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Checking Proportionality (page 3 of 6) Sections: Ratios, Proportions, Checking proportionality, Solving proportions There is some terminology related to proportions that you may need to know. In the proportion: a/b = c/d ...the values in the "b" and "c" positions are called the "means" of the proportion, while the values in the "a" and "d" positions are called the "extremes" of the proportion. A basic defining property of a proportion is that the product of the means is equal to the product of the extremes. In other words, given: a/b = c/d ...it is a fact that ad = bc. This fact is occasionally turned into a homework problem, such as:
For these to be proportional (that is, for them to be a true proportion when they are set equal to each other), it must be that the product of the means is equal to the product of the extremes. In other words, they are wanting you to find the product of the means and the product of the extremes, and then see if these products are equal. So check: 140×30 = 4200
So the answer is that they are not proportional. Because of this property of the two products being equal, many students are taught to solve proportions by "cross-multiplying"; in other words, to multiply the denominators in each ratio up to the other side. For instance, in the ducks-and-geese proportion problem above, the first step would be to multiply the 9 up to the right and the G up to the left, like this:
Then you would proceed as normal, dividing off the 16 to solve for the value of G. This process is called "cross-multiplying" because, when you draw the arrows indicating which number is being multiplied by what other number, you get a cross, or "x", across the "equals" sign. "Cross-multiplying" is standard language, in that it is very commonly used, but it is not technically a mathematical term. You might not see it in your book, but you will almost certainly hear it in your class or study group. The other technical point based on terminology is the finding of the "mean proportional" between two numbers. Mean proportionals are a special class of proportions, where the means of the proportion are equal to each other. An example would be: .1/2 = 2/4 Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved ...because the means are both "2", while the extremes are 1 and 4. This tells you that 2 is the "mean proportional" between 1 and 4. Sometimes you are asked to find the mean proportional between two given values. For instance:
I'll let "x" be the number that I'm looking for. Since x will also be both means, I'll set up my proportion with 3 and 12 as the extremes, and x as both means: .3/x = x/12 Now solve for x: .3/x
= x/12
Since I am looking for the mean proportional of 3 and 12, you would figure that I need to take the positive answer, so the mean proportional would be 6. However, considering the fractions, either way would work: .3/–6 = –6/12 .3/6 = 6/12 So, actually, there are two mean proportionals: –6 and 6 (Your book (or instructor) may want you only to consider the positive mean proportional, since the positive answer is between 3 and 12.)
I'll set this up the same way as before, and solve: .–3/x
= x/–12 So there are again two mean proportionals: –6 and 6 (Your book (or instructor) may only want "–6" as an answer.)
Note the difference is signs; this problem is different from the ones that preceded it. Set up the proportion: .–3/x = x/12 Solve for x: .–3/x
= x/12 Since you can't take the square root of a negative number, there is no solution to this proportion.
At first, you might think that this isn't possible, but it is. I'll just set up the proportion using fractions within fractions, and proceed normally: ( 3/2 )/x
= x/( 3/8 )
So the two mean proportionals are –3/4 and 3/4. (Your book (or instructor) may only be looking for " 3/4 ".) << Previous Top | 1 | 2 | 3 | 4 | 5 | 6 | Return to Index Next >>
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