# Ratios and rates: ratios and rates

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## 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 $\frac{\text{36}}{4}$ and the like denominate numbers $\frac{\text{8 miles}}{\text{2 miles}}$ 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

$\frac{\text{55 miles}}{\text{1 gallon}}\text{and}\frac{\text{40 dollars}}{\text{5 tickets}}$

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

$\frac{a}{b}$ or $\frac{b}{a}$

The ratio $\frac{a}{b}$ is read as " $a$ to $b$ ."

The ratio $\frac{b}{a}$ 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 $\frac{\text{30}}{2}$ . Reducing, we get $\frac{\text{15}}{1}$ .

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

The rate "4 televisions to 12 people" can be expressed as $\frac{\text{4 televisions}}{\text{12 people}}$ . The meaning of this rate is that "for every 4 televisions, there are 12 people."

Reducing, we get $\frac{\text{1 television}}{\text{3 people}}$ . 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."

## Practice set b

Write the following ratios and rates as fractions.

3 to 2

$\frac{3}{2}$

1 to 9

$\frac{1}{9}$

5 books to 4 people

$\frac{\text{5 books}}{\text{4 people}}$

120 miles to 2 hours

$\frac{\text{60 miles}}{\text{1 hour}}$

8 liters to 3 liters

$\frac{8}{3}$

Write the following ratios and rates in the form " $a$ to $b$ ." Reduce when necessary.

$\frac{9}{5}$

9 to 5

$\frac{1}{3}$

1 to 3

$\frac{\text{25 miles}}{\text{2 gallons}}$

25 miles to 2 gallons

$\frac{\text{2 mechanics}}{\text{4 wrenches}}$

1 mechanic to 2 wrenches

$\frac{\text{15 video tapes}}{\text{18 video tapes}}$

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

$\frac{6}{\text{11}}$ is an example of a . (ratio/rate)

$\frac{5}{\text{12}}$ is an example of a . (ratio/rate)

ratio

$\frac{\text{7 erasers}}{\text{12 pencils}}$ is an example of a . (ratio/rate)

$\frac{\text{20 silver coins}}{\text{35 gold coins}}$ is an example of a .(ratio/rate)

rate

$\frac{\text{3 sprinklers}}{\text{5 sprinklers}}$ is an example of a . (ratio/rate)

$\frac{\text{18 exhaust valves}}{\text{11 exhaust valves}}$ is an example of a .(ratio/rate)

ratio

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

$\frac{8}{3}$

$\frac{2}{5}$

two to five

$\frac{\text{8 feet}}{\text{3 seconds}}$

$\frac{\text{29 miles}}{\text{2 gallons}}$

29 mile per 2 gallons or $\text{14}\frac{1}{2}$ miles per 1 gallon

$\frac{\text{30,000 stars}}{\text{300 stars}}$

$\frac{\text{5 yards}}{\text{2 yards}}$

5 to 2

$\frac{\text{164 trees}}{\text{28 trees}}$

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

12 to 5

$\frac{\text{12}}{5}$

81 to 19

42 plants to 5 homes

$\frac{\text{42 plants}}{\text{5 homes}}$

8 books to 7 desks

16 pints to 1 quart

$\frac{\text{16 pints}}{\text{1 quart}}$

4 quarts to 1 gallon

2.54 cm to 1 in

$\frac{2\text{.}\text{54 cm}}{\text{1 inch}}$

80 tables to 18 tables

25 cars to 10 cars

$\frac{5}{2}$

37 wins to 16 losses

105 hits to 315 at bats

$\frac{\text{1 hit}}{\text{3 at bats}}$

510 miles to 22 gallons

1,042 characters to 1 page

1,245 pages to 2 books

## Exercises for review

( [link] ) Convert $\frac{\text{16}}{3}$ to a mixed number.

$5\frac{1}{3}$

( [link] ) $1\frac{5}{9}$ of $2\frac{4}{7}$ is what number?

( [link] ) Find the difference. $\frac{\text{11}}{\text{28}}-\frac{7}{\text{45}}$ .

$\frac{\text{299}}{\text{1260}}$

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

( [link] ) Find the value of $1\text{.}\text{85}+\frac{3}{8}\cdot 4\text{.}1$

3.3875

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Stoney Reply
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Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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s.
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for screen printed electrodes ?
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What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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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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