Section 2.4 – Ratios, Rates and Proportions
§1 Ratios
A ratio is a comparison between two quantities. A ratio is always written in lowest terms. For example, in a
certain room there are 20 girls and 16 boys. What is the ratio of girls to boys?
A ratio can be written three ways. If a and b represent the two quantities, we can write the ratio as
a to b
or
a:b
or a/b
So for the example above, the ratio of girls to boys would be 20 to 16. Remember, however, that the ratio must
be in simplest form. So the final answer is 5 to 4. Note that we can write this as 5:4 or 5/4. This simply means
that for every 5 girls, there are 4 boys.
If I asked for the ratio of boys the girls, what would the answer be?
PRACTICE
1) A certain school has 180 students and 12 teachers. What is the ratio of students to teachers? What is the ratio
of teachers to students?
2) A football team has 60 players and 8 coaches. What is the ratio of players to coaches?
§2 Rates
A rate is also a ratio, but there are units involved. For example, we are all familiar with miles per hour. We
usually express the answer as a unit rate, which means the denominator is equal to 1. For example, what is your
rate if you can travel from Las Vegas to Los Angeles (approximately 280 miles) in 4 hours? We need to write this
as a unit rate, which means how many miles can you drive per hour. As a rate, this is
form, the answer is 70
280 miles
. In reduced
4 hours
miles
, or 70 miles per hour.
hour
PRACTICE
3) A racecar driver can finish a 500 mile race in 3.5 hours. What is the average speed? (round to nearest tenth)
4) A world-class marathon runner can run 26 miles in 2 hours. What is the average speed?
§2 Unit Pricing
The unit price of consumer items is the price per desired unit of measure. We typically use unit pricing to
determine which one is the better deal. For example, a grande (12 ounce) cup of coffee is $1.79 while a venti (16
ounce) of coffee is $1.99. Which one is the better deal? Basically, we want to find how much we are paying per
ounce.
Unit prices are usually rounded to the nearest thousandth. If the unit price is less than $1.00, we round to the
nearest tenth of a cent. The unit price of the grande is then
unit price is
$1.79
= $0.149 per ounce. For the venti, the
12 ounces
$1.99
= $0.124 per ounce . Hence we can see that the venti coffee is a better buy, because it
16 ounces
costs less per ounce. Note that the dollar amount always goes in the numerator.
PRACTICE
5) A 16 ounce carton of milk sells for $4.29 and a 20 ounce carton of milk sells for $5.39. Determine the unit
price of each and state which one is the better buy.
6) A 5-pack of gum sells for $2.99 and a 3-pack of gum sells for $2.19. Determine the unit price of each and state
which one is the better buy.
§2 Proportions
A proportion is a statement that two ratios or rates are equal. For example,
3 12
. Similarly,
=
5 20
300 miles 60 miles
.
=
15 gallons 3 gallons
If one of the values is not given, we can solve for the missing variable by cross-multiplying. To cross-multiply, we
a
b
use the property
that if
=
x 24
c
.
, then ad bc . For example, solve for x if =
=
4 32
d
We cross-multiply to get 32 ⋅ x = 4 ⋅ 24 . We get 32 x = 96, hence x = 3.
PRACTICE
7) Solve the following:
6 18
=
x 12
8) Solve the following:
25 6
=
10 x
§3 Similar Triangles
Similar triangles are triangles whose angles have the same measure. Their sides have different lengths. So they
have the same shape, but not the same size. It turns out that in similar triangles, the lengths of their
corresponding sides are similar to each other! It’s a good idea to get the orientation of the two triangles to be
the same, i.e. they should be ‘facing’ the same direction. That way you can tell what the corresponding sides are .
For example, find the 3 pairs of corresponding sides of the following similar triangles.
9)
10)
Once we know what the corresponding sides are, we can set up a proportion.
For example, look at the following diagram.
To find the height of the three, we can set up a proportion by drawing two similar triangles. The proportion
should be
8 5
= . Use a diagram to help you!
24 x
11) Find the missing length:
12) Find the missing length: