5-6 Graphing Proportional Relationships
Determine whether each relationship is proportional by graphing on the coordinate plane. Explain your
reasoning.
2. SOLUTION: Graph the ordered pairs on the coordinate plane. Then connect the ordered pairs.
The line passes through the origin and is a straight line. So, the number of quarts is proportional to the number of
gallons.
ANSWER: Proportional; the line passes through the origin and is a straight line.
4. The number of students on a school trip is proportional to the number of teachers as shown in the graph.
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a. Find and interpret the constant of proportionality. b. Explain what the points (0, 0), (1, 25) and (4, 100) represent.
SOLUTION: Page 1
the line
passes through theRelationships
origin and is a straight line.
5-6Proportional;
Graphing
Proportional
4. The number of students on a school trip is proportional to the number of teachers as shown in the graph.
a. Find and interpret the constant of proportionality. b. Explain what the points (0, 0), (1, 25) and (4, 100) represent.
SOLUTION: a. Use the point (4, 100) on the graph.
The constant of proportionality, or unit rate, is 25. There are 25 students per teacher.
b. The point (0,0) represents 0 teachers having 0 students on the trip. The point (1, 25) represents 1 teacher having
25 students on the trip. The point (4, 100) represents 4 teachers having 100 students on the trip.
ANSWER: a. 25; There are 25 students per teacher.
b. 0 teachers would have 0 students on the trip, 1 teacher would have 25 students on the trip, and 4 teachers would
have 100 students on the trip.
Determine whether each relationship is proportional by graphing on the coordinate plane. Explain your
reasoning.
6. SOLUTION: Graph the ordered pairs on the coordinate plane. Then connect the ordered pairs.
The graph is a straight line but does not pass through the origin. So, the total cost is nonproportional to the number of
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pizzas.
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ANSWER: a. 25; There are 25 students per teacher.
b. 0 teachers would have 0 students on the trip, 1 teacher would have 25 students on the trip, and 4 teachers would
have 100 students on the trip.
5-6 Graphing Proportional Relationships
Determine whether each relationship is proportional by graphing on the coordinate plane. Explain your
reasoning.
6. SOLUTION: Graph the ordered pairs on the coordinate plane. Then connect the ordered pairs.
The graph is a straight line but does not pass through the origin. So, the total cost is nonproportional to the number of
pizzas.
ANSWER: Nonproportional; the line does not pass through the origin.
8. The number of pounds of walnuts in a nut mix is proportional to the number of pounds of peanuts as shown in the
graph.
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a. Find and interpret the constant of proportionality. b.
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line does not pass through
the origin.
5-6Nonproportional;
Graphing the
Proportional
Relationships
8. The number of pounds of walnuts in a nut mix is proportional to the number of pounds of peanuts as shown in the
graph.
a. Find and interpret the constant of proportionality. b. Explain what the points (0, 0), (1, 0.4) and (7.5, 3) represent.
SOLUTION: a. Use the point (5, 2) on the graph.
The constant of proportionality, or unit rate, is 0.4. For every pound of peanuts, there is 0.4 pounds of walnuts.
b. The point (0,0) represents 0 pounds of peanuts for 0 pounds of walnuts. The point (1, 0.4) represents 1 pound of
peanuts for 0.4 pounds of walnuts. The point (7.5, 3) represents 7.5 pounds of peanuts for 3 pounds of walnuts.
ANSWER: a. 0.4; For every pound of peanuts, there is 0.4 pound of walnuts.
b. For 0 pound of peanuts there are 0 pounds of walnuts, for 1 pound of peanuts there is 0.4 pound of walnuts, and
for 7.5 pounds of peanuts there are 3 pounds of walnuts.
10. Financial Literacy The cost C to place an ad in a newspaper can be found using the formula, C = $5 + $2L, where
L is the number of lines of text. Find the value of C when L = 0. Then explain why this shows that the cost of an ad
is not proportional to the number of lines of text.
SOLUTION: To find the value of C when L = 0, substitute 0 for L in the formula C = $5 + $2L.
So, the value of C is 5 when L = 0. A graph showing the relationship between the number of lines of text and the
cost of an ad would pass through the point (0, 5). This means the graph does not pass through the origin and cannot
be the graph of a proportional relationship.
ANSWER: 5; Sample answer: Since the graph passes through (0, 5) it does not pass through the origin and cannot be the graph
of a proportional relationship.
12. Sonya walks on a treadmill at a constant rate of 3.5 miles per hour. A graph showing the relationship between the
time she walks and the distance she travels passes through the origin and through the point (1, 3.5). Name three
other points that lie on the graph.
SOLUTION: Page 4
Sample answer:
The graph of the relationship is a straight line that passes through the origin and the point (1, 3.5), so the unit rate is
3.5. Use the unit rate to find three other points that lie on the graph.
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be the graph of a proportional relationship.
ANSWER: 5; Sample answer: Since the graph passes through (0, 5) it does not pass through the origin and cannot be the graph
of a proportional relationship.
5-6 Graphing Proportional Relationships
12. Sonya walks on a treadmill at a constant rate of 3.5 miles per hour. A graph showing the relationship between the
time she walks and the distance she travels passes through the origin and through the point (1, 3.5). Name three
other points that lie on the graph.
SOLUTION: Sample answer:
The graph of the relationship is a straight line that passes through the origin and the point (1, 3.5), so the unit rate is
3.5. Use the unit rate to find three other points that lie on the graph.
2 × 3.5 = 7
So, the point (2, 7) lies on the graph.
3 × 3.5 = 10.5
So, the point (3, 10.5) lies on the graph.
4 × 3.5 = 14
So, the point (4, 14) lies on the graph.
ANSWER: Sample answer: (2, 7), (3, 10.5), (4, 14).
Complete each table so that the relationship between the two quantities is proportional. Check your
work by graphing on the coordinate plane.
14. SOLUTION: Find the constant of proportionality.
The constant of proportionality is 6. The cost is 6 dollars for each yard of ribbon.
Use the constant of proportionality to complete the table.
Number of
Total Cost
Tickets
($)
1
6
3
3 × 6 = 18
24
24 ÷ 6 = 4
42
42 ÷ 6 = 7
Graph the ordered pairs on the coordinate plane. Then connect the ordered pairs.
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The graph passes through the origin and is a straight line, so the relationship between the quantities in the table is
proportional.
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1
6
3
3 × 6 = 18
24
24 ÷ 6 = 4
42
42 ÷ 6 = 7
Graph the ordered pairs on the coordinate plane. Then connect the ordered pairs.
5-6 Graphing Proportional Relationships
The graph passes through the origin and is a straight line, so the relationship between the quantities in the table is
proportional.
ANSWER: 16. SOLUTION: Find the constant of proportionality.
The constant of proportionality is 5. For every 1 unit of side length, a regular pentagon has 5 units of perimeter.
Use the constant of proportionality to complete the table.
Perimeter of a
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Side Length (ft) Regular Pentagon
(ft)
10
10 ÷ 5 = 2
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5-6The
Graphing
Proportional
constant of proportionality
is 5. ForRelationships
every 1 unit of side length, a regular pentagon has 5 units of perimeter.
Use the constant of proportionality to complete the table.
Perimeter of a
Side Length (ft) Regular Pentagon
(ft)
10
10 ÷ 5 = 2
4
20
7
7 × 5 = 35
9
9 × 5 = 45
Graph the ordered pairs on the coordinate plane. Then connect the ordered pairs.
The graph passes through the origin and is a straight line, so the relationship between the quantities in the table is
proportional.
ANSWER: 18. Persevere with Problems Explain how the relationship of x and y in the graphs of y = 3x + 1 and y = 3x differ.
SOLUTION: y = 3x
x
y
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0
0
1
3
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5-6 Graphing Proportional Relationships
18. Persevere with Problems Explain how the relationship of x and y in the graphs of y = 3x + 1 and y = 3x differ.
SOLUTION: y = 3x
x
y
0
0
1
3
2
6
3
9
y = 3x + 1
x
y
0
1
1
4
2
7
3
10
Both y = 3x and y = 3x + 1 have graphs that are straight lines. However, only y = 3x has a graph that passes through
the origin. In y = 3x, x and y have a proportional relationship. In y = 3x + 1, x and y have a nonproportional
relationship.
ANSWER: In y = 3x, x and y have a proportional relationship. In y = 3x + 1, x and y have a nonproportional relationship.
20. Building on the Essential Question Explain two ways that you could determine if the following statement
represents a proportional relationship.
A muffin recipe calls for 2 cups of flour per batch.
SOLUTION: Sample answer: One way is to make a table of values using the rate of 2 cups of flour per batch. Graph the values.
If the graph is a straight line and passes through the origin, the relationship is proportional. Another way is to check
that the values in the table all have equal rates.
ANSWER: Sample answer: One way is to make a table of values using the rate of 2 cups of flour per batch. Graph the values.
If the graph is a straight line and passes through the origin, the relationship is proportional. Another way is to check
that the values in the table all have equal rates.
22. Short Response The number of books Jennifer reads is proportional to the number of weeks. Every 20 weeks she
reads 6 books. Find the constant of proportionality.
SOLUTION: Write a ratio comparing the number of books Jennifer reads to the number of weeks.
The constant of proportionality is 0.3.
ANSWER: 0.3
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Write
each
ratio as
fraction
in simplest form.
24. A bag of marbles has 20 red marbles and 45 blue marbles. What is the ratio of red marbles to blue marbles?
SOLUTION: Page 8
The constant of proportionality is 0.3.
5-6ANSWER: Graphing Proportional Relationships
0.3
Write each ratio as a fraction in simplest form.
24. A bag of marbles has 20 red marbles and 45 blue marbles. What is the ratio of red marbles to blue marbles?
SOLUTION: ANSWER: 26. A garden has 8 tomato plants and 4 pepper plants. What is the ratio of pepper plants to total plants in the garden?
SOLUTION: ANSWER: Evaluate each expression.
28. (–y) + y
SOLUTION: The sum of any integer and its opposite is zero.
(–y) + y = 0
ANSWER: 0
Evaluate each expression.
30. 6 ÷ 2 • 3
SOLUTION: ANSWER: 9
32. 15 • 1 + 6
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ANSWER: 21
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ANSWER: 9
5-6 Graphing Proportional Relationships
32. 15 • 1 + 6
SOLUTION: ANSWER: 21
Evaluate each expression. Express the result in scientific notation.
5
9
34. (2.62 × 10 )(6.15 × 10 )
SOLUTION: ANSWER: 15
1.6113 × 10
Order each set of numbers from least to greatest.
5
5
–3
5
–3
36. 3.5 × 10 , 4 × 10 , 3.6 × 10 , 0.004
SOLUTION: 5
3.5 × 10 , 4 × 10 , 3.6 × 10 , 0.004
–3
0.004 written in scientific notation is 4 × 10 . Because two of the numbers have an exponent of –3, compare their
factors.
3.6 < 4
–3
So, 3.6 × 10 is less than 0.004.
Because the other two numbers have an exponent of 5, compare their factors.
3.5 < 4
5
5
So, 3.5 × 10 is less than 4 × 10 .
The numbers with an exponent of –3 are less than the numbers with an exponent of 5.
5
5
–3
The numbers in order from least to greatest are 3.6 × 10 , 0.004, 3.5 × 10 , 4 × 10 .
ANSWER: –3
5
3.6 × 10 , 0.004, 3.5 × 10 , 4 × 10
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