Section 2.5: Ratios, Proportions, and Percent
§1 Ratio
A ratio is a comparison of two quantities using a quotient. We can write a ratio three different ways. The ratio of
a
the number a to the number b can be written as a to b, a : b, or . Only the last case does the quotient need to
b
be in lowest terms. Remember also, the units must be the same. If one quantity is given in hours while the other
quantity is given in minutes, we need to convert one of the units so that it matches the other.
The most common example of a ratio is the number of girls in a class to the number of boys. For example, say a
certain math class has 18 girls and 14 boys. Then ratio of girls to boys can be written as 18 to 14, or as 18:14, or
as 18/14. Remember, however, that the fraction needs to be in lowest terms, so it would really be 9:7.
Another common example is in what we call unit pricing. For example, say a 10 ounce cup of coffee costs $2.50
while a 16 ounce cup of coffee costs $3.75. Which one is the better buy? Really, we mean which size has the
lowest unit price. In these types of problems, the dollar amount always goes in numerator, while the size always
goes in the denominator. So we are figuring out how much the coffee will cost per ounce. It turns out that the
unit price of the 10 ounce cup of coffee is then
$2.50
, which equals $0.25 per ounce while the unit price of
10 ounce
the 16 ounce cup of coffee is
$3.75
which rounds out to $0.23 per ounce. Hence the 16 ounce cup of coffee
16 ounce
is the better buy, because it costs less per ounce.
PRACTICE
1) Find the unit price of each and determine which one is the better buy: 36 ounce carton of juice costs $3.96 or
a 48 ounce carton of juice costs $4.80.
§2 Proportion
A proportion says that two ratios are equal. For example, we can see that the ratio of
3
is equal to the ratio of
5
15
3 15
a c
 . The values of a and d are called the extremes, and the values
, or 
. Generally, we write this as
25
5 25
b d
of b and c are called the means. We can read this as ‘a is to c as b is to d.’ Note that if we cross-multiply, then
8 12
 is true? You
ad  bc . This is a very useful property. For example, how can you show if the proportion
14 21
can try to simplify both ratios to simplest terms, or just cross-multiply. If you get the same product on both sides,
then you know that the proportion is true!
We can also use this property to solve proportions when one of the values is unknown. For example, say we
x6 2
 . We can solve for x by cross-multiplying. We end up with 5  x  6   4 . Solve this for x
want to solve
2
5
26
to get that 5x  30  4 , or x   .
5
PRACTICE
2) Determine if the following proportions are true: a)
3) Solve the following: a)
x  4 x  10

6
8
b)
5
8

35 56
b)
13 1

6 18
c)
4 10

12 40
3x  2 4  x

5
3
§3 Word Problems
We can solve certain word problems by setting up a proportion and solving it. The most common examples are
problems dealing with recipes, maps, and sides of a triangle. You need to remember that when setting up the
a c
proportion, the units must match. Remember the setup  . So the units of a must match the units of c, and
b d
the units of b must match the units of d.
For example, if 6 pens cost $1.32, how much will 8 pens cost? We see that we can set up the proportion as
$1.32 x
6
8
 - it does not matter. Note
 , where x represents the cost of 8 pens. You can also set it up as
6
8
$1.32 x
that when we cross-multiply we end up the same extremes and means. Hence we end up with 6 x  $10.56, or
$1.76. What is the unit price of one pen? We see that one pen costs $0.22.
PRACTICE
4) On a wall map, the distance between Los Angeles and Las Vegas is 5 inches while in actuality the cities are 300
miles apart. On the same wall map, if the distance between San Francisco and Seattle is 14 inches, how far apart
are the cities in real life?
5) A recipe for 24 pancakes calls for 1/3 cup of milk. How many pancakes can be made with 6 cups of milk?
§4 Percent Problems
A percent is actually just a ratio where the second number is always 100. For example, 50% represents the ratio
50
50 to 100, or as a fraction
or as a decimal, 0.50. Similarly, we see that 250% represents the ratio 250 to 100,
100
250
or as a fraction
or 2.5. If the decimal is greater than 1, then the percent is greater than 100%.
100
To convert from a decimal to a percent, we move the decimal two spots to the right and add the percent sign to
the number.
To convert from a percent to a decimal, we move the decimal two spots to the left and drop the percent sign
from the number.
When solving percent problems, we ALWAYS use the decimal form of the percent! We can set up the problem as
amount = percent x base, or a = pb.
Try the following: what is 6% of 80? Here, the percent is given as 80%, so that’s easy. You need to determine if
80 represents the amount or the base. The key word here is ‘of’. Out of 80 means that 80 represents the base.
Hence we can set up our problem as a  0.06  80 , or a  4.8 .
When in doubt, try to remember how exams are scored. Let’ say you received 17 out of 20 on a quiz. As a ratio
17
. How can we find this percent? Well, the amount is 17, and it is out of 20. So we need to solve
this is
20
17  p  20 . We end up with p  0.85. Remember, this is the decimal form! The percent then becomes 85%.
PRACTICE
6) 16% of what number is 12?
7) A shirt that normally costs $30 is now on sale for 20% off. What is the discount amount and what is the sale
price?
8) A TV that normally costs $800 is now on sale for $$680. What is the percent discount?