Decimals, Percents, and Ratios
Key Terms
o Decimal
o Irrational number
o Rational number
o Percent
o Ratio
o Proportion
Decimals
As you have probably noticed, most basic calculators (as well
as many more-advanced calculators) do not deal
with fractions. If you divide 3 by 4, for instance, you will not
get a result of ; instead, you will get 0.75. This
representation of a non-integer number is called
a decimal. Converting from a fraction to a decimal is a
simple matter of performing the long division (or, in some
cases, just using a calculator). The example of is shown
below for the purposes of illustration. Note that careful track
must be kept of the decimal point when performing this type
of division.
In some instances, a fraction cannot be written as a decimal
with a finite (limited) number of decimal places. Consider, for
instance, . Let's look at a portion of the long division for
this fraction.
Clearly, the long division will continue indefinitely, adding
additional sixes to the decimal without end. Such repeating
decimals are occasionally written as, in the case of this
example, . The bar indicates that the 6 repeats unendingly.
A decimal is the same as 0.274274274274.
Converting a decimal to a fraction can be somewhat simpler
(as long as the decimal is not repeating, although even
repeating decimals can be converted to fractions-it just
requires a bit more work). Consider the decimal 0.582, for
instance. If we multiply this decimal by 1,000, we get 582:
Concomitantly, we can divide 582 by 1,000 to get 0.582.
But we can also write this division operation as a fraction:
Reducing to lowest terms yields the following result.
Generally, given some decimal, we can convert to a fraction
by writing in the numerator the decimal, less the decimal
point, and by writing in the denominator 1 followed by the
same number of zeroes as the number of decimal places.
Let's consider another example: 0.64. The fraction
corresponding to this decimal would have a numerator of 64
(we eliminate the decimal point) and a denominator of 100.
The operation of dividing 64 by 100 actually takes the
decimal point (which is located next to the 4--64.0) and
moves it to the left two places, leaving 0.64.
This approach works for any fraction with a finite number of
decimal places, even those that include some number to the
left of the decimal point. For instance,
Decimals with an infinite number of decimal places but no
repeating pattern cannot be converted to a fraction with an
integer numerator and integer denominator-these numbers
are called irrational numbers. Any decimal that can be
converted to a fraction with an integer numerator and integer
denominator is called a rational number; repeating
decimals (even though they have an infinite number of
decimal places) and decimals with a finite number of decimal
places are all rational numbers. The following practice
problems provide you with the opportunity to practice
converting fractions to decimals and vice versa.
Practice Problem: Convert each fraction to a decimal.
a. b. c.
Solution: In each case, do the long division of the numerator
divided by the denominator. In part c, note that the decimal
repeats-you need only perform a couple steps in the division
to recognize this repetition.
a. 0.4375 b. 0.6 c.
Practice Problem: Convert the following decimals to fractions.
a. 0.932 b. 0.34 c. 1.52
Solution: Follow the procedure outlined in the discussion
above. Reduce to lowest terms when possible.
a.
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b.
c.
Percents
You probably have heard the word percent used in casual
conversation and perhaps even in some slightly more
mathematical contexts. For instance, someone might say, "I
am giving 100 percent to the job," or "the sales tax rate is 5
percent." In each of these instances, the figure referred to
using the term percent refers to some portion of a whole
amount. For instance, 100 percent refers to an entirety-
someone who is "giving 100 percent" is "giving it his all." The
term percent actually means "per hundred," and it is often
represented using the symbol %. Thus, 100% and 100
percent are the same thing. A percent is thus a fraction
(note how the symbol % looks suspiciously like a fraction!)
where the number before the symbol represents the portion
per hundred. Thus, for example, 50% of the apples is the
same as of the apples. Percents can be any number,
whether positive or negative.
To convert from a percent to a regular number, simply divide
the percent by 100. Depending on the context, a decimal or a
fraction may be the appropriate representation. If a fraction
is best, write the percent in the numerator and 100 in the
denominator, then reduce to lowest terms. If a decimal is
best, simply move the decimal point of the percent to the left
by two places (the same as dividing by 100). To convert from
a regular number to a percent, simply multiply by 100%.
Thus, for instance, 0.25 is the same as 25%, and 98% is the
same as 0.98 and .
Practice Problem: Convert each fraction to a percent.
a. b. c.
Solution: To convert to a percent, multiply by 100%.
Percents can be written as fractions, but they are typically
written as decimals. In part c, you can do the long division
and write the percent using the bar notation discussed
above.
a. b.
c.
Practice Problem: Convert each percent to a fraction in
lowest terms.
a. 80% b. 35% c. 12.5%
Solution: In each case, divide by 100% and reduce to lowest
terms. To get rid of the decimal in part c, simply multiply
both numerator and denominator by 10 (recall that this is the
same as multiplying the fraction by 1).
a. b.
c.
Percents and Ratios
A percent is not (necessarily) a strict count of, for example,
some number of objects. That is, 50% of a basket of apples
is not necessarily equal to 50 apples. The expression 50%
means half the apples-thus, if the basket contains 24 apples,
50% of the apples is 12. Thus, a percent is actually
a ratio (or proportion), which is a relationship between two
quantities. In the case of our example, we are considering 12
out of 24 apples. For instance, we might be looking at the
number of apples that are bad or the number that exceed a
certain size. In either case, we are considering the
relationship between one number (12 apples of a certain size
or quality, for instance) and another number (24 apples-the
entire lot in the basket). Ratios can be expressed in words
(12 out of 24) or using a colon (12:24) or as a fraction ( ).
Because the representation as a fraction is equivalent to the
other representations, we can reduce the ratio to lowest
terms (in other words, ). Thus, 12 out of 24 is the same
as 1 out of 2, just as 12:24 is the same as 1:2.
A percent is therefore a specific kind of ratio where the
number to which we are comparing is 100. Thus, the ratio
12:24 (or 1:2) is the same as 50%. Simply remember that
the first number in a ratio (given in any of the above-
mentioned forms) corresponds to the numerator of a
fraction, and the second number corresponds to the
denominator. Using the rules that we have studied so far,
you should then be able to convert between ratios, fractions,
percents, and decimals.
Practice Problem: Write each number or percent as a ratio
using colon (:) notation (in lowest terms).
a. b. 48% c. 1.75
Solution: Remember that a ratio (using colon notation) is the
same as a fraction, where the first number is the numerator
and the second the denominator.
a. b.
c.
Using Ratios (Proportions)
The ability to use ratios (or proportions) is a crucial skill in
algebra. Proportions allow us to talk about relative amounts;
for instance, we might talk about performance on a test as a
proportion of questions answered correctly. If we know that a
student correctly answers 95% of the questions on a test, we
have a good indication of her performance regardless of
whether we know how many questions were on the test. If
we do know this number, though, we can also determine how
many she answered correctly. Let's say the test had 200
questions. We know, then, that the quotient of the number
answered correctly divided by 200 is equal to 95% (or 0.95,
or ).
The expression above uses a question mark (?) to represent
the unknown quantity, but we can also use another symbol
corresponding to an unknown quantity. For instance, let's
use x. The letter x is simply a placeholder for a value that is
unknown or that can change.
We now want to find x. One conceptually simple approach is
to convert the fraction on the left into a fraction with a
denominator of 200-the numerators of the fractions would
then be required to be equal.
Thus, we see that x = 190. In other words, if the student
scores 95% correct on a test with 200 questions, then she
has answered 190 questions correctly. Note also that 190 is
simply the product of the percent and the number of
questions.
(Be careful when performing operations using percents. The
best approach is to always convert a percent to a fraction or
decimal before performing the operation.) The above
discussion and example provides a glimpse at proportions
and how to glean information from them.
Practice Problem: Find y to satisfy the expression below.
Solution: This expression equates two proportions. We can
write the known proportion as an equivalent fraction with a
denominator of 16; this allows us to easily find y.
Thus, y = 8.
Practice Problem: An apple picker has learned from
experience that 10% of all apples picked from a certain
orchard have a worm in them. If she has a basket containing
40 apples, how many can she expect to contain a worm?
Solution: This word problem gives us the opportunity to
apply what we know about percents and proportions. First,
let's write 10% as a proportion.
Now, we know that the apple picker has 40 apples in her
basket, and we want to know how many she should expect to
contain a worm. Let's say that the number of apples with a
worm is a (remember, the use of a letter like a is just for the
purposes of holding the place of the unknown number). The
ratio of a to 40 ( ) must be the same as the ratio of apples
with worms to the total number of apples ( ). Let's
therefore calculate a.
