# Proportions

A **proportion** is similar to a ratio, except that it indicates a part of a whole, and so the numerator arises from the denominator. For instance, a researcher might say that, for every ten students in the residence hall, five were women. Five over ten (5 / 10) is a proportion. Proportions must have all of the numbers in the same units, and are frequently written as fractions.

What happens when you want to write a proportion, but the numbers are given in different units? Suppose you were asked to write the proportion of 3 cups to 56 ounces (3 cups out of 56 ounces). You must write a proportion of either cups to cups (cups:cups) or ounces to ounces (ounces:ounces), so you will have to convert one of the numbers so that both numbers are expressed in the same unit of measurement. Let’s convert cups to ounces so we can express the ratio as ounces:ounces. Since there are 8 ounces in one cup, 3 cups are equal to 24 ounces:

**3 cups * 8 ounces/cup = 24 ounces**

Now the two numbers 3 cups and 56 ounces can be written as the following ratio:

**24 ounces to 56 ounces,**

**24:56,**

or, now that the unit of measurement is the same, 24:56 can also be written as a proportion:

**24/56**

You may need to solve some problems involving ratios.

If you divided 36 into two parts in the ratio of 1:2 and one part is a and the other is b, you can find the value of a and b:

You know that

**a+b = 36**

And

**a/b = 1/2**

You can use these equations to solve for a and b, or you can use the following simple method:

Find out how many units are in 1 part of the ratio. To do this, divide the total by the number of parts.

**Number of parts: 1 + 2 = 3.**

**Number of units in each part: 36/3 = 12.**

Then, multiply the number of units in each part by the number of parts in each variable.

**a = 1 * 12 = 12**

**b = 2 * 12 = 24**

As an aside . . .

Percentages are so frequently used that we should spend a little time on them here. The powerful thing about percentages is that they all have the same denominator: 100. When two fractions have the same denominator, comparisons can be made very easily.

**0.21 = 21% = 21 / 100**

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