~ 2.5 ~

Two Equations for Each Relationship

Learning Targets

  • I can find two constants of proportionality for a proportional relationship.

  • I can write two equations representing a proportional relationship described by a table or story.

Notes

If Kiran rode his bike at a constant 10 miles per hour, his distance in miles, d, is proportional to the number of hours, t, that he rode. We can write the equation

d = 10t

With this equation, it is easy to find the distance Kiran rode when we know how long it took because we can just multiply the time by 10.

We can rewrite the equation:

This version of the equation tells us that the amount of time he rode is proportional to the distance he traveled, and the constant of proportionality is 110. That form is easier to use when we know his distance and want to find how long it took because we can just multiply the distance by 110.

When two quantities x and y are in a proportional relationship, we can write the equation

y = kx

and say, “y is proportional to x.” In this case, the number k is the corresponding constant of proportionality. We can also write the equation

and say, “x is proportional to y.” In this case, the number (1/k) is the corresponding constant of proportionality. Each one can be useful depending on the information we have and the quantity we are trying to figure out.

Activities

5.2 Meters and Centimeters

There are 100 centimeters (cm) in every meter (m).


  1. Complete each of the tables.

  2. For each table, find the constant of proportionality.

  3. What is the relationship between these constants of proportionality?

  4. For each table, write an equation for the proportional relationship. Let x represent a length measured in meters and y represent the same length measured in centimeters.

5.3 Filling a Water Cooler

It took Priya 5 minutes to fill a cooler with 8 gallons of water from a faucet that was flowing at a steady rate. Let w be the number of gallons of water in the cooler after t minutes.

  1. Which of the following equations represent the relationship between w and t? Write all that apply.

      • w=1.6t

      • w=0.625t

      • t=1.6w

      • t=0.625w

  2. What does 1.6 tell you about the situation?

  3. What does 0.625 tell you about the situation?

  4. Priya changed the rate at which water flowed through the faucet. Write an equation that represents the relationship of wand t when it takes 3 minutes to fill the cooler with 1 gallon of water.

  5. Was the cooler filling faster before or after Priya changed the rate of water flow? Explain how you know.

Summary

Assignment

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