Many "proportion" word problems can be solved using other methods, so they may be familiar to you. For instance, if you've learned about straight-line equations, then you've learned about the slope of a straight line, and how this slope is sometimes referred to as being "rise over run".
But that word "over" gives a hint that, yes, we're talking about a fraction. And this means that "rise over run" can be discussed within the context of proportions.
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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 by] one-quarter inch. So how much must the guttering decline over the thirty-seven foot span? I'll set up the proportion, using "d" to stand for the distance I'm needing to find.
There is a variable in only one part of my proportion, so I can use the shortcut method to solve.
For convenience sake (because my tape measure isn't marked in decimals), I'll convert this answer to mixed-number form:
The lower end should be inches lower than the high end.
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As is always the case with "solving" exercises, we can check our answers by plugging them back into the original problem. In this case, we can verify the size of the "drop" from one end of the house to the other by checking the products of the means and the extremes (that is, by confirming that the cross-multiplications match) of the completed proportion:
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.
As far as I know, biologists and park managers actually use this technique for estimating populations. The idea is that, after allowing enough time (it is hoped) for the tagged fish to circulate throughout the lake, these fish will then be evenly mixed in with the total population. When the researchers catch some fish later, the ratio of tagged fish in the sample to untagged 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:
Because the variable is in only one part of the proportion, I can use the shortcut method to solve.
This tells me that the estimated population is:
about 1,728 fish
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Another type of "proportion" word problem is unit conversion, which looks like this:
To complete this exercise, I will need conversion factors, which are just ratios. (If you're doing this kind of problem, then you should have access — in your textbook or in 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 my answer is:
88 feet per second
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Take note of how I set up the conversion factors for my multiplicate (above) 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 of "hours" in my conversion factor would cancel off with the "hours" in the original "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.
I've got two elements here for my proportion: grams of flour and number of cookies. I got to "grams" first when reading the exercise, so I'll put "grams" on top in my proportion.
Since the relationship is given to me in terms of grams, not kilograms, I'll need to convert Jade's on-hand measure to "1,206 grams, also. I'll use "c" to stand for the number that I'm trying to figure out for "cookies".
Since I have an unknown in only one spot in this proportion, I can use the shortcut method to solve.
Ohhh! Now I see why the instructions said to round to an "appropriate" whole number: Jade can only make whole cookies; the "point-eight" of a cookie will be an undersized niblet that he'll eat before heading to the party.
While normally I'd round this number up to get my whole-number answer, in this case I need to round down; in other words, in this context (namely, of all the cookies being the same size), I have to ignore the fractional portion (that is, the point-eight decimal part) to get the desired answer.
160 cookies
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They've given me an exchange rate, which is, effectively, just another conversion factor, like the "miles per hour" exercise above. So I'll set up my proportion, with Euros on top, and will use e to stand for the number of Euros he'll need.
I'll use the shortcut method to solve:
Rounding to two decimal places, Kumar will be spending:
₹46.72
Other than for the rate-conversion exercise above, we've been able to solve all of the proportions by the shortcut method. You will likely find this to be the case in your homework, also. But it is always possible that you'll get a question where you'll be better off using cross-multiplication instead.
I know that the ratio is 3 : 7, so I'll be using the fraction 3/7 for one side of my proportion. If the lowest pay rate, in thousands of dollars, is L, then the highest is L + 57. My proportion is:
Because there are variables in two of the parts of this proportion, the shortcut method won't be as useful as cross-multiplication to clear all the fractions. So I'll cross-multiply:
Remembering that I dropped the trailing zeroes and am counting by thousands, the above number means that the lowest salary is (rounded to the nearest thousand) approximately $43,000. Then the highest salary, being around $57K more, is approximately $100,000.
lowest: $43,000
highest: $100,000
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