Download Proportional relationships can be identified in both tables and

Proportional relationships can be identified in both tables and graphs. Today you will have an
opportunity to take a closer look at how graphs and tables for proportional relationships can help you
organize your work to find any missing value quickly and easily.
4-46. Robert’s new hybrid car has a gas tank that holds 12 gallons of gas. When the tank is full, he can
drive 420 miles. Assume that his car uses gas at a steady rate.
A. Is the relationship between the number of gallons of gas used and the number of miles that can
be driven proportional? For example, does it change like Sonja’s birdseed prediction, or is it
more like Gustavo’s college savings? Explain how you know.
B. Show how much gas Robert’s car will use at various distances by copying and completing the
table below.
A. Robert decided to graph the situation, as shown below. The distance Robert can travel using one
gallon of gas is called the unit rate. Use Robert’s graph to predict how far he can drive using
one gallon of gas. That is, find his unit rate.
B. While a graph is a useful tool for estimating, it is often difficult to find an exact answer on a
graph. What is significant about the point labeled (1, y)? How can you calculate y?
C. Use the table in part (b) and your result in part (d) to find Robert’s unit rate.
D. Work with your team to write the equation to find the exact number of miles Robert can drive
with any number of gallons of gas. Be prepared to share your strategy.
E. Use your equation to find out how many gallons of gas Robert will need to drive 287 miles. .
4-47. THE YOGURT SHOP
Jell E. Bean owns the local frozen yogurt shop. At her store, customers serve themselves a bowl of
frozen yogurt and top it with chocolate chips, frozen raspberries, and any of the different treats
available. Customers must then weigh their creations and are charged by the weight of their bowls.
Jell E. Bean charges $32 for five pounds of dessert, but not many people buy that much frozen
yogurt. She needs you to help her figure out how much to charge her customers. She has customers that
are young children who buy only a small amount of yogurt as well as large groups that come in and pay
for everyone’s yogurt together.
A. Is it reasonable to assume that the weight of the yogurt is proportional to its cost? How can you
tell?
B. Assuming it is proportional, make a table that lists the price for at least ten different weights of
yogurt. Be sure to include at least three weights that are not whole numbers.
C. What is the unit rate of the yogurt? (Stores often call this the unit price.) Use the unit rate to
write an equation that Jell E. Bean can use to calculate the amount any customer will pay.
D. If Jell E. Bean decided to start charging $0.50 for each cup before her customers started filling it
with yogurt and toppings, could you use the same equation to find the new prices? Why or why
not?
4-48. Lexie claims that she can send 14 text messages in 22 minutes. Her teammates Kenny and Esther
are trying to predict how many text messages Lexie can send in a 55-minute lunch period if she keeps
going at the same rate.
a. Is the relationship between the number of text messages and time in minutes proportional? Why
or why not?
b. Kenny represented the situation using the table shown below. Explain Kenny’s strategy for
using the table.
i.
c. Esther wants to solve the problem using an equation. Help her write an equation to determine
how many text messages Lexie could send in any number of minutes.
d. Find the missing value in Kenny’s table.
e. Solve Esther’s equation. Will she get the same answer as Kenny?
f. What is Lexie’s unit rate? That is, how many text messages can she send in 1 minute?
4-49. Additional Challenge: Use your reasoning skills to compute each unit rate (the price per pound).
A. $4.20 for
pound of cheese
B. $1.50 for
pound of bananas
C. $6.00 for
pound of deli roast beef
D. $7.50 for
pound of sliced turkey
Proportional Relationships
A relationship is proportional if one quantity is a multiple of the other. This relationship can be
identified in tables, graphs, and equations.
Table: Equivalent ratios of
(or
) can be seen in a table.
Graph: A straight line through the origin.
Equation: An equation of the form y = kx where k is the constant of proportionality.
Example: Three pounds of chicken costs $7.00. What is the cost for x pounds?
Equation:
The relationship between pounds and cost is proportional. The table has equivalent ratios (
), the graph is a straight line through the origin, and the equation is of the form y = kx.
Example: The county fair costs $5.00 to enter and $1.00 per ride. Equation: y = 1x + 5
The relationship between rides and cost is not proportional, because the table does not contain
equivalent ratios (
addition.
), the graph does not pass through the origin, and the equation contains