# Lesson 2: Proportional relationships

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## Equations: For Students

More in this series
Lesson 1.
Lesson 3.
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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