Proportional Relationships
Is the relationship between cups of juice and total cost proportional if each cup costs $2?
Yes.
Each cup costs the same amount, so the cost per cup stays constant.
In y = 5x, what is the constant of proportionality?
5
If y = 3x, and x = 4, what is y?
y = 12
Substitute 4 for x: y = 3 x 4
Using the coordinates (1,4) and (2,8) is it proportional?
Yes
The ratio is 4 in both sets.
What point must the graph of a proportional relationship pass through?
The origin, (0,0)
A student walks 3 miles in 2 hours.
Is distance proportional to time?
Yes.
The distance-to-time ratio stays the same: 3 miles per hour.
A table shows 2 books cost $6 and 5 books cost $15.
Find the cost per book.
$3 per book
Divide the cost by the number of books.
Each ticket costs $7.
Write an equation for the total cost y of x tickets
y = 7x
Multiply the number of tickets by the $7 cost per ticket.
Using a table, if:
x = 1, 2, 3
y =3, 6, 10
Is it proportional? Why or Why not?
No, not all ratios are equal.
3/1 = 3
6/2 = 3
10/3 is not 3
A graph has the point (2, 10) on a proportional line, What is the constant of proportionality?
The CoP is 5.
10/2=5
A recipe uses 2 cups of flour for 5 cups of oats.
How much flour for 15 cups of oats?
Six cups of flour.
Fifteen cups of oats is 3 times 5 cups, so multiply the flour by 3.
If 12 pages are read in 3 minutes at a constant rate, what is the rate in pages per minute.
4 pages per minutes
Divide pages by minutes. 12/3 = 4
The cost y is proportional to the number x of pounds at $4 per pound.
Write an equation and find the cost for 6 pounds.
y = 4x
y = 4(6)
y = $24
Compare the relationships below. Which one has the greater Unit Rate?
Relationship A: y = 4x
Relationship B: x = 2 y = 6
Relationship A has a unit rate of 4 and is higher
than relationship B, which has a unit rate of 3.
A line through (0, 0) and (3,12) represents a proportional relationship.
Find k and write the equation.
Cop take 12 divided by 3: k = 4
equation: y = 4x
A car travels 150 miles in 3 hours at a steady speed.
How far will it travel in 5 hours?
250 miles
Find the steady rate, then multiply by 50 MPH.
A proportional relationship includes (4, 18).
Find k (constant of proportionality) and the value of y when x=10.
k = 4.5 y = 45
Divide 18 by 4 to find k.
Then multiply k by 10.
A club charges a $5 sign-up fee plus $2 per meeting. Is the total cost proportional to the meetings?
Use an equation to justify your answer.
No.
y = 2x + 5
Table A: Table B:
x y x y
2 8 2 10
4 16 4 18
6 24 6 26
Which table is Proportional?
Table A, the constant is 4.
Graph A passes through (2, 6) and Graph B passes through (2, 8). Both pass through the origin.
Which line is steeper and has the larger constant?
Graph A has a rate of 3 and Graph B has a rate of 4.
Graph B is steeper because it has the larger constant of proportionality.
A drink mix uses 3 scoops for 8 cups of water.
How many scoops are needed for 20 cups of water?
7.5 Scoops
The rate is 3/8 scoop per cup, then multiply that rate by 20 cups.
A graph has a proportional line through (5, 35).
Find k, write its equation, and find y when x = 9.
35 divided by 5 to find k. k = 7
The equation is y = 7x
y = 7 x 9 y = 63
A cyclist rides at a constant speed of 12 miles per hour.
Write an equation for distance d after t hours and find the time for 54 miles.
d = 12t
54/12
t = 4.5 hours
Relationship A: y = 2.5x
Relationship B: x = 4 y = 12
Compare their unit rates and state which relationship grows faster.
Relationship B has the greater rate and grows faster.
Unit rate for A: 2.5
Unit rate for B: 3
A proportional graph passes through (0,0) and (4, 14).
Find the constant, write the equation, and explain what the point (6, 21) means.
14/4 = 3.5
21/6 = 3.5 When x is 6, y = 21.
k = 3.5
equation: y = 3.5x