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Ratios and Rates

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses ratios and rates. By the end of the module students should be able to distinguish between denominate and pure numbers and between ratios and rates.

Section Overview

  • Denominate Numbers and Pure Numbers
  • Ratios and Rates

Denominate Numbers and Pure Numbers

Denominate Numbers, Like and Unlike Denominate Numbers

It is often necessary or convenient to compare two quantities. Denominate num­bers are numbers together with some specified unit. If the units being compared are alike, the denominate numbers are called like denominate numbers. If units are not alike, the numbers are called unlike denominate numbers. Examples of denominate numbers are shown in the diagram:

8 gallons, 32 cents, and 54 miles, all labeled as the denominations.

Pure Numbers

Numbers that exist purely as numbers and do not represent amounts of quantities are called pure numbers. Examples of pure numbers are 8, 254, 0, 21582158 size 12{"21" { {5} over {8} } } {}, 2525 size 12{ { {2} over {5} } } {}, and 0.07.

Numbers can be compared in two ways: subtraction and division.

Comparing Numbers by Subtraction and Division

Comparison of two numbers by subtraction indicates how much more one number is than another.
Comparison by division indicates how many times larger or smaller one number is than another.

Comparing Pure or Like Denominate Numbers by Subtraction

Numbers can be compared by subtraction if and only if they both are like denominate numbers or both pure numbers.

Sample Set A

Example 1

Compare 8 miles and 3 miles by subtraction.

8 mile-3 miles = 5 miles 8 mile-3 miles = 5 miles size 12{"8 mile -3 miles "="5 miles"} {}

This means that 8 miles is 5 miles more than 3 miles.

Examples of use: I can now jog 8 miles whereas I used to jog only 3 miles. So, I can now jog 5 miles more than I used to.

Example 2

Compare 12 and 5 by subtraction.

12 5 = 7 12 5 = 7 size 12{"12" - 5=7} {}

This means that 12 is 7 more than 5.

Example 3

Comparing 8 miles and 5 gallons by subtraction makes no sense.

8 miles-5 gallons = ? 8 miles-5 gallons = ? size 12{"8 miles -5 gallons "=?} {}

Example 4

Compare 36 and 4 by division.

36 ÷ 4 = 9 36 ÷ 4 = 9 size 12{"36" div 4=9} {}

This means that 36 is 9 times as large as 4. Recall that 36÷4=936÷4=9 size 12{"36" div 4=9} {} can be expressed as 364=9364=9 size 12{ { {"36"} over {4} } =9} {}.

Example 5

Compare 8 miles and 2 miles by division.

8 miles 2 miles = 4 8 miles 2 miles = 4 size 12{ { {"8 miles"} over {"2 miles"} } =4} {}

This means that 8 miles is 4 times as large as 2 miles.

Example of use: I can jog 8 miles to your 2 miles. Or, for every 2 miles that you jog, I jog 8. So, I jog 4 times as many miles as you jog.

Notice that when like quantities are being compared by division, we drop the units. Another way of looking at this is that the units divide out (cancel).

Example 6

Compare 30 miles and 2 gallons by division.

30 miles 2 gallons = 15 miles 1 gallon 30 miles 2 gallons = 15 miles 1 gallon size 12{ { {"30 miles "} over {"2 gallons "} } = { {" 15 miles "} over {"1 gallon"} } } {}

Example of use: A particular car goes 30 miles on 2 gallons of gasoline. This is the same as getting 15 miles to 1 gallon of gasoline.

Notice that when the quantities being compared by division are unlike quantities, we do not drop the units.

Practice Set A

Make the following comparisons and interpret each one.

Exercise 1

Compare 10 diskettes to 2 diskettes by

  1. (a) subtraction:
  2. (b) division:

Solution

  1. (a) 8 diskettes; 10 diskettes is 8 diskettes more than 2 diskettes.
  2. (b) 5; 10 diskettes is 5 times as many diskettes as 2 diskettes.

Exercise 2

Compare, if possible, 16 bananas and 2 bags by

  1. (a) subtraction:
  2. (b) division:

Solution

  1. (a) Comparison by subtraction makes no sense.
  2. (b) 16 bananas2 bags=8 bananasbag16 bananas2 bags=8 bananasbag, 8 bananas per bag.

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 364364 size 12{ { {"36"} over {4} } } {} and the like denominate numbers 8 miles2 miles8 miles2 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 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 aa size 12{a} {} and bb size 12{b} {}. This means that we're letting aa size 12{a} {} represent some number and bb size 12{b} {} represent some, perhaps different, number. With this agreement, we can write the ratio of the two numbers aa size 12{a} {} and bb size 12{b} {} as

a b a b or b a b a

The ratio abab size 12{ { {a} over {b} } } {} is read as " aa size 12{a} {} to bb size 12{b} {}."

The ratio baba size 12{ { {b} over {a} } } {} is read as " bb size 12{b} {} to aa size 12{a} {}."

Since a ratio or a rate can be expressed as a fraction, it may be reducible.

Sample Set B

Example 7

The ratio 30 to 2 can be expressed as 302302 size 12{ { {"30"} over {2} } } {}. Reducing, we get 151151 size 12{ { {"15"} over {1} } } {}.

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

Example 8

The rate "4 televisions to 12 people" can be expressed as 4 televisions12 people 4 televisions12 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 television3 people1 television3 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.

Exercise 3

3 to 2

Solution

3232 size 12{ { {3} over {2} } } {}

Exercise 4

1 to 9

Solution

1919 size 12{ { {1} over {9} } } {}

Exercise 5

5 books to 4 people

Solution

5 books4 people5 books4 people size 12{ { {"5 books"} over {"4 people"} } } {}

Exercise 6

120 miles to 2 hours

Solution

60 miles1 hour60 miles1 hour size 12{ { {"60 miles"} over {"1 hour"} } } {}

Exercise 7

8 liters to 3 liters

Solution

8383 size 12{ { {8} over {3} } } {}

Write the following ratios and rates in the form "aa size 12{a} {} to bb size 12{b} {}." Reduce when necessary.

Exercise 8

9595 size 12{ { {9} over {5} } } {}

Solution

9 to 5

Exercise 9

1313 size 12{ { {1} over {3} } } {}

Solution

1 to 3

Exercise 10

25 miles2 gallons25 miles2 gallons size 12{ { {"25 miles"} over {"2 gallons"} } } {}

Solution

25 miles to 2 gallons

Exercise 11

2 mechanics4 wrenches2 mechanics4 wrenches size 12{ { {"2 mechanics"} over {"4 wrenches"} } } {}

Solution

1 mechanic to 2 wrenches

Exercise 12

15 video tapes18 video tapes15 video tapes18 video tapes size 12{ { {"15 video tapes"} over {"18 video tapes"} } } {}

Solution

5 to 6

Exercises

For the following 9 problems, complete the statements.

Exercise 13

Two numbers can be compared by subtraction if and only if

                              
.

Solution

They are pure numbers or like denominate numbers.

Exercise 14

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

               
.

Exercise 15

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

               
.

Solution

rate

Exercise 16

611611 size 12{ { {6} over {"11"} } } {} is an example of a

               
. (ratio/rate)

Exercise 17

512512 size 12{ { {5} over {"12"} } } {} is an example of a

               
. (ratio/rate)

Solution

ratio

Exercise 18

7 erasers12 pencils 7 erasers12 pencils is an example of a

               
. (ratio/rate)

Exercise 19

20 silver coins 35 gold coins20 silver coins 35 gold coins is an example of a

               
.(ratio/rate)

Solution

rate

Exercise 20

3 sprinklers5 sprinklers 3 sprinklers5 sprinklers is an example of a

               
. (ratio/rate)

Exercise 21

18 exhaust valves 11 exhaust valves 18 exhaust valves 11 exhaust valves is an example of a

               
.(ratio/rate)

Solution

ratio

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

Exercise 22

8383 size 12{ { {8} over {3} } } {}

Exercise 23

2525 size 12{ { {2} over {5} } } {}

Solution

two to five

Exercise 24

8 feet 3 seconds 8 feet 3 seconds

Exercise 25

29 miles 2 gallons 29 miles 2 gallons

Solution

29 mile per 2 gallons or 14121412 size 12{"14" { {1} over {2} } } {} miles per 1 gallon

Exercise 26

30,000 stars 300 stars 30,000 stars 300 stars

Exercise 27

5 yards 2 yards 5 yards 2 yards

Solution

5 to 2

Exercise 28

164 trees 28 trees 164 trees 28 trees

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

Exercise 29

12 to 5

Solution

125125 size 12{ { {"12"} over {5} } } {}

Exercise 30

81 to 19

Exercise 31

42 plants to 5 homes

Solution

42 plants5 homes42 plants5 homes size 12{ { {"42"" plants"} over {5"homes"} } } {}

Exercise 32

8 books to 7 desks

Exercise 33

16 pints to 1 quart

Solution

16 pints1 quart16 pints1 quart size 12{ { {"16 pints"} over {"1 quart"} } } {}

Exercise 34

4 quarts to 1 gallon

Exercise 35

2.54 cm to 1 in

Solution

2.54 cm1 inch2.54 cm1 inch size 12{ { {2 "." "54cm"} over {"1 inch"} } } {}

Exercise 36

80 tables to 18 tables

Exercise 37

25 cars to 10 cars

Solution

5252 size 12{ { {5} over {2} } } {}

Exercise 38

37 wins to 16 losses

Exercise 39

105 hits to 315 at bats

Solution

1 hit3 at bats1 hit3 at bats size 12{ { {"1 hit"} over {"3 at bats"} } } {}

Exercise 40

510 miles to 22 gallons

Exercise 41

1,042 characters to 1 page

Solution

1,042   characters1  page1,042   characters1  page size 12{ { {1,"042"" characters"} over {1" page"} } } {}

Exercise 42

1,245 pages to 2 books

Exercises for Review

Exercise 43

((Reference)) Convert 163163 size 12{ { {"16"} over {3} } } {} to a mixed number.

Solution

513513 size 12{5 { {1} over {3} } } {}

Exercise 44

((Reference)) 159159 of 247247 is what number?

Exercise 45

((Reference)) Find the difference. 11287451128745 size 12{ { {"11"} over {"28"} } - { {7} over {"45"} } } {}.

Solution

29912602991260 size 12{ { {"299"} over {"1260"} } } {}

Exercise 46

((Reference)) Perform the division. If no repeating patterns seems to exist, round the quotient to three decimal places: 22.35÷1722.35÷17 size 12{"22" "." "35"¸"17"} {}

Exercise 47

((Reference)) Find the value of 1.85+384.11.85+384.1 size 12{1 "." "85"+ { {3} over {8} } cdot 4 "." 1} {}

Solution

3.3875

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