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A R I T H M E T I C

PROPORTIONALITY

Lesson 18  Section 2

 3. What does it mean to say that two quantities are inversely proportional? By whatever ratio one quantity changes, the other will change in the inverse ratio.

This means that if one of the quantities doubles, then the other will become half as large.  For the inverse of the ratio 2 to 1 ("doubles") is the ratio 1 to 2 ("half").  The terms are exchanged.

Example 1.   Let us suppose that the time it takes to do a job is inversely proportional to the number of workers.  The more workers, the shorter the time.

Specifically:  If 6 workers can do a job in 4 days, then how long will it take 12 workers?

Answer.  The number of workers has doubled , going from 6 to 12.  Therefore it will take only half as many days.  It will take only 2 days.

Example 2.   The speed that a car can achieve in 10 seconds is inversely proportional to its weight.  (That is, the more the car weighs, the slower it will be going.)

After 10 seconds, a car that weighs 2400 pounds can achieve a speed of 44 miles per hour.  If the car weighed 1600 pounds, how fast would it be going?

Answer.  What ratio has the new weight to the original weight -- 1600 pounds to 2400 pounds?  1600 is two thirds of 2400:

1600 is to 2400  as  16 is to 24  as  2 is to 3.

(After ignoring the 0's, we see that both 16 and 24 have a common divisor 8.  Lesson 16, Question 7.)

Now, the inverse ratio of 2 to 3 is the ratio 3 to 2.  And since 3 ÷ 2 = 1 R 1, then

3 is one and a half times 2.

The speed therefore will be one and a half times 44:

44 + 22 = 66 miles per hour.

When quantities are inversely proportional, we say that one of them varies inversely as the other.  Thus the speed that a car can achieve in a given time varies inversely as its weight.

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