1. The document discusses ratios, rates, proportions, and using unit analysis to solve problems involving conversions between measurement units. 2. It provides examples of writing ratios in different formats and explains what a rate is. Proportions are introduced as a way to represent equal ratios and the cross product and reciprocal properties for solving proportions. 3. An example problem demonstrates converting between time units to calculate the number of seconds in 75 years using successive unit conversions and proportions.

2. 1. Use ratios and rates to solve real-life problems.
2. Solve proportions.

3. A ratio is the comparison of two sets of numbers.
Three ways to write ratios
1. Write the ratio using the word “to” between the two
number being compared.
For example: There are 8 girls and 5 boys in my class.
What is the ration of girls to boys?
The ratio is 8 girls to 5 boys
8 to 5

4. 2. Write the ratio using a colon between the two number
being compared.
For example: There are 3 apples and 4 oranges in the
basket. What is the ratio of apples to oranges?
The ratio is : 3 apples to 4 oranges
3 : 4

5. For example: Your school’s basketball team has won 7 games
and lost 3 games. What is the ratio of wins to
losses?
Because we are comparing wins to losses the first
number in our ratio should be the number of wins and
the second number is the number of losses.
The ratio is games won___________
games lost
=
7 games_______
3 games
=
7__
3
3. Write ratio as a fraction

6. In a ratio, if the numerator and denominator are measured in
different units then the ratio is called a rate.
A unit rate is a rate per one given unit, like 60 miles per 1 hour.
Example: You can travel 120 miles on 60 gallons of gas.
What is your fuel efficiency in miles per gallon?
Rate =
120 miles________
60 gallons
=
________20 miles
1 gallon
Your fuel efficiency is 20 miles per gallon.

7. Writing the units when comparing each unit of a rate is called unit
analysis.
You can multiply and divide units just like you would multiply and
divide numbers. When solving problems involving rates, you can
use unit analysis to determine the correct units for the answer.
Example: How many minutes are in 5 hours?
To solve this problem we need a unit rate that relates minutes to
hours. Because there are 60 minutes in an hour, the unit rate we
choose is 60 minutes per hour.
5 hours • 60 minutes________
1 hour
= 300 minutes

8. An equation in which two ratios are equal is called a proportion.
A proportion can be written using colon notation like this
a:b::c:d
or as the more recognizable (and useable) equivalence of two fractions.
a___ ___
=
b
c
d

9. a:b::c:d
a___ ___
=
b
c
d
When Ratios are written in this order, a and d are the extremes, or
outside values, of the proportion, and b and c are the means, or
middle values, of the proportion.
Extremes
Means

10. To solve problems which require the use of a proportion we can use one
of two properties.
The reciprocal property of proportions.
If two ratios are equal, then their reciprocals are equal.
The cross product property of proportions.
The product of the extremes equals the product of the means

11. Example:
x
35
3
5

355
3 x

Write the original proportion.
Use the reciprocal property.
35
355
3
35 
x Multiply both sides by 35 to isolate
the variable, then simplify.
x21

12. Example:
9
62

x
x 629
Write the original proportion.
Use the cross product property.
6
6
6
18 x

Divide both sides by 6 to isolate the
variable, then simplify.
x3

13. If the average person lives for 75 years, how long would that be
in seconds?

14. If the average person lives for 75 years, how long would that be
in seconds?
To solve this problem we need to convert 75 years to seconds. We
can do this by breaking the problem down into smaller parts by
converting years to days, days to hours, hours to minutes and
minutes to seconds.
There are 365.25 days in one year, 24 hours in one day, 60
minutes in 1 hour, and 60 seconds in a minute.

minute1
seconds60
hour1
minutes60
day1
hours24
year1
days25.365
years75
Multiply the fractions, and use unit analysis to determine the
correct units for the answer.
2366820000
seconds

15. John constructs a scale model of a building. He says that 3/4th
feet of height on the real building is 1/5th inches of height on
the model.
What is the ratio between the height of the model and the
height of the building?
If the model is 5 inches tall, how tall is the actual building in
feet?

16. What is the ratio between the height of the model and the
height of the building?
What two pieces of information does the problem give you
to write a ratio?
For every 3/4th feet of height on the building…
the model has 1/5th inches of height.
Therefore the ratio of the height of the model to the height
of the building is…

feet
4
3
inches
5
1

3
4
5
1
feet15
inches4
This is called a scale factor.

17. If the model is 5 inches tall, how tall is the actual building in
feet? To find the actual height of the building, use the ratio
from the previous step to write a proportion to represent
the question above.
x
inches5
feet15
inches4

1554  x
4
75
4
4

x
feet75.18x
Use the cross product.
Isolate the variable, then simplify.
Don’t forget your units.