimates that two out of every three students will attend the class party.
She knows there are 750 students in her class. Set up and solve a proportion to
help her estimate how many people will attend.
To set up the proportion, be sure both ratios make the same comparison.
Use a to represent the number of students who will attend.
Students who will attend
Students who will attend
a
2
3 750
Students who are invited
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CHAPTER 2 Proportional Reasoning and Variation
Students who are invited
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In the proportion, when a is divided by 750, the answer is 23.
2
750 3 a
Multiply by 750 to undo the division.
500 a
Multiply and divide.
Jennifer can estimate that 500 students will attend the party.
EXAMPLE C
After the party, Jennifer found out that 70% of the class attended. How many
students attended?
Solution
70% is 70 out of 100. So write and solve a proportion to answer the question
“If 70 students out of 100 attended the party, how many students out of 750
attended?”
History
Let s represent the number of students who attended.
The Pythagoreans, a group
of philosophers begun by
Pythagoras in about 520 B.C.E.,
realized that not all numbers
are rational. For example, for a
square one unit on a side, the
diagonal 2 is irrational.
Another irrational number
is pi, or , the ratio of the
circumference of a circle to
its diameter.
70
s
100
750
70
750 100 s
525 s
Write the proportion.
Multiply by 750 to undo the division.
Multiply and divide.
525 out of 750 students attended the party.
You have worked with ratios and proportions in this lesson. Numbers that can be
written as the ratio of two integers are called rational numbers.
EXERCISES
You will need your graphing calculator for Exercise 14.
Practice Your Skills
1. List these fractions in increasing order by estimating their values. Then use your
calculator to find the decimal value of each fraction.
7
a. 8
13
b. 2
0
13
c. 5
52
d. 2
5
2. Ms. Lenz collected information about the students in her class.
Eye Color
Brown eyes
Blue eyes
Hazel eyes
9th graders
9
3
2
8th graders
11
4
1
Write these ratios as fractions.
a. ninth graders with brown eyes to ninth graders a
b. eighth graders with brown eyes to students with brown eyes
c. eighth graders with blue eyes to ninth graders with blue eyes a
d. all students with hazel eyes to students in both grades
LESSON 2.1 Proportions
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3. Phrases such as miles per gallon, parts per million (ppm),
and accidents per 1000 people indicate ratios. Write each
ratio named below as a fraction. Use a number and a unit
in both the numerator and the denominator.
a. In 2000, the McLaren was the fastest car produced. Its
top speed was recorded at 240 miles per hour. a
b. Pure capsaicin, a substance that makes hot peppers taste
hot, is so strong that 10 ppm in water can make your
tongue blister. a
c. In 2000, women owned approximately 350 of every
thousand firms in the United States. a
d. The 2000 average income in Philadelphia, Pennsylvania,
was approximately $35,500 per person.
4. What number should you multiply by to solve for the unknown in each proportion?
T
24
a
a. 4
0 30
49
R
b. 5
6 32
5. Find the value of the unknown number in each proportion.
T
24
49
R
52
42
b. 5
c. 91 S a
a. 4
0 30
6 32
M
87
6
62
c
36
e. 1
f. n g. 1
6 232
217
5 13
M
87
a
c. 1
6 232
100
7
d. 3
0 x
220
60
h. 3
3 W
Reason and Apply
6. APPLICATION Write a proportion for each problem, and
solve for the unknown number.
a. Leaf-cutter ants that live in Central and South America
weigh about 1.5 grams (g). One ant can carry a 4 g
piece of leaf that is about the size of a dime. If a
person could carry proportionally as much as the
leaf-cutter ant, how much could a 55 kg algebra
student carry?
b. The leaf-cutter ant is about 1.27 cm long and takes
strides of 0.84 cm. If a person could take proportionally
equivalent strides, what size strides would a 1.65 m tall
algebra student take?
c. The 1.27 cm long ants travel up to 0.4 km from home
each day. If a person could travel a proportional
distance, how far would a 1.65 m tall person travel?
7. Write three other true proportions using the four values in
each proportion.
2
10 a
a. 5 2
5
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CHAPTER 2 Proportional Reasoning and Variation
a
12
b. 9 2
7
j
l
c. k m
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8. APPLICATION Jeremy has a job at the
movie theater. His hourly wage is $7.38.
Suppose 15% of his income is withheld
for taxes and Social Security.
a. What percent does Jeremy get to keep?
b. What is his hourly take-home wage?
9. In a resort area during the summer
months, only one out of eight people is a
year-round resident. The others are there
on vacation. If the year-round population
of the area is 3000, how many people are
there in the summer? a
10. APPLICATION To make three servings of
Irish porridge, you need 4 cups of water and 1 cup of steel-cut oatmeal. How much
of each ingredient will you need for two servings? For five servings?
11. APPLICATION When chemists write formulas for chemical compounds, they indicate
how many atoms of each element combine to form a molecule of that compound.
For instance, they write H2O for water, which means there are two hydrogen atoms
and one oxygen atom in each molecule of water. Acetone (or nail polish remover)
has the formula C3H6O. The C stands for carbon.
carbon
hydrogen
oxygen
hydrogen
oxygen
Model of acetone molecule
Model of water molecule
a. How many of each atom are in one molecule of acetone? a
b. How many atoms of carbon must combine with 470 atoms of oxygen to form
acetone molecules? How many atoms of hydrogen are required? a
c. How many acetone molecules can be formed from 3000 atoms of carbon,
3000 atoms of hydrogen, and 1000 atoms of oxygen? a
Review
12. In the dot plot below, circle the points that represent values for the five-number
summary. If a value is actually the mean of two data points, draw a circle around the
two points.
0
4
8
12
16
20
LESSON 2.1 Proportions
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13. The Forbes Celebrity 100 (www.forbes.com) listed these ten people (and their incomes for
2004 in millions) among the celebrities that got the most media attention. Find the
three measures of center for their incomes, and explain why they are so different.
210)
Mel Gibson ($
($210)
Oprah Winfrey
147)
J.K. Rowling ($
acher ($80)
Michael Schum
80)
Tiger Woods ($
g ($75)
Steven Spielber
)
Jim Carrey ($66
en ($64)
Bruce Springste
60)
Nora Roberts ($
eld ($57)
David Copperfi
[Data set: CELEB]
Tiger Woods
14. Use the order of operations to evaluate these expressions. Check your results on
your calculator.
a. 5 4 8
b. 12 (7 4)
d. 18(3) 81
c. 3 6 25 30
THE GOLDEN RATIO
In this project you’ll research the amazing number mathematicians call the
golden ratio. (There are plenty of books and web sites on the topic. Find links
at www.keymath.com/DA .) Your project should include
102
Basic information on the golden ratio, such as its exact value, why it represents
a mathematically “ideal” ratio, and how to construct a golden rectangle. (Its
length-to-width ratio is the golden ratio.)
Some history of the golden ratio, including its role in ancient Greek architecture.
At least one other interesting mathematical fact about the golden ratio, such as
its relationship to the Fibonacci sequence or its own reciprocal.
A report on where to find the
golden ratio in the environment,
architecture, or art. List items
and their measurements or
include prints from photographs,
art, or architecture on which you
have drawn the golden rectangle.
CHAPTER 2 Proportional Reasoning and Variation
Once you’ve learned how
to construct the golden
ratio, The Geometer’s
Sketchpad is an ideal tool
for further exploration.
You can create a Custom
Tool for dividing segments
into the golden ratio, and
then use this tool to help
you construct the golden
rectangle or even the
golden spiral.