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3.1.1: Ratios and Rates

  • Page ID
    116770
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    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:

    8 gallons, 32 cents, and 54 miles, all labeled as the denominations.

    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.

    Sample Set A

    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.

    Sample Set A

    Compare 12 and 5 by subtraction.

    Solution

    \(12 - 5 = 7\)

    This means that 12 is 7 more than 5.

    Sample Set A

    Comparing 8 miles and 5 gallons by subtraction makes no sense.

    Solution

    \(\text{8 miles - 5 gallons = ?}\)

    Sample Set A

    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 \(\dfrac{36}{4} = 9\).

    Sample Set A

    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).

    Sample Set A

    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

    1. subtraction:
    2. 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

    1. subtraction:
    2. 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

    Definition: Ratio

    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.

    Definition: Rate

    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.

    Sample Set B

    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.

    Sample Set B

    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


    This page titled 3.1.1: Ratios and Rates is shared under a CC BY license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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