Ratios
Purplemath
What is a ratio?
A "ratio" is just a comparison between, or a relating of, two different things. Ratios are used to create proportions by setting two ratios equal to each other and solving for some unknown, and ratios can also be used to find per-unit rates such as how many mile a car can drive "per liter" or how many hours the average student at a given university spends studying "per week".
What is an example of finding a ratio?
As an example of a ratio, 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 rest are women, so:
35 − 15 = 20
...there are twenty women in the group. The language "the ratio of (this) to (that)" means that (this) comes before (that) in the comparison. So, if one were to express "the ratio of men to women", then the ratio, in English words, would be "15 men to 20 women" (or just "15 to 20").
The order of the items in a ratio is very important, and must be respected; whichever word came first in the ratio (when expressed in words), its number must come first in the ratio. If the expression had been "the ratio of women to men", then the in-words expression would have been "20 women to 15 men" (or just "20 to 15").
Expressing the ratio of men to women as "15 to 20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:
odds notation: 15 : 20
fractional notation:
You should be able to recognize all three notations; you will probably be expected to know them, and how to convert between them, on the next test. For example:
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There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese as a ratio with a colon, as a fraction (do not reduce), and in words.
They want "the ratio of ducks to geese", so the number for the ducks comes first (or, for the fractional form, on top). So my answer is:
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Consider the above park, with 16 ducks and 9 geese. Express the ratio of geese to ducks in all three formats.
This time, they want me to give them "the ratio of geese to ducks". I'll be using the exact same numbers but, in this case, the number of geese comes first (or, for the fractional form, on top). So my answer is:
The numbers were the same in each of the two exercises above, but the order in which they were listed differed, varying according to the order in which the elements of the ratio were expressed. In ratios, order is 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, . This fraction reduces to . This means that we can also express the ratio of men to women in the group as being 3 : 4, or "3 to 4".
This points out something important about ratios: the numbers used in the ratio might not be the absolute measured values. The ratio "15 to 20" refers to the absolute numbers of men and women, respectively, in the group of thirty-five people. The simplified or reduced ratio "3 to 4" tells us only that, for every three men, there are four women. The simplified ratio also tells us that, in any representative set of seven people (3 + 4 = 7) from this group, three will be men. In other words, the men comprise of the people in the group. These relationships and reasoning are what we use to solve many word problems:
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In a certain class, the ratio of passing grades to failing grades is 7 to 5. How many of the 36 students failed the course?
The ratio, "7 to 5" (or 7 : 5 or ), tells me that, of every representative group of students, five failed. By "representative group", I mean a group which has the same ratio of students as are in the entire class. I can figure out the size of this group by using the ratio they've given me. The size of the representative group will be the sum of its parts:
7 + 5 = 12
So the representative group has 12 students in it, of which 7 passed and 5 failed. In particular, the fraction of the group that failed is given by dividing the number of flunking students by the total number of students in the representative group. That is:
So of the group flunked and, because this group is representative, of the entire class flunked. This means that I can now find the number of students in the entire class that flunked (this exercise is depressing!) by multiplying the fraction from the representative group by the size of the whole class:
So, of the class of 36 students, the class was not passed by:
15 students
The ratio from a representative group can also be used to provide percentage information.
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In the class above, what percentage of students passed the class? (Round your answer to one decimal place.)
I already know that the representative group contains 12 students, of which 7 passed the class. Converting this to a percentage (by dividing, and then moving the decimal point, as explained here), I get:
7/12 = 0.583333... = 58.3333...%
They want the answer rounded to one decimal place, so my answer is:
58.3% passed
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In the park mentioned previously, the ratio of ducks to geese is 16 to 9. How many of the 300 birds are geese?
The ratio tells me that, of every representative group of 16 + 9 = 25 birds, 9 are geese. That is, of the birds are geese. I can use this fraction from the representative group to find the answer for the entire group:
This is the number they're wanting. In the entire park, there are:
108 geese
Generally, ratio problems will just be a matter of stating ratios or simplifying them. For instance:
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Express the following ratio in simplest form: $10 to $45
This exercise wants me to write the ratio as a reduced fraction. So first I'll form the fraction, and then I'll do the cancelling that leads to "simplest form".
The dollar signs cancelled off, too, because they were the same, top and bottom, in the ratio's fractional form. So my answer is:
This reduced fraction is the ratio's expression in simplest fractional form. The units (being the "dollar" signs) cancelled on the fraction because the units (namely, the "$" symbols) were the same on both values.
When both values in a ratio have the same unit or designator, there should be no unit or designator on the reduced form of the ratio. The units aren't factors, exactly, but they'll cancel in the same manner as do factors.
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Express the following ratio in simplest form: 240 miles to 8 gallons
The terms in this ratio have different units, so they won't cancel off; there will be units on my simplified ratio. My simplification looks like this:
(240 miles) / (8 gallons)
= (30 miles) / (1 gallon)
This particular ratio of units, "(miles)/(gallon)", has its own simplified form; namely, "miles per gallon", which is abbreviated as "mpg". So, in standard English, my answer is:
30 mpg
In contrast to the answer to the previous exercise, this exercise's answer did need to have units on it, since the units on the two parts of the ratio (namely, the "miles" and the "gallons") were not the same, and thus did not cancel each other off. When a ratio ends up with units (or dimensions) on it, the ratio may also be referred to as a "rate". In the case of the exercise above, the rate was the distance covered per unit-volume of fuel.
Conversion factors are simplified ratios, so they might be covered around the same time that you're studying ratios and proportions. For instance:
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What is the length, in feet, of the playing area of an American football field (that is, the length of the field exclusive of the "end" zones)?
I know that the length of an American football field, exclusive of the "end" zones, is 100 yards. I also know that 3 feet are equal to 1 yard. I can set up this equality as a ratio. Because they've given me a measurement in "yards", I'll want the unit of "yards" to cancel off in my multiplication. Because of this, I'll state my ratio (of yards to feet) with the "feet" on top:
(3 feet)/(1 yard)
Now I can multiply the length they've given me by my conversion factor (being the ratio above), and simplify:
[(3 feet)/(1 yard)][100 yards]
= (3 feet)(100) = 300 feet
This value is the number they're wanting, so my answer is:
300 feet
For more on this topic, look at the "Cancelling / Converting Units" lesson.
Ratios are the comparison 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.
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