MATH
STUDENT BOOK
8th Grade | Unit 4
Unit 4 | Proportional Reasoning
Math 804
Proportional Reasoning
Introduction |3
1.Proportions
5
Proportions |5
Applications |11
Direct Variation |16
SELF TEST 1: Proportions |25
2.Percents
27
Fraction, Percent, and Decimal Equivalents |27
Solving Percent Problems |35
Applications |42
More Applications |48
SELF TEST 2: Percents |55
3. Measurement/Similar Figures
57
Unit Conversion within Customary Units |57
Unit Conversion within Metric Units |65
Corresponding Parts |71
Indirect Measure |78
Models and Scales |82
SELF TEST 3: Measurement/Similar Figures |89
4.Review
93
LIFEPAC Test is located in the
center of the booklet. Please
remove before starting the unit.
Section 1 |1
Proportional Reasoning | Unit 4
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2| Section 1
Unit 4 | Proportional Reasoning
Proportional Reasoning
Introduction
In this unit, ratio and proportion are defined and the different notations for ratio
are given. Proportions are used to solve problems, such as unit pricing and rate.
Students convert between fractions, decimals, and percents. Students also learn
how things are measured indirectly. Word problems require students to use their
knowledge of similar figures to set up and solve a proportion. Lastly, students
use their knowledge of similar figures and scales to solve problems involving
scale drawings.
Objectives
Read these objectives. The objectives tell you what you will be able to do when
you have successfully completed this LIFEPAC. When you have finished this
LIFEPAC, you should be able to:
zz Use proportions to solve for a missing value.
zz Solve direct variation problems.
zz Convert and compare fractions, decimals, and percents.
zz Solve percent problems.
zz Convert customary units.
zz Convert metric units.
zz Use similar figures to solve for a missing measure and to measure
indirectly.
Section 1 |3
Proportional Reasoning | Unit 4
Survey the LIFEPAC. Ask yourself some questions about this study and write your questions here.
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4| Section 1
Unit 4 | Proportional Reasoning
1.Proportions
Proportions
A ratio is used to compare two numbers
and a proportion says that two ratios are
equal.
Objectives
z Write ratios and proportions.
z Determine
z Solve
if an equation is a proportion.
for a missing value in a proportion.
Vocabulary
cross multiplication—multiplying diagonally across the equal sign in a proportion
proportion—a statement that says two ratios are equal
ratio—a comparison of two quantities or numbers as a quotient
Ratios
A caterer is calculating the amount of food
he needs to purchase for an upcoming
party. According to his serving chart, he will
need 2 ounces of a certain kind of cheese
for every 3 people at the party. If there are
supposed to be 102 people attending the
party, how many ounces of cheese does the
caterer need to purchase?
A ratio is a comparison of two quantities or
numbers. The caterer needs to compare
the amount of cheese to the number of
people it will serve. This ratio is 2 to 3, or 2
ounces for every 3 people.
Connections! Do you remember the
definition of a rational number? It gets its
name from the word ratio. Any number that
can be expressed as the ratio of two integers
is considered rational.
Ratios may be expressed by using the
word “to,” a colon, or a division symbol.
And, they should always be expressed in
lowest terms. Each of the following ratios is
equivalent, or means the same thing:
„ 2
to 3
„ 2:3
„
„ 2
÷3
It is important to write the ratio in the
correct order. If you are asked to find the
ratio of ounces of cheese to people, then
the number of ounces must be the first
number, or in the numerator, and the
number of people must be the second
number, or in the denominator.
is not
equivalent to . The ratio 3 to 2 would be
stating that you need 3 ounces of cheese
for every 2 people, causing the caterer to
order much more than is needed!
Section 1 |5
Proportional Reasoning | Unit 4
Reminder! The second number, or
denominator, in a ratio cannot be zero,
because division by zero is undefined.
Although ratios are usually written as
a fraction, ratios are not the same as
fractions. Fractions can be expressed
as mixed numbers because fractions
represent how many parts of a whole that
you have. Ratios represent a comparison
of two numbers, so they should not be
expressed as mixed numbers. A single part
of the ratio can be a mixed number, but the
entire ratio should always be expressed as a
proper or improper fraction.
For example! You can express the fraction
of pizza as 1 pizzas, since it is a single
number. By contrast, you cannot express the
ratio 3 pizzas for every two people ( ) as
1 because the mixed fraction leaves out
the number of people. However, you can
convert the
ratio of pizzas to people to
dimes to coins is 2 to 6. Let’s try another
one.
Example:
►
In a book club, the ratio of women to
total members is 4 to 7. What is the
ratio of women to men?
Solution:
►
The given ratio tells us that for every
7 members, there are 4 women.
That means that the remaining 3
members (7 - 4) must be men. So, the
ratio of women to men is 4 to 3.
So, how does being able to write a ratio
help us to solve the caterer’s problem from
the beginning of the lesson?
Since we know the ratio of ounces to
people has to be 2 to 3, we can write an
equivalent ratio that will help us find the
number of ounces for 102 people. Again,
it is very important that the order of the
ratios be consistent. Both ratios should
express the number of ounces to the
number of people.
, since 3 pizzas for every 2 people is the
same as 1
Let’s look at some other ratios. Suppose
you know that a jar has only nickels and
dimes in it. And, the ratio of nickels to
dimes is 4 to 2. What other ratios could
you state from that information? Well, the
ratio of dimes to nickels would be the other
way around, or 2 to 4. That’s an easy one.
What about the ratio of nickels to the total
number of coins in the jar? You know from
the ratio that for every 4 nickels, there are 2
dimes. So, we can say that for every 6 coins
(4 + 2) there are 4 nickels and 2 dimes. Now
we can form two more ratios! The ratio of
nickels to coins is 4 to 6, and the ratio of
6| Section 1
=
pizzas for every 1 person.
=
The letter n is used to represent the
unknown value that we are trying to find.
So, now we can solve for n to find the
number of ounces of cheese the caterer
needs. What would we multiply the 3 by in
order to get 102? 102 divided by 3 is 34, so
we can rename this fraction by multiplying
both the numerator and the denominator
by 34.
·
=
The caterer will need 68 ounces of cheese.
Unit 4 | Proportional Reasoning
Proportions
We just used a proportion to solve the
caterer’s problem. A proportion is a
statement where two ratios are equal. The
equation
=
is a proportion.
Some other examples of proportions are:
„
=
„
=
„
=
for solving the previous example,
Try cross multiplying.
= • The numerator stayed the same but
the denominator was multiplied by 2.
= • The numerator was multiplied by 2,
but the denominator by 3.
If two ratios are not equivalent, the equation
is not a proportion.
=
=
.
=
„
„ 6
· 24 = 144 • Cross product of 6 and 24.
„ 16
· 9 = 144 • Cross product of 16 and 9.
The cross products both equal 144, so the
equation is a proportion. If the products are
not equal when you cross multiply, then the
equation is not a proportion.
Example:
►
Example:
Is
In a proportion, the results of cross
multiplication are always equal.
Knowing this fact gives us another option
Look at the following equations. These are
not proportions. Can you see why?
►
Cross Multiplication
Another approach to determining if an
equation is a proportion is to use cross
multiplication. By multiplying diagonally
across the equal sign, we can determine if
two ratios are equivalent.
Determine if
=
is a proportion.
Solution:
a proportion?
►
Solution:
►
Both ratios can be reduced to
►
The ratios are equivalent, so it is a
proportion.
=
.
Think about it! The fraction bar represents
division. A different way to determine if
two ratios are equivalent is to divide the
numerators by the denominators. If the
two quotients are equal, then the ratios are
equivalent. In this example, 6 ÷ 16 = 0.375,
and 9 ÷ 24 = 0.375
Cross multiply.
Original equation.
2 · 5 = 10 Cross product of 2 and 5.
3 · 4 = 12 Cross product of 3 and 4.
10 ≠ 12 The cross products are unequal.
►
The cross products are not equal, so
the ratios are not equivalent. This is
not a proportion.
Let’s see how cross multiplication can
be used to solve the problem from the
beginning of this lesson.
Section 1 |7
Proportional Reasoning | Unit 4
We had a ratio of cheese to people that was
2 to 3. We needed to find the amount of
cheese for 102 people. The proportion we
set up was =
. We know that using
cross multiplication, the products have to
be equal. So, set up an equation showing
this.
2 · 102 = 3n Cross multiply.
204 = 3n Simplify.
68 = n Divide both sides by 3.
When we solve the equation, we find that
n = 68. Our answer shows that the caterer
needs to order 68 ounces of cheese for
the upcoming party. If you look back in
the lesson, you’ll see that this is the same
answer we got before. In math, you can
often solve problems in more than one
way!
Be Careful! Make sure that the two ratios
are set up consistently. The number of wins
should be in the numerator of both ratios,
and the number of losses should be in the
denominator of both ratios.
Example:
►
Six is to 1
►
Find a.
Solution:
►
6a =
Joe’s basketball team’s win to loss
ratio was 5 to 2. If they lost 4 games,
how many games did they win? How
many total games did they play?
=
Equation for the proportion.
20 = 2w Cross multiply.
10 = w Solve the equation.
►
8| Section 1
The team had 10 wins this season.
To find the total number of games
played, add the number of wins to
the number of losses. Ten wins plus
four losses equals 14 total games
played.
·
6a =
Convert the mixed number (1
) to an improper fraction.
Multiply the fractions.
6a = 12 Simplify the fraction.
a = 2 Divide both sides by 6.
Solution:
Set up a proportion using w to
represent the number of games they
won.
· 9 Cross multiply.
6a = 1
Example:
►
Six is to 1 represents the first ratio.
Nine is to a represents the second
ratio. The word “as” tells us that
these two ratios are proportional.
Equation for the proportion.
Let’s look at a couple more examples.
►
as nine is to a.
This might help! Remember that to multiply
mixed numbers, you first need to change
each number to an improper fraction. Then,
multiply straight across.
In addition to “as,” a double semi-colon is
also used to show a proportion. Here are
some other examples of what proportions
can look like.
„ a
is to b as c is to d
„ a
: b as c : d
„ a
: b :: c : d
Unit 4 | Proportional Reasoning
Let’s Review
Before going on to the practice problems,
make sure you understand the main points
of this lesson.
„ Ratios
show a comparison between
quantities.
„ Two
equivalent ratios form a proportion.
„ In
a proportion, the products of cross
multiplying are always equal.
„ The
order of the quantities, in both
ratios and proportions, is very
important.
Complete the following activities.
1.1
Solve for the variable in the following proportion.
x = _________
1.2
Solve for the variable in the following proportion.
y = _________
=
1.3
Solve for the variable in the following proportion.
m = _________
=
1.4
Solve for the variable in the following proportion. n :
n = _________
1.5
Solve for the variable in the following proportion.
b = _________
1.6
Two numbers are in the ratio of 2 to 3. If the smaller number is 18, the larger
number is _____.
21
27
36
54
…
…
…
…
1.7
All of the following are equivalent except _____.
5 is to 3
…
…
…
=
as 6 : 1
is to 1
as 2 is to b
7.5:4.5
…
Section 1 |9
Proportional Reasoning | Unit 4
1.8
In a group of students, the ratio of girls to boys is 3 to 2. If there are 15 girls, how
many total students are there?
10
20
25
30
…
…
…
…
1.9
On a field trip, there are 12 adults and 14 students. What is the ratio of the number
of adults to the total number of people on the field trip?
6 to 13
12 to 14
26 to 12
6 to 7
…
…
…
…
1.10 If 2d = 5c, then all of the following are true except.
… =
… =
… =
… =
… =
… =
1.11 Which of the following is not a proportion?
… =
… =
1.12 If a soccer team won 7 of its 13 games, what was their ratio of wins to losses?
Assume that there were no tie games.
7 to 6
7 to 20
7 to 13
1 to 2
…
…
…
…
10| Section 1
Unit 4 | Proportional Reasoning
Applications
The customer in the cartoon offered to pay
more than the farmer was asking for the
apples. Can you see why? In this lesson,
you’ll learn how to calculate rates, which
include unit prices.
Objectives
z Determine unit rate or unit price.
z Use
proportional reasoning to solve problems.
Vocabulary
rate—a type of ratio that compares two different kinds of quantities or numbers
unit price—a rate showing the price for 1 item
unit rate—a rate with a denominator of 1; a rate which shows an amount of something
compared to 1 of something else
Rates
This apple farmer would love it if every
customer was as naïve as the character in
the comic strip. Sure, 85¢ sounds cheap
compared to $1.29, but is it really? If both
prices were getting us the same number
of apples, then 85¢ would definitely be
cheaper! But, is 3 apples for 85¢ really a
better deal than 5 apples for $1.29?
We can use a type of ratio, called a rate, to
find the answer.
A rate is a ratio that compares two different
kinds of numbers or measurements. For
example, distance traveled in a certain
amount of time is a rate. Or, the cost of a
certain weight of something would also be
a rate.
Section 1 |11
Proportional Reasoning | Unit 4
One rate that is fairly common is the
Farmer’s
Customer’s
comparison of miles, or kilometers, to
Price:
Offer:
hours. Everyone who has been in a car is
probably familiar with how fast the car is
Divide by 5:
Divide by 3:
going or its rate of speed. Traveling at a rate
of 50 miles per hour is faster than a rate of
Unit Price:
Unit Price:
30 miles per hour. These rates of speed are
called unit rates because the rate is given
The unit prices are $0.258 per apple and
for 1 unit of time, an hour.
$0.283 per apple.
This might help! The word “per” means to
divide. So, “miles per hour” means that miles
are in the numerator of the rate and hours
are in the denominator. This is the same
in unit price, which means “price per item.”
The price should always be in the numerator
when calculating unit price.
A unit rate is a rate with 1 in the
denominator.
Keep In Mind! Round up when the next
place value is greater than or equal to 5. If
the next place value is less than 5, keep the
number the same. Also, when comparing,
make sure to round rates to the same place
value and wait until the end of the problem to
do any rounding; otherwise, the calculation
could be incorrect.
So, the farmer was selling the apples for
about 26 cents each. The customer offered
In our apple problem, we have two different
to pay about 28 cents each. The man
rates for buying apples: $1.29 for 5 apples
would have paid more for the apples at his
and 85¢ for 3 apples. To determine which
“bargain price”!
one is the better rate, we need to express
each as a unit rate. When the unit rate is a
Rates can also be used to find the value of
price, we usually refer to it as a unit price.
one quantity when another is known. For
In other words, how much does it cost for 1 example, suppose you need to find the cost
apple in each case?
to buy 100 apples from the farmer at his
Set up each rate as a ratio in the form of
“price per apple.” The price should be in
the numerator, and the number of apples
should be in the denominator. Also, in
order to compare the two rates, the prices
must be in the same units. So, change 85¢
to $0.85. Once the rates are set up, divide
to get the denominator to be 1.
original price.
Let’s use the rate to write a proportion, and
then cross multiply to solve.
=
Set up the proportion.
$1.29(100) = 5x Cross multiply.
$129.00 = 5x Multiply on the left side.
$25.80 = x Divide both sides by 5.
At the rate of $1.29 for 5 apples, it will cost
$25.80 for 100 apples.
12| Section 1
Unit 4 | Proportional Reasoning
ratio of boys to students to set up
the proportion we need. There are 4
boys for every 3 girls, so for every 7
students, there are 4 boys. The ratio
of boys to students in the school is 4
to 7.
Let’s look at a few more examples that use
proportions to solve problems.
Example:
►
Sally drinks one and a half cans of
soda for every two hours she is at
work. At this rate, how many cans of
soda does Sally drink in a 40-hour
workweek?
4 · 182 = 7n Cross multiply.
Solution:
728 = 7n
Set up the proportion.
1
· 40 = 2x Cross multiply.
·
Change 1
to an
= 2x improper
fraction.
►
►
30 = x Divide both sides by 2.
Solution:
►
Be Careful! Notice that on each side of the
proportion we have cans to hours. Also,
notice that in the previous examples, the
proportions have been set up consistently.
When you set up a proportion, make sure
that the rates are written in the same order.
►
The ratio of boys to girls in a middle
school is 4 to 3. If there are 182
students in the school, how many are
boys?
Solution:
►
The ratio we’re given is of boys to
girls. Since we’re given how many
total students there are, we need the
During the cross-country season,
Megan ran a 3-mile race in of an
hour. She ran a 2-mile race in 11
minutes during the track season. In
which race did Megan have a faster
average speed?
sides.
Example:
There are 104 boys in the school.
Example:
Complete the
Sally will drink 30 cans of soda per 40
hours of work.
Complete the
multiplication on the left
side.
104 = n Divide both sides by 7.
60 = 2x multiplication on both
►
Set up the proportion.
=
Find Megan’s unit rate of speed for
each race. Notice that the times
were given in two different units of
measurement. The units must be the
same in order to make an accurate
comparison. One option is to change
of an hour into minutes.
►
There are 60 minutes in an hour, so
of 60 or
►
3-mile race:
=
►
=
2-mile race:
=
►
· 60 is 20 minutes.
=
Megan had a faster average speed in
the 2-mile race.
Section 1 |13
Proportional Reasoning | Unit 4
Let’s Review
Before going on to the practice problems,
make sure you understand the main points
of this lesson.
„ In
„ Rates
„ Proportions
express a relationship between
two different types of measurement.
unit rates, the second number, or the
denominator, is 1.
„ In
order to compare rates, numbers
must be expressed in common units.
can be used to solve
practical problems.
Complete the following activities.
1.13 Match each item with the correct unit price.
40 cents each
_________ 20 for $1.56
63 cents each
_________ $3.78 for 6
78 cents each
_________ 2 for one dollar
$1.05 each
_________ 5 at $3.90
7.8 cents each
_________ 1
for $1.58
1.14 As a unit rate, 137.5 miles in 2.5 hours would be _____.
5.5 miles per hour
137 miles per 2 hours
…
…
55 miles per hour
…
343.75 miles per hour
…
1.15 The produce market sells corn at the rate of 5 ears for $1.30. At this rate, how much
will a dozen ears cost?
$3.90
$3.85
$3.12
$2.60
…
…
…
…
1.16 An office worker can type at the rate of 57 words per minute. Which equation could
be used to solve for the number of words he can type in 6 minutes?
… =
… =
… =
… =
1.17 A certain product has a unit price of 9.5 cents per ounce. The product sells in
bottles of 14 ounces and is priced at $1.30. As compared to the unit price, the $1.30
price of the 14-ounce bottle _____.
is too low
is correct
…
…
is too high
…
14| Section 1
can’t be determined
…
Unit 4 | Proportional Reasoning
1.18 All of the following have the same unit price except _____.
2 for $5.08
…
for $1.27
5 for $12.70
…
…
2
…
for $6.25
1.19 Which of the following cars is traveling at the fastest rate of speed?
141 miles in 3 hours
97 miles in 2 hours
…
…
205 miles in 5 hours
…
172 miles in 4 hours
…
1.20 In order to get a certain shade of blue paint, a mixer must have 5 parts white paint
to 3 parts blue. If 4 gallons of paint must be mixed, how many gallons of white paint
must be used?
3.7 gallons
2.5 gallons
1.9 gallons
1.5 gallons
…
…
…
…
1.21 Which of the following boxes of cereal is the best buy?
14-ounce box for $2.19
18-ounce box for $3.07
…
…
15.5-ounce box for $2.28
…
24-ounce box for $3.95
…
1.22 Myles mows lawns during the summer to earn money. Since he is trying to save
money for a car, he tries not to spend very much of it. In fact, his goal is to save $5
for every $7 he makes. If Myles made $128 this week, how much can he spend?
$91.43
$3.66
$25.60
$36.57
…
…
…
…
Section 1 |15
Proportional Reasoning | Unit 4
Direct Variation
Objectives
z Recognize a relationship as a direct variation.
z Calculate
the constant of variation.
z Calculate
a missing value in a direct variation problem.
z Use
the constant of variation to determine the equation of a direct variation.
Vocabulary
constant of variation—the rate of change in a direct variation; also known as constant of
proportionality
direct variation—a function of the form y = kx, where k is not zero
directly proportional—two quantities that increase or decrease (change) by the same
factor; also known as directly related
Direct Variations
This movie problem seems pretty simple. If
it costs $5 for one person, then it will cost
$10 for two people, $15 for three people,
and $20 for four people.
But, wait! We just assumed that the cost per
person, or the unit price, would stay the
same, even though the word problem didn’t
16| Section 1
say that the cost for each person would be
$5. Look at the following ad:
Unit 4 | Proportional Reasoning
theaters have a constant rate that they
charge for admission. For each additional
person that goes to the movie, the total
price increases by the same amount.
Key point! In direct variations, the rate
of change is constant, or stays the same.
If the rate of change is not constant, the
relationship is not directly related and is not a
direct variation.
Is the cost per person constant?
Vocabulary! The word constant means to
stay the same.
No! The unit price decreases as the number
of people increases:
Relationships that have a constant rate
of change are called direct variations—if
one quantity changes, the other quantity
changes by the same factor. And, we say
the two quantities are directly related or
directly proportional.
Number
of People
Cost
Rate
Unit
Price
1
$5.00
=5
$5.00
Keep in mind! There are many ways
to express that a relationship is a direct
variation. Some other common phrases are
“directly related to,” “directly proportional to,”
and “varies directly as.”
2
$9.00
= 4.5
$4.50
We can solve direct variation problems using
proportions.
3
$12.00
=4
$4.00
4
$15.00
= 3.75
$3.75
5
$15.00
=3
$3.00
Because the unit rate changes depending
on the number of people in the car,
this relationship is not proportional. In
our original answer to this problem, we
assumed that the total cost was directly
related to the number of people. Or, that
the cost for each person, the unit price,
stayed the same for any number of people.
And, in fact, many relationships do have
a constant rate of change. Most movie
Let’s look at some examples of direct
variation problems.
Example:
►
The amount of money Mariah earns
is directly related to the number of
hours she works. When Mariah works
40 hours, she earns $537. How much
will she earn if she works 30 hours?
Solution:
►
Since the money (m) earned is a
direct variation of the hours (h)
worked, set up a proportion to solve.
Section 1 |17
Proportional Reasoning | Unit 4
=
=
Set up the proportion.
537 · 30 = 40m Cross multiply.
16,110 = 40m on the left side.
402.75 = m Divide both sides by 40.
What is the constant of variation in the
camping problem?
Mariah earns $402.75 if she works 30
hours.
dividing the cost by the hours: = = 3.
The constant of variation is 3, which means
that the ratio of cost to hours should always
be 3 to 1.
Complete the multiplication
►
Example:
►
You go camping and rent a mountain
bike to ride the trails. It costs $6.00 to
rent the bike for 2 hours. If the cost,
c, varies directly as the time, h, how
many hours will you have the bike if
you pay $15?
Solution:
=
=
Set up the proportion.
6h = 2 · 15 Cross multiply.
Complete the multiplication
6h = 30 on the right side.
h = 5 Divide both sides by 6.
►
When a relationship is a direct variation, y
varies directly as x, and k is the constant of
variation (k ≠ 0).
You will have the bike for 5 hours.
Remember that in a direct variation, the
ratio between the two quantities stays
constant. That’s why we’re able to use a
proportion to solve. That constant ratio has
a special name - the constant of variation or
the constant of proportionality.
►
18| Section 1
=k
The constant of variation is found by
What would the unit price for this problem
be? It is $3 per hour, equal to the constant
of variation! Notice that we find the unit
price (or rate) the same way that we find
the constant of variation. In fact, unit rate
and constant of variation represent the
same idea. However, unit rate is expressed
as a comparison of two quantities (a ratio),
while the constant of variation is expressed
as a single factor.
Example:
►
A quantity, m, is directly related to
p. If m is 4 when p is 16, what is the
constant of variation?
Solution:
►
►
►
This is a direct variation. So, the
constant of variation is equal to the
ratio of m to p.
=
=
The constant of variation is
.
Unit 4 | Proportional Reasoning
Complete the following activities.
1.23 The amount that Susan charges per hour for babysitting is directly proportional to
the number of children she is watching. She charges $4.50 for 3 kids. How much
would she charge to babysit 5 kids?
$1.50
$3.33
$6.50
$7.50
…
…
…
…
1.24 The time (t) it takes to clean an office building is directly related to the number
of people (p) working. Which of the following equations can be used to find the
constant of variation (k)?
… = k
… = p
… = k
… = t
1.25 Assume that r varies directly as p. What is the constant of proportionality if r = 3
when p = 15.
k=5
k=
k=3
k = 45
…
…
…
…
1.26 The relationship between the amount of bleach (in quarts) and the amount of
water (in gallons) of a certain cleaning product is a direct variation. The constant of
variation is . How much water should be used to dilute 5 quarts of bleach?
1.67 gallons
15 gallons
…
…
5 gallons
…
45 gallons
…
1.27 A quantity, y, varies directly as x. When y = 10, x = 6. Find x when y = 14.
4.3
8.4
23.3
60
…
…
…
…
1.28 The circumference of a circle varies directly as the diameter of a circle. When the
diameter of a circle is 5, the circumference of the circle is approximately 15.7. What
is the approximate circumference of a circle with a diameter of 9? Round answer to
nearest tenth.
2.9
8.7
28.3
706.5
…
…
…
…
1.29 The perimeter of a square is directly proportional to the length of one of its
sides. The perimeter is 28 when the length of a side is 7. What is the constant of
proportionality?
4
14
196
…
…
…
…
Section 1 |19
Proportional Reasoning | Unit 4
Graphing Direct Variations
We said that a direct variation has the form
Now, graph your ordered pairs on a
coordinate graph.
= k. Can you express a direct variation
any other way? Yes! You can express k as a
fraction by putting it over 1. Now you have
a proportion that you can simplify using
cross multiplication.
„
=
„ y
·1=k·x
„ y
= kx
So, another way to express a direct
Let’s also graph the movie theatre problem
variation is y = kx. And, y is actually a
function of x. For any number that we put in from the beginning of the lesson. Then we
can compare the graphs to see what direct
for x, we will get exactly one value for y.
variation functions have in common.
Since direct variations are a type of
The unit rate of the movie problem was
function, let’s try graphing one of our
$5. It costs $5 for one person, $10 for two
earlier examples. In the camping example
people, $15 for three people, and so on.
from Section 1, the cost to rent a bike, c,
Since the cost (c) is directly related to the
is directly related to the number of hours,
number of people (p), and the constant of
h, that it is rented. Since it costs $6 to rent
variation is 5, we can write the function as
the bike for 2 hours, we calculated that the
constant of variation is 3. We can represent c = 5p.
this relationship using the form y = kx. So, in
the camping problem, y = 3x, or c = 3h. Let’s Our table of ordered pairs would be:
make a table of values, where h is the first
ordered
number, and c is the second number.
p
c
pair
0
0
(0, 0)
This might help! In a function table, x is
1
5
(1, 5)
always the first number and y is always the
2 10
(2, 10)
second number. So, when using different
3 15
(3, 15)
letters, the variable that takes the x’s place in
the form y = kx will be the first number.
h
0
1
2
3
20| Section 1
c
0
3
6
9
ordered pair
(0, 0)
(1, 3)
(2, 6)
(3, 9)
And, the coordinate graph looks like this:
Unit 4 | Proportional Reasoning
„ Both
graphs go through the point (0,0).
Do all direct variation functions go
through the point (0,0)? Yes! Why? We
know that all direct variations have to be
in the form y = kx. No matter what the
constant of variation, k, is, if we put 0 in
for x, we will always get 0 for y, because
0 multiplied by any number is 0.
Key point! All direct variation graphs are
straight lines and go through the point (0,0),
which is the origin.
Let’s put the two functions on the same
graph, so we can compare them more
easily.
What is different about the two graphs?
„ The
blue line is steeper than the red
line. That’s because the functions
have a different constant of variation.
Although the rate of change in each line
is constant, the rates of change of the
two lines are different. So, one line will
increase more quickly than the other.
The constant of variation determines
which direction the line will go (up or
down) and how steep the line will be.
This might help! The only number that k
cannot equal is 0. That means that k can be
any positive or negative real number other
than 0.
Example:
►
What is similar about the two graphs?
Look at the values in the following
table. Is this a direct variation?
x
6
9
12
y
-2
-3
-4
„ Both
graphs are straight lines. Are all
direct variation functions straight lines?
Yes! That’s because direct variations
increase or decrease at a constant rate.
Section 1 |21
Proportional Reasoning | Unit 4
Solution:
►
►
Calculate the constant of variation
for each ordered pair.
•
=
=-
•
=
=-
•
=
=-
Since k = - for all of the ordered
pairs, it is a direct variation.
Think about it! We can’t choose (0,0), the
origin, to determine the constant of variation
because every direct variation goes through
that point. The origin won’t help us determine
the constant of variation for a specific line.
►
=
=4
►
=
=4
►
Example:
►
Is the following graph a direct
variation? If so, what is its equation?
The constant of variation, or k, is 4.
Substitute 4 for k in y = kx. So, the
equation of this direct variation is
y = 4x.
Let’s Review
Before going on to the practice problems,
make sure you understand the main points
of this lesson.
„ In
a direct variation, the ratio between
the two quantities is constant.
„ The
constant ratio between the two
quantities is called the constant
of variation or the constant of
proportionality.
„ Proportions
can be used to solve direct
variation problems.
Solution:
►
►
The graph is a straight line, and it
goes through the point (0,0). Yes, it is
a direct variation.
The equation of a direct variation
is in the form y = kx, where k is the
constant of variation. To find what
number the constant is, let’s choose
a point (other than the origin) and
find the ratio of y to x.
22| Section 1
„ Direct
variations are a special type of
function that have the form y = kx.
„ The
graph of a direct variation is always
a straight line and always goes through
the origin.
Unit 4 | Proportional Reasoning
Complete the following activities.
1.30 In a direct variation, the constant of proportionality is -3. Which of the following
would be its equation?
y=x-3
-3y = x
y = -3x
-3 = xy
…
…
…
…
1.31 Select the graph that is not a direct variation.
…
…
…
…
1.32 Which of the following sets of points is a direct variation?
…(0, 0); (2, 4); (5, 15)
…(3, 6); (-2, 4); (-1, 2)
…(0, 0); (-1, -2); (1, 2)
…(0, 5); (1, 6); (2, 7)
1.33 What is the equation of the following direct variation?
y = -2x
…
y=- x
…
y = 2x
…
y=
…
x
Section 1 |23
Proportional Reasoning | Unit 4
1.34 What is the equation of a direct variation that goes through the points (-6, 2) and
(9, -3)?
y = -3x
3-y=x
…
…
y=x-3
…
y=- x
…
1.35 Determine whether the following graph is a direct variation. Explain how you came
to your conclusion.
Review the material in this section in preparation for the Self Test. The Self Test
will check your mastery of this particular section. The items missed on this Self Test will
indicate specific area where restudy is needed for mastery.
24| Section 1
Unit 4 | Proportional Reasoning
SELF TEST 1: Proportions
Complete the following activities (6 points, each numbered activity).
1.01If x:6 as 3:9, then x is equal to ___________ .
1.02 S varies directly as T. If S is 20 when T is 4, then T is ___________ when S is 30.
1.03If
=
, then x is ___________ .
1.04 Y is directly related to X, and Y is 81 when X is 27. The constant of variation is
___________ .
1.05 The ratio of boys to girls is 5 to 4. There are 80 boys. How many girls are there?
4
16
64
100
…
…
…
…
1.06 At what rate is a car traveling, if it goes 157.5 miles in 2.5 hours?
15 miles per hour
73 miles per hour
…
…
63 miles per hour
…
393.75 miles per hour
…
1.07 Which of the following direct variations has a constant of variation that is equal to -3?
…
…
…
…
1.08 All of the following ratios are equivalent except _____.
2:3
…
…
…
8 to 12
…
Section 1 |25
Proportional Reasoning | Unit 4
1.09 The following table shows a direct variation. Find y.
0
…
3 15 18 27
y
5
6
9
1
…
2
…
3
…
1.010 Which of the following equations is not a proportion?
… =
… =
… =
… =
1.011 A certain paint is to be mixed with 3 parts yellow to 2 parts blue. What is the ratio of
yellow paint to the total amount of mixed paint?
3 to 2
3 to 5
2 to 3
3 to 1
…
…
…
…
1.012 A certain paint is to be mixed with 3 parts yellow to 2 parts blue. Twelve gallons of
mixed paint are needed. How many gallons of blue must be used?
4
4.8
7.2
10
…
…
…
…
1.013 Which of the following prices is the lowest price per pound?
2 pounds for $2.75
4 pounds for $5.60
…
…
3 pounds for $4.05
…
$1.36 per pound
…
1.014 Which of the following sets of ordered pairs is not a direct variation?
…(0, 0); (-2, 4); (3, -6)
…(10, 2); (15, 3); (20, 4)
…(1, 1); (-2, -2); (3, 3)
…(0, 0); (1, 3); (2, 4)
1.015 Which of the following is the equation of a direct variation that has a constant of
variation equal to y=x…
?
- y=x
…
y = -2x
…
y=- x
…
1.016 A double recipe of cookies calls for 5 cups of flour. Which of the following
proportions could be used to find the amount of flour for a triple recipe?
… =
77
26| Section 1
96
… =
SCORE
… =
TEACHER
… =
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