Proportional Relationships and Unit Rate
What is a Directly Proportional Relation?
In a directly proportional relation, as one amount increases, then another amount increases at the same rate.
This is the symbol for “directly proportional:”
Example 1: You are paid $10 an hour
The amount of money you earn is directly proportional to how many hours you work
When you work for more hours, you get more money
Earnings Hours worked:
2 hours = $20
3 hours = $30
Constant of Proportionality
The constant of proportionality is the value that relates the two amounts. From example 1 above, this would be the amount of money you are paid each hour.
Earnings = 10 x Hours worked
This can be written as: y = kx, where k is the constant of proportionality. In this case, k = 10
Example 1: If y is directly proportional to x, and when x = 4 then y = 16. What is the constant of proportionality?
● y = kx
● 16 = k(4)
● k = 4
Answer: The constant of proportionality (k) is 4, y = 4x
What is an Indirectly Proportional Relation?
In an indirectly proportional relation, as one amount decreases, then another amount increases at the same rate.
When y is inversely proportional to x, this can be expressed as an equation as such: y = k /x
Example 1: 4 people can paint a fence in 3 hours. How long will it take 6 people to paint it?
● As the number of people painting goes up, the painting time goes down
● As the number of people painting does down, the painting time goes up
We can use y = k / x to solve this problem.
● t = k / n
t = number of hours
k = constant of proportionality
n = number of people
“4 people can paint a fence in 3 hours”
● 3 = k / 4
● k = 12
“How long will it take 6 people to paint it?
● t = 12 / 6 = 2
Answer: It will take 6 people 2 hours to paint the fence.
In terms of price, comparing unit rates, or unit prices, is a good way of finding which item is the “best buy” or most bang for your buck.
Example 1: What is the better deal: 2 liters of milk at $3.80 or 1.5 liters of milk at $2.70?
● $3.80 / 2 liters = $1.90 per liter
● $2.70 / 1.5 liters = $1.80 per liter
The deal with the lower unit price is the 1.5 liters of milk at $2.70.
Example 2: What is the better deal: 10 pencils for $4.00 or 6 pencils for $2.70?
● $4.00 / 10 pencils = $0.40 per pencil
● $2.70 / 6 pencils = $0.45 per pencil
The deal with the lower unit price is the 10 pencils for $4.00.