7.1: Ratios and Rates
Learning Objectives
- be able to distinguish between denominate and pure numbers and between ratios and rates
Denominate Numbers and Pure Numbers
Denominate Numbers, Like and Unlike Denominate
Numbers
It is often necessary or convenient to compare two quantities
. Denominate numbers
are numbers together with some specified unit. If the units being compared are alike, the denominate numbers are called
like denominate numbers
. If units are not alike, the numbers are called
unlike denominate numbers
. Examples of denominate numbers are shown in the diagram:
Pure
Numbers
Numbers that exist purely as numbers and do
not
represent amounts of quantities are called
pure numbers
. Examples of pure numbers are 8, 254, 0, \(21 \dfrac{5}{8}\), \(\dfrac{2}{5}\), and 0.07.
Numbers can be compared in two ways: subtraction and division.
Comparing Numbers by Subtraction and
Division
Comparison
of two numbers by subtraction
indicates how
much more
one number is than another.
Comparison by division
indicates how
many times
larger or smaller one number is than another.
Comparing Pure or Like Denominate Numbers by Subtraction
Numbers can be compared by subtraction if and only if they both are like denominate numbers or both pure numbers.
Compare 8 miles and 3 miles by subtraction.
Solution
\(\text{8 miles - 3 miles = 5 miles}\)
This means that 8 miles is 5 miles more than 3 miles.
Examples of use : I can now jog 8 miles whereas I used to jog only 3 miles. So, I can now jog 5 miles more than I used to.
Compare 12 and 5 by subtraction.
Solution
\(12 - 5 = 7\)
This means that 12 is 7 more than 5.
Comparing 8 miles and 5 gallons by subtraction makes no sense.
Solution
\(\text{8 miles - 5 gallons = ?}\)
Compare 36 and 4 by division.
Solution
\(36 \div 4 = 9\)
This means that 36 is 9 times as large as 4. Recall that \(36 \div 4 = 9\) can be expressed as \(\dfrac{36}{4} = 9\).
Compare 8 miles and 2 miles by division.
Solution
\(\dfrac{\text{8 miles}}{\text{2 miles}} = 4\)
This means that 8 miles is 4 times as large as 2 miles.
Example of use : I can jog 8 miles to your 2 miles. Or, for every 2 miles that you jog, I jog 8. So, I jog 4 times as many miles as you jog.
Notice that when like quantities are being compared by division, we drop the units. Another way of looking at this is that the units divide out (cancel).
Compare 30 miles and 2 gallons by division.
Solution
\(\dfrac{\text{30 miles}}{\text{2 gallons}} = \dfrac{\text{15 miles}}{\text{1 gallon}}\)
Example of use : A particular car goes 30 miles on 2 gallons of gasoline. This is the same as getting 15 miles to 1 gallon of gasoline.
Notice that when the quantities being compared by division are unlike quantities, we do not drop the units.
Practice Set A
Make the following comparisons and interpret each one.
Compare 10 diskettes to 2 diskettes by
- subtraction:
- division:
- Answer
-
a. 8 diskettes; 10 diskettes is 8 diskettes more than 2 diskettes.
b. 5; 10 diskettes is 5 times as many diskettes as 2 diskettes.
Practice Set A
Compare, if possible, 16 bananas and 2 bags by
- subtraction:
- division:
- Answer
-
a. Comparison by subtraction makes no sense.
b. \(\dfrac{\text{16 bananas}}{\text{2 bags}} = \dfrac{\text{8 bananas}}{\text{bag}}\), 8 bananas per bag.
Ratios and Rates
A comparison, by division, of two pure numbers or two like denominate numbers is a ratio .
The comparison by division of the pure numbers \(\dfrac{36}{4}\) and the like denominate numbers \(\dfrac{\text{8 miles}}{\text{2 miles}}\) are examples of ratios.
A comparison, by division, of two unlike denominate numbers is a rate .
The comparison by division of two unlike denominate numbers, such as
\(\dfrac{\text{55 miles}}{\text{1 gallon}}\) and \(\dfrac{\text{40 dollars}}{\text{5 tickets}}\)
are examples of rates.
Let's agree to represent two numbers (pure or denominate) with the letters \(a\) and \(b\). This means that we're letting \(a\) represent some number and \(b\) represent some, perhaps different, number. With this agreement, we can write the ratio of the two numbers \(a\) and \(b\) as
\(\dfrac{a}{b}\) or \(\dfrac{b}{a}\)
The ratio \(\dfrac{a}{b}\) is read as "\(a\) to \(b\)."
The ratio \(\dfrac{b}{a}\) is read as "\(b\) to \(a\)."
Since a ratio or a rate can be expressed as a fraction, it may be reducible.
The ratio 30 to 2 can be expressed as \(\dfrac{30}{2}\). Reducing, we get \(\dfrac{15}{1}\).
The ratio 30 to 2 is equivalent to the ratio 15 to 1.
The rate "4 televisions to 12 people" can be expressed as \(\dfrac{\text{4 televisions}}{\text{12 people}}\). The meaning of this rate is that "for every 4 televisions, there are 12 people."
Reducing, we get \(\dfrac{\text{1 television}}{\text{3 people}}\). The meaning of this rate is that "for every 1 television, there are 3 people.”
Thus, the rate of "4 televisions to 12 people" is the same as the rate of "1 television to 3 people."
Practicee Set B
Write the following ratios and rates as fractions.
3 to 2
- Answer
-
\(\dfrac{3}{2}\)
Practicee Set B
1 to 9
- Answer
-
\(\dfrac{1}{9}\)
Practicee Set B
5 books to 4 people
- Answer
-
\(\dfrac{\text{5 books}}{\text{4 people}}\)
Practicee Set B
120 miles to 2 hours
- Answer
-
\(\dfrac{\text{60 miles}}{\text{1 hour}}\)
Practicee Set B
8 liters to 3 liters
- Answer
-
\(\dfrac{8}{3}\)
Write the following ratios and rates in the form "\(a\) to \(b\)." Reduce when necessary.
Practicee Set B
\(\dfrac{9}{5}\)
- Answer
-
9 to 5
Practicee Set B
\(\dfrac{1}{3}\)
- Answer
-
1 to 3
Practicee Set B
\(\dfrac{\text{25 miles}}{\text{2 gallons}}\)
- Answer
-
25 miles to 2 gallons
Practicee Set B
\(\dfrac{\text{2 mechanics}}{\text{4 wrenches}}\)
- Answer
-
1 mechanic to 2 wrenches
Practicee Set B
\(\dfrac{\text{15 video tapes}}{\text{18 video tapes}}\)
- Answer
-
5 to 6
Exercises
For the following 9 problems, complete the statements.
Exercise \(\PageIndex{1}\)
Two numbers can be compared by subtraction if and only if .
- Answer
-
They are pure numbers or like denominate numbers.
Exercise \(\PageIndex{2}\)
A comparison, by division, of two pure numbers or two like denominate numbers is called a .
Exercise \(\PageIndex{3}\)
A comparison, by division, of two unlike denominate numbers is called a .
- Answer
-
rate
Exercise \(\PageIndex{4}\)
\(\dfrac{6}{11}\) is an example of a . (ratio/rate)
Exercise \(\PageIndex{5}\)
\(\dfrac{5}{12}\) is an example of a . (ratio/rate)
- Answer
-
ratio
Exercise \(\PageIndex{6}\)
\(\dfrac{\text{7 erasers}}{\text{12 pencils}}\) is an example of a . (ratio/rate)
Exercise \(\PageIndex{7}\)
\(\dfrac{\text{20 silver coins}}{\text{35 gold coins}}\) is an example of a . (ratio/rate)
- Answer
-
rate
Exercise \(\PageIndex{8}\)
\(\dfrac{\text{3 sprinklers}}{\text{5 sprinklers}}\) is an example of a . (ratio/rate)
Exercise \(\PageIndex{9}\)
\(\dfrac{\text{18 exhaust valves}}{\text{11 exhaust valves}}\) is an example of a . (ratio/rate)
- Answer
-
ratio
For the following 7 problems, write each ratio or rate as a verbal phrase.
Exercise \(\PageIndex{10}\)
\(\dfrac{8}{3}\)
Exercise \(\PageIndex{11}\)
\(\dfrac{2}{5}\)
- Answer
-
two to five
Exercise \(\PageIndex{12}\)
\(\dfrac{\text{8 feet}}{\text{3 seconds}}\)
Exercise \(\PageIndex{13}\)
\(\dfrac{\text{29 miles}}{\text{2 gallons}}\)
- Answer
-
29 mile per 2 gallons or \(14 \dfrac{1}{2}\) miles per 1 gallon
Exercise \(\PageIndex{14}\)
\(\dfrac{\text{30,000 stars}}{\text{300 stars}}\)
Exercise \(\PageIndex{15}\)
\(\dfrac{\text{5 yards}}{\text{2 yards}}\)
- Answer
-
5 to 2
Exercise \(\PageIndex{16}\)
\(\dfrac{\text{164 trees}}{\text{28 trees}}\)
For the following problems, write the simplified fractional form of each ratio or rate.
Exercise \(\PageIndex{17}\)
12 to 5
- Answer
-
\(\dfrac{12}{5}\)
Exercise \(\PageIndex{18}\)
81 to 19
Exercise \(\PageIndex{19}\)
42 plants to 5 homes
- Answer
-
\(\dfrac{\text{42 plants}}{\text{5 homes}}\)
Exercise \(\PageIndex{20}\)
8 books to 7 desks
Exercise \(\PageIndex{21}\)
16 pints to 1 quart
- Answer
-
\(\dfrac{\text{16 pints}}{\text{1 quart}}\)
Exercise \(\PageIndex{22}\)
4 quarts to 1 gallon
Exercise \(\PageIndex{23}\)
2.54 cm to 1 in
- Answer
-
\(\dfrac{\text{2.54 cm}}{\text{1 inch}}\)
Exercise \(\PageIndex{24}\)
80 tables to 18 tables
Exercise \(\PageIndex{25}\)
25 cars to 10 cars
- Answer
-
\(\dfrac{5}{2}\)
Exercise \(\PageIndex{26}\)
37 wins to 16 losses
Exercise \(\PageIndex{27}\)
105 hits to 315 at bats
- Answer
-
\(\dfrac{\text{1 hit}}{\text{3 at bats}}\)
Exercise \(\PageIndex{28}\)
510 miles to 22 gallons
Exercise \(\PageIndex{29}\)
1,042 characters to 1 page
- Answer
-
\(\dfrac{\text{1,042 characters}}{\text{1 page}}\)
Exercise \(\PageIndex{30}\)
1,245 pages to 2 books
Exercises for Review
Exercise \(\PageIndex{31}\)
Convert \(\dfrac{16}{3}\) to a mixed number.
- Answer
-
\(5 \dfrac{1}{3}\)
Exercise \(\PageIndex{32}\)
\(1 \dfrac{5}{9}\) of \(2 \dfrac{4}{7}\) is what number?
Exercise \(\PageIndex{33}\)
Find the difference. \(\dfrac{11}{28} - \dfrac{7}{45}\).
- Answer
-
\(\dfrac{299}{1260}\)
Exercise \(\PageIndex{34}\)
Perform the division. If no repeating patterns seems to exist, round the quotient to three decimal places: \(22.35 \div 17\)
Exercise \(\PageIndex{35}\)
Find the value of \(1.85 + \dfrac{3}{8} \cdot 4.1\)
- Answer
-
3.3875