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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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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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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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