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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.

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."

Practice set b

Write the following ratios and rates as fractions.

3 to 2

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

1 to 9

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

5 books to 4 people

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

120 miles to 2 hours

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

8 liters to 3 liters

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

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

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

1 to 3

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

25 miles to 2 gallons

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

1 mechanic to 2 wrenches

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

5 to 6

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.

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

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

rate

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

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

ratio

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

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

rate

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

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

ratio

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

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

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

two to five

8 feet 3 seconds

29 miles 2 gallons

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

30,000 stars 300 stars

5 yards 2 yards

5 to 2

164 trees 28 trees

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

12 to 5

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

81 to 19

42 plants to 5 homes

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

8 books to 7 desks

16 pints to 1 quart

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

4 quarts to 1 gallon

2.54 cm to 1 in

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

80 tables to 18 tables

25 cars to 10 cars

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

37 wins to 16 losses

105 hits to 315 at bats

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

510 miles to 22 gallons

1,042 characters to 1 page

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

1,245 pages to 2 books

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} } } {}

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

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

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

( [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"} {}

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

3.3875

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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Prasenjit
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Azam
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Prasenjit
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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