Math 40
Prealgebra
Section 6.1 – Ratios and Rates
6.1 Ratios and Rates
In this section we will learn how ratios and rates help us to see important relationships.
Ratios
Ratios are a way of comparing two numbers (or quantities) with the same units by the
operation of division.
There are three ways in which we can express a ratio:
-
Words
“the ratio of a to b”
-
Colon
-
Fraction
a:b
a
b
CAUTION: Order is important!
The quantity mentioned first (before the colon/“to”) is the numerator.
The quantity mentioned second (after the colon/“to”) is the denominator.
In this class using a fraction to express a ratio is most common. Now would be a good time to learn how to
properly use that fraction key on your scientific calculator! Before you start this section, make sure to ask
your instructor or tutor to show you how to use your calculator to do the following:
1)
2)
3)
4)
Enter fractions and mixed numbers
Reduce
Divide fractions and mixed numbers
Convert back and forth between mixed numbers and improper fractions.
Example 1: Write each ratio as a fraction in lowest terms.
a) 18 :15
b) 21 to 5
1
4
Solution:
a) 18 :15
18 is mentioned first, so 18 is the numerator.
15 is mentioned second, so 15 is the denominator.
18
Now reduce.
15
6

5
Notice that we leave the final answer as an improper fraction! Writing it as a mixed number
is incorrect. This is because a ratio is a comparison of two quantities with the same units.

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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
b) 21 to 5
1
4
21 is mentioned first, so 21 is the numerator.
5
1
1
is mentioned second, so 5 is the denominator.
4
4
21
Now divide.
1
5
4
4

1
We really need to write the 1 in the denominator! Writing the answer as just 4 would be incorrect.
This is because a ratio is a comparison of two quantities with the same units.

You Try It 1: Write each ratio as a fraction in lowest terms.
a) 36 to 28
2
b) 2 :15
9
Example 2: Write each ratio as a fraction in lowest terms.
a) $16 to $56
b) 44 feet : 12 feet
Solution:
a) $16 to $56
$16 is mentioned first, so $16 is the numerator.
$56 is mentioned second, so $56 is the denominator.
$16

Now reduce.
$56
2
We treat the dollar sign ($) as a variable so it divides out.

7
same thing
1
In fact, the units always divide out in a ratio since
same thing
b) 44 feet : 12 feet
44 feet is mentioned first, so 44 feet is the numerator.
12 feet is mentioned second, so 12 feet is the denominator.
44 feet
Now reduce.
12 feet
11

We treat the units (feet) as a variable so they divide out.
3
Notice that we leave the final answer as an improper fraction! Writing it as a mixed number
is incorrect. This is because a ratio is a comparison of two quantities with the same units.

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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
You Try It 2: Write each ratio as a fraction in lowest terms.
1
1
a) 3 days to 5 days
2
2
b) 48 yards : 24 yards
Example 3: Write the following ratio as a fraction in lowest terms and interpret the results.
A solution contains 30 ounces of medicine and 120 ounces of water. What is the ratio of water to
medicine?
Solution:
Water is mentioned first, so 120 ounces is the numerator.
Medicine is mentioned second, so 30 ounces is the denominator.
120 ounces
30 ounces
40

1
Now reduce.
We interpret the results as for every 40 ounces of water, there is 1 ounce of medicine.
You Try It 3: Write the following ratio as a fraction in lowest terms and interpret the results.
Your monthly expenses are $840 while your monthly income is $820. What is the ratio of your
monthly expenses to your monthly income?
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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
Example 4: Write the following ratio as a fraction in lowest terms: 12 seconds to 2 minutes
Solution:
WAIT! Before we set up a fraction, we need to make sure that both quantities have the same
units.
What do we know about the relationship between seconds and minutes?
There are 60 seconds in 1 minute.
To avoid fractions or mixed numbers in our calculations, we rewrite the “larger” unit in terms of
the “smaller” unit.
A minute is longer (“larger”) than a second so we must convert 2 minutes to seconds.
We do this by multiplying:
2 minutes  60 seconds 

  2  60 seconds   120 seconds
1
 1 minute 
(You may find the above set up a bit confusing. Don’t worry! We will explain it in more detail in
Section 6.3. As long as you understand that you multiply 2 by 60 seconds, you should be ok in this
section.)
So, 12 seconds to 2 minutes turns into 12 seconds to 120 seconds.
Now we are able to set up the ratio as a fraction. 12 seconds to 120 seconds.
12 seconds
120 seconds
1

10

Remember, the seconds divide out since
same thing
1
same thing
You Try It 4: Write the following ratio as a fraction in lowest terms: 2 hours to 25 minutes
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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
Many of the problems from this part of the section will involve relationships from the U.S. Customary System.
If you are not familiar with these relationships, you will want to use following table:
U.S. Customary System Relationships
Length
12 inches (in.) = 1 foot (ft)
3 feet (ft) = 1 yard (yd)
5280 feet (ft) = 1 mile (mi)
Mass (Weight)
16 ounces (oz) = 1 pound (lb)
2000 pounds (lbs) = 1 ton (T)
Capacity (Volume)
8 fluid ounces (fl oz) = 1 cup (c)
2 cups (c) = 1 pint (pt)
2 pints (pts) = 1 quart (qt)
4 quarts (qts) = 1 gallon (gal)
Note: We recommend studying the table above so you can get really comfortable with the relationships and
unit abbreviations. Trust us! It will be worth your time since these will come back in Section 6.3.
Example 5: Write the following ratio as a fraction in lowest terms: 5 yards to 4 feet
Solution:
WAIT! Before we set up a fraction, we need to make sure that both quantities have the same
units.
What do we know about the relationship between yards and feet?
There are 3 feet in 1 yard.
To avoid fractions or mixed numbers in our calculations, we rewrite the “larger” unit in terms of
the “smaller” unit.
A yard is longer (“larger”) than a foot so we must convert 5 yards to feet.
We do this by multiplying:
5 yards  3 feet

 1 yard
1


  5  3 feet   15 feet


So, 5 yards to 4 feet turns into 15 feet to 4 feet.
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Math 40
Prealgebra
Section 6.1 – Ratios and Rates
Now we are able to set up the ratio as a fraction. 15 feet to 4 feet.
15 feet
4 feet
15

4

Remember, the feet divide out since
same thing
1
same thing
You Try It 5: Write the following ratio as a fraction in lowest terms: 1 mile to 6540 feet
Example 6: Write the following ratio as a fraction in lowest terms: 12 ounces to 4 pounds
Solution:
WAIT! Before we set up a fraction, we need to make sure that both quantities have the same
units.
What do we know about the relationship between ounces and pounds?
There are 16 ounces in 1 pound.
To avoid fractions or mixed numbers in our calculations, we rewrite the “larger” unit in terms of
the “smaller” unit.
A pound is heavier (“larger”) than an ounce so we must convert 4 pounds to ounces.
We do this by multiplying:
4 pounds  16 ounces 

  4 16 ounces   64 ounces
 1 pound 
1


So, 12 ounces to 4 pounds turns into 12 ounces to 64 ounces.
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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
Now we are able to set up the ratio as a fraction. 12 ounces to 64 ounces.
12 ounces
64 ounces
3

16

Remember, the ounces divide out since
same thing
1
same thing
You Try It 6: Write the following ratio as a fraction in lowest terms: 6 pints to 8 cups
In many cases you will come across comparisons where no relationship between units exist. This makes it
impossible to rewrite each quantity with the same units. Such cases are called rates.
Rates
Rates are a way of comparing two numbers (or quantities) with different units by the
operation of division.
In a rate, one of the following words separates the quantities that are being compared:
“in”
“for”
“on”
“per”
“from”
These key words behave similar to the word “to” in a ratio in the sense that they indicate where the fraction bar
is located. Again, order is important!
The quantity mentioned first (before the key word) is the numerator.
The quantity mentioned second (after the key word) is the denominator.
You will find that you use the same procedure to write a rate as a fraction in lowest terms as you would to write
a ratio as a fraction in lowest terms. The only difference is that the units do not divide out. Make sure that
you always write the units of your rate in your final answer.
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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
Example 7: Write the following rate as a fraction in lowest terms and interpret the results.
4830 students for 92 faculty
Solution:
4830 students is mentioned first, so 4830 students is the numerator.
92 faculty is mentioned second, so 92 faculty is the denominator.
4830 students
92 faculty
Now reduce.

105 students
2 faculty
Notice that the units are written in the final answer. This is extremely important for interpretation!
Now interpret. There are 105 students for every 2 faculty.
You Try It 7: Write the following rate as a fraction in lowest terms and interpret the results.
160 patients for 48 nurses
Example 8: Write the following rate as a fraction in lowest terms and interpret the results.
224 miles traveled on 12 gallons of gas
Solution:
224 miles is mentioned first, so 224 miles is the numerator.
12 gallons is mentioned second, so 12 gallons is the denominator.
224 miles
12 gallons
Now reduce.

56 mi
3 gallons
Notice that the units are written in the final answer. This is extremely important for interpretation!
Now interpret. For every 56 miles traveled, 3 gallons gas were used..
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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
You Try It 8: Write the following rate as a fraction in lowest terms and interpret the results.
180 kilometers traveled on 14 liters of gas
Example 9: Write the following rate as a fraction in lowest terms and interpret the results.
$576 earned for 48 hours of work
Solution:
$576 is mentioned first, so $576 is the numerator.
48 hours is mentioned second, so 48 hours is the denominator.
$576
48 hours
Now reduce.

$12
1 hour
Notice that the units are written in the final answer. This is extremely important for interpretation!
Now interpret. $12 were earned for every hour worked.
You Try It 9: Write the following rate as a fraction in lowest terms and interpret the results.
$477 earned for 53 hours of work
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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
Notice that the results were a little nicer to interpret from Example 9 and You Try It 9 since the denominator in
each of these rates was 1. Such rates are called unit rates.
Unit Rates
Rates with 1 as the denominator are called unit rates.
$12
and interpreted it as “$12 earned for every hour of work”. In future
1 hour
explanations, you may also notice that “per” is often used when interpreting unit rates. So, we could have also
$12
interpreted the rate
as “$12 per hour”.
1 hour
In Example 9, we obtained the rate
Fact: Any rate can be written as a unit rate to help us better interpret the comparisons between the two
quantities of the rate.
Finding a Unit Rate
1) Set up the rate as a fraction as you normally would.
2) Instead of reducing the fraction to lowest terms, use your calculator to divide the
numerator by the denominator.
numerator  denominator
3) Don’t forget to include the units in your final answer.
Should I write all of the rates as unit rates just to be safe? No! The directions of the problem will specify if
you need to write your final answer as a unit rate.
Example 10: Write the following as a unit rate and interpret the results.
1015 calories in 2 candy bars
Solution:
1015 calories is mentioned first, so 1015 calories is the numerator.
2 candy bars is mentioned second, so 2 candy bars is the denominator.
1015 calories
2 candy bars
Now use your calculator to divide. numerator  denominator
1015  2  507.5
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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
So,
1015 calories 507.5 calories
.

2 candy bars
1 candy bar
Now interpret. 507.5 calories per candy bar
You Try It 10: Write the following as a unit rate and interpret the results.
24-pound turkey for 15 people
Example 11: Write the following as a unit rate and interpret the results.
Round to the nearest tenth if necessary.
320 miles on 12 gallons of gas
Solution:
320 miles is mentioned first, so 320 mile is the numerator.
12 gallons is mentioned second, so 12 gallons is the denominator.
320 miles
12 gallons
Now use your calculator to divide. numerator  denominator
320 12  26.7 (Rounded to the nearest tenth)
So,
320 miles 26.7 miles
.

12 gallons
1 gallon
Now interpret. 26.7 miles per gallon, or 26.7 mpg.
You Try It 11: Write the following as a unit rate and interpret the results.
304 miles on 9 gallons of gas
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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
You may have recognized the results from Example 11 and You Try It 11 as a calculation of gas mileage. This
is an especially important topic as the gas has become more expensive in the last 10 years. Let’s find out how
we can apply this concept!
Example 12: A Honda Accord is estimated to travel 354 miles on 13 gallons of gas while a Toyota Camry is
estimated to travel 422 miles on 15 gallons of gas. Use unit rates to determine which car gets the
better gas mileage.
Solution:
Set up and calculate the unit rate for each car.
Accord:
354 miles 27.2 miles

 27.2 mpg
13 gallons
1 gallon
Camry:
422 miles 28.1 miles

 28.1 mpg
15 gallons
1 gallon
The car with the greatest miles per gallon is considered to be the car with the best mileage.
Since the Camry has a higher unit rate, it gets the better gas mileage.
You Try It 12: A Volkswagen Jetta Hybrid is estimated to travel 723 miles on 16 gallons of gas while a Honda
Civic Hybrid is estimated to travel 798 miles on 18 gallons of gas. Use unit rates to determine
which car gets the better gas mileage.
When shopping for groceries, household supplies, and health and beauty products, we often come across many
different brands and package sizes. We can use unit rates to help save money by calculating the lowest cost per
unit. We call this finding the best buy.
Cost Per Unit
Cost per unit is a rate that tells how much you pay for one item (or unit).
Cost per unit 
cost
$

1 unit units
In other words, we can think if unit cost as a unit rate with money in the numerator.
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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
Example 13: Use unit rates to find the best buy. 6-pack of soda for $1.99
12-pack of soda for $3.49
24-pack of soda for $7
Solution:
Set up and calculate the unit cost for each package. In this case, unit cost would be cost per can.
6-pack:
cost $1.99

 $0.332 per can
units 6 cans
12-pack:
cost
$3.49

 $0.291 per can
units 12 cans
24-pack:
cost
$7

 $0.583 per can
units 12 cans
We rounded to the nearest thousandth just in case the unit rates ended up being close in value.
The package with the lowest cost per can is considered to be the best buy.
Since the 12-pack has the lowest unit cost, it is the best buy.
You Try It 13: Use unit rates to find the best buy.
a) 24 ounces of syrup for $1.82
36 ounces of syrup for $2.72
b) 2 quarts of oil for $3.25
3 quarts of oil for $4.95
4 quarts of oil for $6.48.
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2015 Campeau
Math 40
Prealgebra
Section 6.1 – Ratios and Rates
Finding the best buy may sometimes be more complicated. Coupons or how much use you will get out of each
unit can affect the cost per unit.
Example 14: A small bag of potatoes costs $0.19 per pound and a large bag costs $0.15 per pound. Suppose
there are only two people in your family and half of the large bag would probably spoil before
you used it up. Which bag is the best buy? Explain.
Solution:
The unit cost of the small bag is given as $0.19 per pound.
If your family is only able to use half of the larger bag, you would really be paying $0.30 per
pound:
$0.15
$0.15
$0.30
becomes


1
1 pound
pound 1 pound
2
This would mean that the smaller bag is the best buy.
You Try It 14: Which brand is the best buy, assuming the claim below is true? Explain.
Duracell: 4-pack of AA batteries for $2.79
Energizer: One AA battery for $1.19 that lasts twice as long as one Duracell AA battery
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