25
f.
25
Use the equation to help Carina figure out how many people will probably vote
for Mr. Mears if 350,125 people vote in the election. What do x and y
represent in your equation? [ 238,085 votes. x is the number of potential
voters, y is the number of votes for Mr. Mears. ]
g. What information would the unit rate give Carina? What is the unit rate? How
CC 2.5 Chapter 8does
the unit rate compare to the slope? [ The unit rate would tell her how
CC 3 Chapter 7 many votes to expect per one potential voter. 1 potential
0.68 votes
voter . The unit rate is
the same as the slope with a denominator of 1 unit. ]
7-­‐78 to 7-­‐80 7-78.
In two minutes, Stacie can write her name 17 times.
a.
Use a proportion (write equivalent ratios) to find how long it will take Stacie to
write her name 85 times. [ 172 = 85x , x = 10 minutes ]
b.
Make a graph of this situation. Which piece of information
is on the dependent axis? [ See graph at right; the number
of times is the dependent axis. ]
c.
7-78.
Another name for the slope in a proportional relationship is
the constant of proportionality. Find the constant of
proportionality and write an equation in y = mx + b form for Stacie’s situation.
What do x and y represent in this situation? [ The constant of
proportionality is 172 ; y = 172 x . x is the time (minutes) and y is the number
of times Stacie can write her name. ]
Problem continued from previous page.
d.
Use the equation to find how many minutes it will take Stacie to write her name
Problem continues on next page. →
85 times. [ 85 = 172 x , x = 10 minutes ]
e.
What is the advantage of the equation in part (d) compared to the proportion
(equivalent ratios) in part (a)? What is the advantage of the proportion?
[ Advantage of equation: can find number of times for any number of
minutes, can easily see unit rate. Advantage of proportion: can quickly
solve for a specific situation. ]
Chapter 7: Slope and Association
f.
7-79. What is the unit rate at which Stacie writes her name? [
17 names
2
1 minute
=
595
8.5 names
1 minute
]
Beth and her father have a large model railroad in their basement. Beth figured out
that the equation y = 260
3 x relates the length of an item on her model railroad to the
length of the real thing, where x is the size on the model and y is the real object’s
size. Beth’s uncle has a different model railroad in his home. It has a different scale.
He put together the following table for Beth.
Human
Tree
Apartment
Building
Height on Model Railroad
(inches)
0.94
4.70
Height of Real Object
(inches)
60
300
7.52
480
a.
In decimal notation, what is the unit rate for Beth’s railroad? For her uncle’s
prototype
railroad? Make sure you show units on your unit rate. [ Beth: 86.71 inch
,
inch model
63.8 inch prototype
Uncle: 1 inch model ]
b.
On which model would a horse be taller? Use the unit rates to justify your
answer without making any calculations. [ Possible answer: on Beth’s
railroad, a real 87-inch horse is about an inch on the model; on the uncle’s
railroad, a real 64-inch horse is about an inch, so an 87-inch horse would be
more than an inch; so any horse is taller on the Uncle’s model. ]
minutes, can easily see unit rate. Advantage of proportion: can quickly
solve for a specific situation. ]
f.
7-79.
What is the unit rate at which Stacie writes her name? [
17 names
2
1 minute
=
8.5 names
1 minute
]
Beth and her father have a large model railroad in their basement. Beth figured out
that the equation y = 260
3 x relates the length of an item on her model railroad to the
length of the real thing, where x is the size on the model and y is the real object’s
size. Beth’s uncle has a different model railroad in his home. It has a different scale.
He put together the following table for Beth.
Human
Tree
Apartment
Building
Height on Model Railroad
(inches)
0.94
4.70
Height of Real Object
(inches)
60
300
7.52
480
a.
In decimal notation, what is the unit rate for Beth’s railroad? For her uncle’s
prototype
railroad? Make sure you show units on your unit rate. [ Beth: 86.71 inch
,
inch model
63.8 inch prototype
Uncle: 1 inch model ]
b.
On which model would a horse be taller? Use the unit rates to justify your
answer without making any calculations. [ Possible answer: on Beth’s
railroad, a real 87-inch horse is about an inch on the model; on the uncle’s
railroad, a real 64-inch horse is about an inch, so an 87-inch horse would be
more than an inch; so any horse is taller on the Uncle’s model. ]
596
Eight of 29 students in your class want to attend the Winter Ball.
a.
Find the constant of proportionality, and then write an equation in y = mx + b
attend
8
form. What do x and y represent? [ 829students
total students , y = 29 x , x is the total
number of students and y is the number attending ]
b.
If your class represents the entire school, how many of the 1490 students will
probably attend the dance? Solve your equation from part (b).
[ x ≈ 411 students ]
Core Connections, Course 3
ETHODS AND MEANINGS
MATH NOTES
7-80.
Proportional Equations
A proportional relationship can be seen in a table or a graph, as you
saw in Section 1.2 of Chapter 1. The equation for a proportion is y = kx ,
where k is the constant of proportionality, or the slope of the line. The
starting point of the linear equation is always zero, because a proportional
relationship always passes through the origin.
The constant of proportionality, when written as a fraction with a
denominator of 1, is the unit rate.
$7.00
For example, if the constant of proportionality (the slope) is 3 pounds
,
chicken
then the equation relating weight to cost is y = 73 x , where x is the weight (lbs)
and y is the cost ($).
The unit rate is
$7
3
! $2.33 per pound.