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# Ratios and Proportions

Knowing how to work with ratios and proportions is very handy in chemistry classes, especially when working with different units of measurement. Let's start with some dictionary definitions:

Ratio: The relative size of two quantities expressed as the quotient of one divided by the other; the ratio of a to b is written as a:b or a/b.
Proportion: An equality between two ratios.

So what are these things really? Consider the following situation.

In 1995, 78 women were enrolled in chemistry at a certain high school while 162 men were enrolled. What was the ratio of women to men? Men to women?

Let's answer the questions using the definition of ratio. Filling in what we know:

women : men is 78:162 or 78/162
men : women is 162:78 or 162/78

We could have reduced the fractions (cancelling out a factor of 6) or used our calculators to get a decimal equivalent for these fractions using the divide key:

women : men is 78 ÷ 162 or 13 ÷ 27 or 0.481481481
men : women is 162 ÷ 78 or 27 ÷ 13 or 2.07692308

By writing the answer in these ways we have lost information, namely the specific number of men and women. Be careful! When given a ratio such as 13:27, the fractions may have been reduced so the original quantities could have been larger. This brings us to the idea of proportion.

Ratios are said to be in proportion when their corresponding fractions are equal. What we really did above was notice that the fraction 78/162 was equal to the fraction 13/27 - because we could divide both numbers in the first ratio by six (6) to get the second ratio - so the ratios are equal as well, i.e.,

78/162 = 13/27,
or using the colon form
78 : 162 = 13 : 27.

These two (2) equalities are examples of proportions (equal ratios); it is as simple as that. How are proportions used? Let's add another question to our problem:

In 1996 the number of men enrolled was 193 while the ratio of women to men enrolled in chemistry stayed the about same as in 1995. How many women were enrolled in chemistry in 1996?

To answer this, we build a proportion equating the ratio of women to men in the two (2) years:

78 : 162 = ?? : 193

These are easiest to solve when written in fraction form. Notice that we've used x for the unknown number of women since it is common in algebra to use the letter x as the variable.

 78 162
=
 x 193

We can then use the "cross-multiply" technique:

78 * 193 = 162 x
multiply on our calculator:
15054 = 162 x
and then divide on our calculator:
92.9259259 = x

So the number of women enrolled in 1996 was 92.9259259??? Use your common sense! There are no fractions of women walking around, so we will report 93 (rounding 92.9259259 correctly). It is probably wise to say "There are about 93 women enrolled in chemistry in 1996" since we were told the proportions were about the same. The issue of when to round will come up in chemistry, too. Often the objects being counted will be atoms or atomic particles like protons, neutrons and electrons; only whole number answers make sense in this case. Sometimes, however, we're measuring amounts (such as masses) which can be fractional. Then you should use the rules for correct rounding and significant figures reviewed in Session 1.

## Try It Out

1. The table below is printed on the side of a box of pancake mix:

pancakes amount of mix amount of water
6 1 cup 3/4 cup
12 2 cups 1 1/2 cups
18 3 cups 2 1/4 cups

(a)
What is the ratio of mix to water in each case?
(b)
What quantities of mix and water should be used if we want to make 15 pancakes?
(c)
What quantities of mix and water should be used if we want to make 100 pancakes?
2. Ammonia is a compound consisting of a 1 : 3 ratio of nitrogen and hydrogen atoms.

(a)
If a sample of ammonia contains 1563 nitrogen atoms, how many atoms of hydrogen are present?
(b)
If a sample of ammonia contains 1425 hydrogen atoms, how many nitrogen atoms are present?
3. Methane is a compound consisting of a 1 : 4 ratio of carbon and hydrogen atoms.

(a)
If a sample of methane contains 1565 atoms, how many carbon and hydrogen atoms are present?
(b)
Can a sample of methane contain a total of 1566 atoms?
4. The mass to charge ratio of an isotope of a particular element was determined to be 1.97x10-7 kilograms per coulomb. If the charge on the isotope is 1.614x10-19 coulombs, what is the mass of the isotope?

[Numbers and their Properties] [Numbers in Science] [Ratios and Proportions]
[Units, Dimensions, and Conversions] [Percents] [Simple Statistics] [Logarithms]

Developed by Shodor
in cooperation with the Department of Chemistry,
The University of North Carolina at Chapel Hill