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Ratios (page 1 of 6) Sections: Ratios, Proportions, Checking proportionality, Solving proportions Proportions are built from ratios. A "ratio" is just a comparison between two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the ratio of men to women is 15 to 20. (Notice that, in the expression "the ratio of men to women", "men" came first. This order is very important, and must be respected: whichever word came first, its number must come first. If the expression had been "the ratio of women to men", then the ratio would have been "20 to 15".) There are two other notations for this "15 to 20" ratio: 15 : 20 .15/20 You should be able to recognize all three notations; you will probably be expected to know them for your test. Given the numbers, you should be able to write down the ratios. For example:
16 : 9, 16/9, 16 to 9
9 : 16, 9/16, 9 to 16 (Note that the numbers were the same in both of the above problems, but the order differed, depending on the order in which the ratio was expressed. In ratios, order can be very important!) Let's return to the 15 men and 20 women in our original group. I had expressed the ratio as a fraction, namely, 15/20. This fraction reduces to 3/4. This means that you can also express the ratio of men to women as 3/4, 3 : 4, or "3 to 4". This points out something important about ratios: the numbers used in the ratio might not be the absolute references. The ratio "15 to 20" refers to the absolute numbers of men and women, respectively. But "3 to 4" just tells you that, for every three men, there are four women. This also tells you that, in any representative set of seven people (3 + 4 = 7) from this group, three will be men. In other words, the men comprise 3/7 of the people in the group. This is what you use to solve many word problems:
The ratio, "7 to 5" (or 7 : 5 or 7/5), tells you that, of every 7 + 5 = 12 students, five failed. That is, 5/12 of the class flunked. Then ( 5/12 )(36) = 15 students failed.
The ratio tells you that, of every 16 + 9 = 25 birds, 9 are geese. That is, 9/25 of the birds are geese. Then there are ( 9/25 )(300) = 108 geese. Generally, though, ratio problems will just be a matter of stating ratios or simplifying them. For instance:
This means that you should write the ratio as a fraction, and you should then reduce the fraction: .10/45 = 2/9. This reduced fraction is the ratio's expression in simplest form. Note that the units "canceled" on the fraction, since the units, "$", were the same on both values. So there is no unit on the answer.
Depending on the text (or instructor), you may be supposed to keep the units on a ratio. In this case, you would have (240 miles)/(8 gallons) = (30 miles)/(1 gallon), or, in more common language, 30 miles per gallon. Properly, this answer should have units on it, since the units, "miles" and "gallons", do not cancel out. Conversion factors are simplified ratios. For instance, suppose you are asked how many feet long an American football field is. You know that it is 100 yards. You would then use the relationship of 3 feet to 1 yard, and multiply by 3 to get 300 feet. (For more on this topic, look at the "Cancelling / Converting Units" lesson.) Ratios are the comparing of one thing to another (miles to gallons, feet to yards, ducks to geese, et cetera). But their true usefulness comes in the setting up and solving of proportions. Top | 1 | 2 | 3 | 4 | 5 | 6 | Return to Index Next >>
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