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Ratios and rates

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 36 4 size 12{ { {"36"} over {4} } } {} and the like denominate numbers 8 miles 2 miles size 12{ { {"8 miles"} over {"2 miles"} } } {} are examples of ratios.

Rate

A comparison, by division, of two unlike denominate numbers is a rate .

The comparison by division of two unlike denominate numbers, such as

55 miles 1 gallon and 40 dollars 5 tickets size 12{ { {"55 miles"} over {"1 gallon"} } `"and"` { {"40 dollars"} over {"5 tickets"} } } {}

are examples of rates.

Let's agree to represent two numbers (pure or denominate) with the letters a size 12{a} {} and b size 12{b} {} . This means that we're letting a size 12{a} {} represent some number and b size 12{b} {} represent some, perhaps different, number. With this agreement, we can write the ratio of the two numbers a size 12{a} {} and b size 12{b} {} as

a b or b a

The ratio a b size 12{ { {a} over {b} } } {} is read as " a size 12{a} {} to b size 12{b} {} ."

The ratio b a size 12{ { {b} over {a} } } {} is read as " b size 12{b} {} to a size 12{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 30 2 size 12{ { {"30"} over {2} } } {} . Reducing, we get 15 1 size 12{ { {"15"} over {1} } } {} .

The ratio 30 to 2 is equivalent to the ratio 15 to 1.

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The rate "4 televisions to 12 people" can be expressed as 4 televisions 12 people size 12{ { {4" televisions"} over {"12"`"people"} } } {} . The meaning of this rate is that "for every 4 televisions, there are 12 people."

Reducing, we get 1 television 3 people size 12{ { {1" televisions"} over {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."

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Practice set b

Write the following ratios and rates as fractions.

3 to 2

3 2 size 12{ { {3} over {2} } } {}

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1 to 9

1 9 size 12{ { {1} over {9} } } {}

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5 books to 4 people

5 books 4 people size 12{ { {"5 books"} over {"4 people"} } } {}

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120 miles to 2 hours

60 miles 1 hour size 12{ { {"60 miles"} over {"1 hour"} } } {}

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8 liters to 3 liters

8 3 size 12{ { {8} over {3} } } {}

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Write the following ratios and rates in the form " a size 12{a} {} to b size 12{b} {} ." Reduce when necessary.

9 5 size 12{ { {9} over {5} } } {}

9 to 5

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1 3 size 12{ { {1} over {3} } } {}

1 to 3

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25 miles 2 gallons size 12{ { {"25 miles"} over {"2 gallons"} } } {}

25 miles to 2 gallons

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2 mechanics 4 wrenches size 12{ { {"2 mechanics"} over {"4 wrenches"} } } {}

1 mechanic to 2 wrenches

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15 video tapes 18 video tapes size 12{ { {"15 video tapes"} over {"18 video tapes"} } } {}

5 to 6

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Exercises

For the following 9 problems, complete the statements.

Two numbers can be compared by subtraction if and only if .

They are pure numbers or like denominate numbers.

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A comparison, by division, of two pure numbers or two like denominate numbers is called a .

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A comparison, by division, of two unlike denominate numbers is called a .

rate

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6 11 size 12{ { {6} over {"11"} } } {} is an example of a . (ratio/rate)

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5 12 size 12{ { {5} over {"12"} } } {} is an example of a . (ratio/rate)

ratio

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7 erasers 12 pencils is an example of a . (ratio/rate)

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20 silver coins 35 gold coins is an example of a .(ratio/rate)

rate

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3 sprinklers 5 sprinklers is an example of a . (ratio/rate)

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18 exhaust valves 11 exhaust valves is an example of a .(ratio/rate)

ratio

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For the following 7 problems, write each ratio or rate as a verbal phrase.

8 3 size 12{ { {8} over {3} } } {}

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2 5 size 12{ { {2} over {5} } } {}

two to five

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29 miles 2 gallons

29 mile per 2 gallons or 14 1 2 size 12{"14" { {1} over {2} } } {} miles per 1 gallon

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For the following problems, write the simplified fractional form of each ratio or rate.

12 to 5

12 5 size 12{ { {"12"} over {5} } } {}

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42 plants to 5 homes

42 plants 5 homes size 12{ { {"42"" plants"} over {5"homes"} } } {}

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16 pints to 1 quart

16 pints 1 quart size 12{ { {"16 pints"} over {"1 quart"} } } {}

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2.54 cm to 1 in

2 . 54 cm 1 inch size 12{ { {2 "." "54cm"} over {"1 inch"} } } {}

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25 cars to 10 cars

5 2 size 12{ { {5} over {2} } } {}

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105 hits to 315 at bats

1 hit 3 at bats size 12{ { {"1 hit"} over {"3 at bats"} } } {}

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1,042 characters to 1 page

1, 042   characters 1   page size 12{ { {1,"042"" characters"} over {1" page"} } } {}

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Exercises for review

( [link] ) Convert 16 3 size 12{ { {"16"} over {3} } } {} to a mixed number.

5 1 3 size 12{5 { {1} over {3} } } {}

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( [link] ) 1 5 9 of 2 4 7 is what number?

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( [link] ) Find the difference. 11 28 7 45 size 12{ { {"11"} over {"28"} } - { {7} over {"45"} } } {} .

299 1260 size 12{ { {"299"} over {"1260"} } } {}

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( [link] ) Perform the division. If no repeating patterns seems to exist, round the quotient to three decimal places: 22 . 35 ÷ 17 size 12{"22" "." "35"¸"17"} {}

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( [link] ) Find the value of 1 . 85 + 3 8 4 . 1 size 12{1 "." "85"+ { {3} over {8} } cdot 4 "." 1} {}

3.3875

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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