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# Percents

Most students learn to use percents in middle school, but many people still have trouble interpreting information involving percents even as adults. Just what is a percent (%)?

The word percent comes from the Latin per centum meaning "out of one hundred," so we can think of 22% as "22 out of 100." Thus a percent is a symbol representing a ratio of the part -- in this case 22 -- to the whole -- 100. A percent must be changed to a number (fraction or decimal) before we can compute with it. Hence 22% is the ratio 22 : 100, which gives:

 22 100
= .22

Other examples:

percent notation ratio notation number notation
30% 30 : 100 30 / 100 = 0.3
8% 8 : 100 8 / 100 = 0.08
63.7% 63.7 : 100 63.7 / 100 = 0.637
100% 100 : 100 100 / 100 = 1
212% 212 : 100 212 / 100 = 2.12
5 3/4% 5 3/4 : 100 or 5.75 : 100 5.75 / 100 = 0.0575

We will most often use the decimal form (in bold above) because it is easy to find on a calculator by dividing by 100. Let's make some notes from these examples.

• The mechanics of changing a percent to a decimal number involves moving the decimal point two places to the left. The percent symbol can help us remember this: • When we change percents smaller than 10 percent to numbers, we need to place holding 0s -- as with 8% and 5 3/4%.

• 100% is 1, reinforcing the idea that x% describes x's part of "the whole."

• Percents representing more than 1 are possible (212%) -- they are used to talk about something growing really large rather than as "part out of the whole."

Before looking at how percents come up in science, we need to be able to answer questions like those below. Click on the question to look at how to find the answer. We give several tips on using percents on these pages, so you might want to look even if you got the answer right!

What is 20% of 35? It's 7.
11 is what percent of 20? It's 55%.
6.5% of what number is 80.02? It's 1231.076923 (or 1200 to two sig figs).
What is 10 after being increased by 15% ? It's 11.5.
What is 201.1 after being decreased by 25% ? It's 150.825.

Percents come up several times in chemistry. Remember that the key is to change a percent to a number before calculating with it and, if the answer is to be stated as a percent, convert the number to a percent before giving the answer.

So what else do you need to know to be ready for chemistry?

• When measuring a sample for its constituent parts, the amounts of each part are often stated as %s -- you'll see this in percent abundance of isotopes and in percent composition of compounds. Don't forget: part / whole. For example:

A sample of lead was tested in a mass spectrometer, and four isotopes were found along with their % abundances: 204 at 1.4%, 206 at 24.1%, 207 at 22.1% and 208 at 52.4%.

How do we read these? 1.4% of the sample was isotope 204, 24.1% of the sample was 206, etc. Notice that the percents add up to 100. All the parts together should total up to the whole (within some error).

• When conducting an experiment to synthesize a chemical compound, you'll compare the amount you should get (according to the theory of how chemicals bind together) to the actual amount you did get from your experiment -- percent yield = experiment / theory. For example:

Suppose we know that if we take formic acid and geraniol, we can make a synthetic rose perfume. If we start with 1000.0 g of geraniol added to formic acid, the theory of chemical reactions (stoichiometry!) can be used to calculate a theoretical yield of 1182.2 g of the rose essence. Our experiment actually produces 871.2 g. What is the percent yield?

experiment / theory = 871.2 / 1182.2 = 0.736931145 = 73.69% to four sig figs.

• When working with measurements there is often some associated error -- usually measured as percent error:

% error = true - approximate true For example, suppose we test a new thermometer for accuracy by using it to find the boiling point of pure water. The boiling point of pure water is 212°F but the thermometer measures 212.9°F. What is the % error of the reading?

% error = 212 - 212.9 212 = -.9 212 = .004245283

which is 0.425% to three sig figs. Note the absolute values; that's why the - sign was dropped.

## Try It Out

1. Find the missing quantity in each of the following:
(a)
What is 10% of 672.3 kg?
(b)
42 is what percent of 132.3 g?
(c)
18.7 g is .5% of what amount?
2. In 80.042 g (1 mole) of ammonium nitrate, there are 28.014 g of nitrogen, 47.997 g of oxygen and the rest is hydrogen.

(a)
What percent of ammonium nitrate is nitrogen?
(b)
What percent of ammonium nitrate is oxygen?
(c)
What percent of ammonium nitrate is hydrogen?
3. We performed an experiment in which the theoretical yield of the desired chemical is 143.2 L. If the percent yield of the actual experiment is 62%, how much of the chemical did we really end up with?
4. A sample thought to be caffeine is tested and the resulting composition was 53% carbon, 4% hydrogen, 30% nitrogen and 13% oxygen. We know from theory that one mole of caffeine contains 96.08 g carbon, 10.08 g hydrogen, 56.04 g nitrogen and 32.00 g oxygen. Was the sample really caffeine?

[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