# 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

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