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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.
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.
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."
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
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
$\frac{5}{\text{12}}$ is an example of a
ratio
$\frac{\text{7 erasers}}{\text{12 pencils}}$ is an example of a
$\frac{\text{20 silver coins}}{\text{35 gold coins}}$ is an example of a
rate
$\frac{\text{3 sprinklers}}{\text{5 sprinklers}}$ is an example of a
$\frac{\text{18 exhaust valves}}{\text{11 exhaust valves}}$ is an example of a
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
$\frac{\mathrm{1,}\text{042}\text{characters}}{1\text{page}}$
1,245 pages to 2 books
( [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: $\text{22}\text{.}\text{35}\xf7\text{17}$
( [link] ) Find the value of $1\text{.}\text{85}+\frac{3}{8}\cdot 4\text{.}1$
3.3875
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