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Solving Proportions: Examples (page 5 of 7)

Sections: Ratios, Proportions, Checking proportionality, Solving proportions

• You are installing rain gutters across the back of your house. The directions say that the gutters should decline ¹/₄ inch for every four feet. The gutters will be spanning thirty-seven feet. How much lower than the starting point (that is, the high end) should the low end be?

Rain gutters have to be slightly sloped so the rainwater will drain toward and then down the downspout. As I go from the high end of the guttering to the low end, for every four-foot length that I go sideways, the gutters should decline [be lower lower by] one-quarter inch. So how much must the guttering decline over the thirty-seven foot span? I'll set up the proportion.




9.25 = 4d
2.3125 = d

For convenience sake (because my tape measure isn't marked in decimals), I'll convert this answer to fractional form: Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

The lower end should be 2 ⁵/₁₆ inches lower than the high end.

As is always the case with "solving" exercises, you can check your answer by plugging it back into the original problem. In this case, you can verify the size of the "drop" from one end of the house to the other by checking the means and the extremes. Converting the "one-fourth" to "0.25", we get:

(0.25)(37) = 9.25
(4)(2.3125) = 9.25

Since the values match, then the proportionality must have been solved correctly, and the solution must be right.

• Biologists need to know roughly how many fish live in a certain lake, but they don't want to stress or otherwise harm the fish by draining or dragnetting the lake. Instead, they let down small nets in a few different spots around the lake, catching, tagging, and releasing 96 fish. A week later, after the tagged fish have had a chance to mix thoroughly with the general population, the biologists come back and let down their nets again. They catch 72 fish, of which 4 are tagged. Assuming that the catch is representative, how many fish live in the lake?

As far as I know, this is a technique that biologists and park managers actually use. The idea is that, after allowing the tagged fish to circulate, they are evenly mixed in with the total population. When the researchers catch some fish later, the ratio of tagged fish in the sample is representative of the ratio of the 96 fish that they tagged with the total population.

I'll use " f " to stand for the total number of fish in the lake, and set up my ratios with the numbers of "tagged" fish on top. Then I'll set up and solve the proportion:





f  × 4 = 72 × 96 
4f = 6912
f = 1728

There are about 1728 fish in the lake.

A related type of problem is unit conversion, which looks like this:

• How many feet per second are equivalent to 60 mph?

For this, I will need conversion factors, which are just ratios. If you're doing this kind of problem, then you should have access (in your text or a handout, for instance) to basic conversion factors. If not, then your instructor is probably expecting that you have these factors memorized. I'll set everything up in a long multiplication so that the units cancel:

Then the answer is 88 feet per second.

Note how I set up the conversion factors in not-necessarily-standard ways. For instance, one usually says "sixty minutes in an hour", not "one hour in sixty minutes". So why did I enter the hour-minute conversion factor (in the second line of my computations above) as "one hour per sixty minutes"? Because doing so lined up the fractions so that the unit "hour" would cancel off with the "hours" in "60 miles per hour". This cancelling-units thing is an important technique, and you should review it further if you are not comfortable with it.

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