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Lesson 18

# PROPORTIONALITY

In this Lesson, we will answer the following:

1. What does it mean to say that two quantities are directly proportional (or simply, proportional)?
2. How do we solve problems of proportionality?

Section 2

3. What does it mean to say that two quantities are inversely proportional?

 1. What does it mean to say that two quantities are directly proportional (or simply, proportional)? By whatever ratio one quantity changes, the other changes in the same ratio.

For example, let us say that the distance you travel is proportional to the time.  This means that if you travel twice as long, you will go twice as far.  If you travel three times as long, you will go three times as far.  While if you travel half as long, you will go half as far.

By whatever ratio the time changes, the distance will change proportionally, that is, in the same ratio.

 2. How do we solve problems of proportionality? Form ratios between the things of the same kind.

To begin with, we can only form ratios between things of the same kind:  length to length, time to time, dollars to dollars, and so on.  A ratio between things of different kinds ("This amount of money is half of this amount of time") makes no sense.

When we relate things of different kinds, as in so many "dollars per hour," that is not called a ratio but a rate.

Example 1.   In 4 hours, you can travel 110 miles.  How far can you travel in 8 hours?

Answer.  Since you travel twice as many hours -- 8 hours are twice as many as 4 hours -- then you will travel twice as many miles.

2 × 110 miles = 220 miles.

Proportionally, upon forming ratios between things of the same kind:

4 hours are to 8 hours  as  110 miles are to 220 miles.

Example 2.   Maria can earn \$70 in 6 hours.  How much will she earn in 18 hours? in 3 hours? in 9 hours?

Solution.  Earnings are directly proportional to time.  If Maria works three times as long -- 18 hours -- then she will earn three times as much.  She will earn 3 × \$70 = \$210.

As for 3 hours, they are half of 6 hours: therefore she will earn half of \$70; she will earn \$35.   (Lesson 15, Question 3.)

Next, 9 hours.  What ratio have 9 hours to 6 hours?

9 hours are 3 hours more than 6 hours.  That is, 9 hours are one and a half times 6 hours.  (Lesson 17, Question 6.)  Therefore she will earn one and a half times \$70.  She will earn

\$70 + \$35 = \$105.

When two quantities are directly proportional, we say that one of them varies directly as the other.  In this example, wages vary directly as time.

Example 3.   Which is a better value: 12 ounces for \$5.00, or 48 ounces for \$22.00?

Answer.  48 ounces are four times 12 ounces.  Therefore we should expect the price to be four times more.  But four times \$5 is \$20 -- which is less than \$22.  Therefore 12 ounces for \$5 is a better value.

Example 4.   Which is a better value: 15 ounces for \$9.00, or 20 ounces for \$11.00?

Answer.  20 ounces are 5 ounces more than 15 ounces.  That is, 20 ounces are one third more.  Now, one third more than \$9.00 is

\$9.00 + \$3.00 = \$12.00

Therefore, 20 ounces for \$11.00 is a better value.  You save \$1.00.

At this point, please "turn" the page and do some Problems.

or

Continue on to the next Section.

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