## 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

• etc....

### 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?

Inverse proportion:

● 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.

### Unit Rates

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.