Question

wednesday homework
topic: constant of proportionality (unit rate)
directions: find the constant of proportionality (k). state what the unit rate means.

1. 40 packs of gum cost $20

k = _ meaning: _

1. a runner travels 24 miles in 3 hours

k = _ meaning: _

1. 150 photos are uploaded in 5 minutes

k = _ meaning: _

1. 42 notebooks cost $28

k = _ meaning: _

Explanation:

Step1: Recall the formula for the constant of proportionality

The formula for the constant of proportionality \(k\) in a relationship \(y = kx\) (where \(y\) and \(x\) are related quantities) is \(k=\frac{y}{x}\).

Step2: Solve for \(k\) in each case

Case 1: 40 packs of gum cost $20

Let \(y\) be the cost (\(y = 20\)) and \(x\) be the number of packs (\(x = 40\)). Then \(k=\frac{20}{40}=0.5\). The unit rate \(k = 0.5\) means that for each pack of gum (\(x = 1\)), the cost \(y\) is \(0.5\) dollars.

Case 2: A runner travels 24 miles in 3 hours

Let \(y\) be the distance (\(y = 24\)) and \(x\) be the time (\(x = 3\)). Then \(k=\frac{24}{3}=8\). The unit rate \(k = 8\) means that in one hour (\(x = 1\)), the runner travels \(8\) miles.

Case 3: 150 photos are uploaded in 5 minutes

Let \(y\) be the number of photos (\(y = 150\)) and \(x\) be the time (\(x = 5\)). Then \(k=\frac{150}{5}=30\). The unit rate \(k = 30\) means that in one minute (\(x = 1\)), \(30\) photos are uploaded.

Case 4: 42 notebooks cost $28

Let \(y\) be the cost (\(y = 28\)) and \(x\) be the number of notebooks (\(x = 42\)). Then \(k=\frac{28}{42}=\frac{2}{3}\). The unit rate \(k=\frac{2}{3}\) means that for each notebook (\(x = 1\)), the cost \(y\) is \(\frac{2}{3}\) dollars.

Answer:

1. \(k = 0.5\), meaning: Each pack of gum costs \(0.5\) dollars.
2. \(k = 8\), meaning: The runner travels \(8\) miles per hour.
3. \(k = 30\), meaning: \(30\) photos are uploaded per minute.
4. \(k=\frac{2}{3}\), meaning: Each notebook costs \(\frac{2}{3}\) dollars.