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Lesson 2: Proportional relationships

A tray of cupcakes.

Courtesy of Wikimedia Commons

By Michael Avidon, math editor

Equations: For Students

More in this series
Lesson 1.
Lesson 3.
If you want to help your child,
use our version for guardians.

Performance Expectations (CCSS)

This lesson covers the following parts of 7.RP.A.2:

Recognize and represent proportional relationships between quantities.

  1. Identify the constant of proportionality (unit rate) in … equations … of proportional relationships.
  2. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

In the previous lesson, you looked at this proportional relationship:

Cupcakes
Number Cost
1 $3.50
2 $7.00
3 $10.50

You saw that the ratio was a constant $3.50 per cupcake. If C is the cost of the cupcakes, and n is the number of cupcakes, then this ratio is represented by Cn . So
Cn = 3.50

Multiplying both sides by n yields the equation C = 3.5n. This represents the proportional relationship. For example, if n = 3, then C = 3.5(3) = 10.5. That is, the cost of ___ cupcakes is $_________, as it says in the third row of the table.

In general, if there is a proportional relationship between a quantity y and a quantity x with a constant of proportionality k, then
yx = k, which implies y = kx

A proportional relationship between a quantity y and a quantity x that has a constant of proportionality k is represented by the equation y = kx.

If an equation in a different form can be rewritten as above, then it is a proportional relationship. If it cannot be rewritten as above, then it is not a proportional relationship.

We can take any table or verbal description of a proportional relationship and turn it into an equation. We just need to first determine the constant of proportionality or unit rate.

Example 1

Brick machine
Time
(hours)
Number
of bricks
2 1100
5 2750
8 4400

You saw in the previous lesson that the output of this brick-making machine follows a proportional relationship. Write an equation to represent the relationship. (To print this lesson out, click on the printable version.)

Solution:

Let N = the number of bricks and t = the time in hours. The unit rate is _____ bricks per hour. The equation is N = 550t.

Example 2

A 10-pound wheel of baby gouda cheese sells for $64. Write an equation relating the cost to the weight of the cheese.

Solution:

The cheese sells for $_________ per pound. Let C = the cost of the cheese in dollars and w = the weight of the cheese in pounds. Then the equation is C = 6.4w.

Remember that a proportional relationship works “both ways.” So in the above example, not only is C proportional to w, but w is also proportional to C. You can take the equation above and divide both sides by 6.4 to write the relationship the other way:

164C = w → w = 0.15625C

Example 3

The equation K = 0.4536P represents the relationship between pounds, P, and kilograms, K. Identify the constant of proportionality and give a verbal description of the relationship. Also, give a verbal description of the reverse relationship.

Solution:

The constant of proportionality is 0.4536. This means for every 1 pound there is 0.4536 kilogram. To reverse the relationship, you need the reciprocal of the constant. In this case, it is about 2.2. So for every 1 kilogram, there are about 2.2 pounds.

Example 4

x y
3 18
5 30
7 42

Represent y = 6x as a table, with rows for x = 3, 5 and y = 42.
Solution:

When x = 3, y = 6(3) = _____.

When x = 5, y = 6(____) = ______.

When y = 42, 42 = 6x → x = 7.

Exercises for lesson 2

Represent each proportional relationship with an equation.
1.

Area, A
(sq. ft.)
Cost, C
90 $81
180 $162
360 $324

2.

Production
Time, t (hours) Widgets, w
3 1005
5 1675
7 2345

3.

T-shirts
Number, n Cost, C
2 $23
4 $46
5 $57.50

4. An artist buys unprimed canvas in rolls. Each roll has 90 square feet of canvas and costs $54. Let C = the cost of the canvas in dollars and A = the area of the canvas in square feet.
5. A 3-pound package of ground beef sells for $12. Let B = the cost of the ground beef in dollars and w = the weight of the ground beef in pounds.
6. A mason is building a brick wall. So far, he has laid 5 rows of brick (and cement) that reaches a height of 45 cm. Let n = the number of rows of brick and h = the height of the wall in centimeters.

For each equation, (a) represent the proportional relationship by a table with input values 1, 2, and 3, and (b) write the equation for the reverse relationship.
7. y = 4x
8. y = 0.2x
9. y = x3
10. Does each equation represent a proportional relationship?
(a) y = x + 5              (b) y = x2                  (c) y = 2(x – 5) + 10
11. The equation y = 2.54x represents the relationship between inches, x, and centimeters, y. Give a verbal description of the relationship.
12. The equation y = 33.8x represents the relationship between liters, x, and fluid ounces, y. Give a verbal description of the relationship.

13. Challenge Problem:
While the constant of proportionality is usually positive in real-world problems, it could be negative. Let h = the number of hours after midnight, and T = the temperature in degrees Celsius. Explain what the relationship means in real-world terms at midnight and beyond.

NEXT LESSON: Head to Lesson 3 or look back at Lesson 1

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