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Proportions

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 proportions. By the end of the module students should be able to describe proportions and find the missing factor in a proportion and be able to work with proportions involving rates.

Section Overview

  • Ratios, Rates, and Proportions
  • Finding the Missing Factor in a Proportion
  • Proportions Involving Rates

Ratios, Rates, and Proportions

Ratio, Rate

We have defined a ratio as a comparison, by division, of two pure numbers or two like denominate numbers. We have defined a rate as a comparison, by division, of two unlike denominate numbers.

Proportion

A proportion is a statement that two ratios or rates are equal. The following two examples show how to read proportions.

Three fourths equals six eighths. 3 is to four as six is to eight. 25 miles divided by 1 gallon equals 50 miles divided by 2 gallons. 25 miles is to 1 gallon as 50 miles is to 2 gallons.

Sample Set A

Write or read each proportion.

Example 1

35=122035=1220 size 12{ { {3} over {5} } = { {"12"} over {"20"} } } {}

3 is to 5 as 12 is to 20

Example 2

10 items5 dollars=2 items1 dollar10 items5 dollars=2 items1 dollar size 12{ { {"10 items"} over {"5 dollars"} } = { {"2 items"} over {"1 dollar"} } } {}

10 items is to 5 dollars as 2 items is to 1 dollar

Example 3

8 is to 12 as 16 is to 24.

8 12 = 16 24 8 12 = 16 24 size 12{ { {8} over {"12"} } = { {"15"} over {"24"} } } {}

Example 4

50 milligrams of vitamin C is to 1 tablet as 300 milligrams of vitamin C is to 6 tablets.

50 1 = 300 6 50 1 = 300 6 size 12{ { {"50"} over {1} } = { {"300"} over {6} } } {}

Practice Set A

Write or read each proportion.

Exercise 1

38=61638=616 size 12{ { {3} over {8} } = { {6} over {"16"} } } {}

Solution

3 is to 8 as 6 is to 16

Exercise 2

2 people1 window=10 people5 windows2 people1 window=10 people5 windows size 12{ { {"2 people"} over {"1 window"} } = { {"10 people"} over {"5 windows"} } } {}

Solution

2 people are to 1 window as 10 people are to 5 windows

Exercise 3

15 is to 4 as 75 is to 20.

Solution

154=7520154=7520 size 12{ { {"15"} over {4} } = { {"75"} over {"20"} } } {}

Exercise 4

2 plates are to 1 tray as 20 plates are to 10 trays.

Solution

2 plates1 tray=20  plates10  trays2 plates1 tray=20  plates10  trays size 12{ { {"2 plates"} over {"1 tray"} } = { {"20"" plates"} over {"10"" trays"} } } {}

Finding the Missing Factor in a Proportion

Many practical problems can be solved by writing the given information as propor­tions. Such proportions will be composed of three specified numbers and one unknown number. It is customary to let a letter, such as xx size 12{x} {}, represent the unknown number. An example of such a proportion is

x 4 = 20 16 x 4 = 20 16 size 12{ { {x} over {4} } = { {"20"} over {"16"} } } {}

This proportion is read as " xx size 12{x} {} is to 4 as 20 is to 16."

There is a method of solving these proportions that is based on the equality of fractions. Recall that two fractions are equivalent if and only if their cross products are equal. For example,

Three fourths equals six eighths. Next to this equation are the same two fractions, with arrows pointing from the denominators to the opposite fraction's numerator, indicating a cross product. The cross product is 3 times 8 equals 6 times 4, or 24 equals 24.

Notice that in a proportion that contains three specified numbers and a letter representing an unknown quantity, that regardless of where the letter appears, the following situation always occurs.

(number)⋅(letter)=(number)⋅(number)︸(number)⋅(letter)=(number)⋅(number)︸

We recognize this as a multiplication statement. Specifically, it is a missing factor statement. (See (Reference) for a discussion of multiplication statements.) For example,

x4=2016 means that  16⋅x=4⋅20 4x=1620 means that  4⋅20=16⋅x 54=x16 means that  5⋅16=4⋅x 54=20x means that  5⋅x=4⋅20 x4=2016 means that  16⋅x=4⋅20 4x=1620 means that  4⋅20=16⋅x 54=x16 means that  5⋅16=4⋅x 54=20x means that  5⋅x=4⋅20

Each of these statements is a multiplication statement. Specifically, each is a missing factor statement. (The letter used here is xx size 12{x} {}, whereas MM size 12{M} {} was used in (Reference).)

Finding the Missing Factor in a Proportion

The missing factor in a missing factor statement can be determined by dividing the product by the known factor, that is, if xx size 12{x} {} represents the missing factor, then
x = ( product ) ÷ ( known factor ) x = ( product ) ÷ ( known factor ) size 12{x= \( "product" \) div \( "known factor" \) } {}

Sample Set B

Find the unknown number in each proportion.

Example 5

x4=2016x4=2016 size 12{ { {x} over {4} } = { {"20"} over {"16"} } } {}. Find the cross product.

16⋅x = 20⋅4 16⋅x = 80 Divide the product 80 by the known factor 16. x = 80 16 x = 5 The unknown number is 5. 16⋅x = 20⋅4 16⋅x = 80 Divide the product 80 by the known factor 16. x = 80 16 x = 5 The unknown number is 5.

This mean that 54=201654=2016 size 12{ { {5} over {4} } = { {"20"} over {"16"} } } {}, or 5 is to 4 as 20 is to 16.

Example 6

5x=20165x=2016 size 12{ { {5} over {x} } = { {"20"} over {"16"} } } {}. Find the cross product.

5⋅16 = 20⋅x 80 = 20⋅x Divide the product 80 by the known factor 20. 80 20 = x 4 = x The unknown number is 4. 5⋅16 = 20⋅x 80 = 20⋅x Divide the product 80 by the known factor 20. 80 20 = x 4 = x The unknown number is 4.

This means that 54=201654=2016 size 12{ { {5} over {4} } = { {"20"} over {"16"} } } {}, or, 5 is to 4 as 20 is to 6.

Example 7

163=64x163=64x size 12{ { {"16"} over {3} } = { {"64"} over {x} } } {} Find the cross product.

16⋅x = 64⋅3 16⋅x = 192 Divide 192 by 16. x = 192 16 x = 12 The unknown number is 12. 16⋅x = 64⋅3 16⋅x = 192 Divide 192 by 16. x = 192 16 x = 12 The unknown number is 12.

The means that 163=6412163=6412 size 12{ { {"16"} over {3} } = { {"64"} over {"12"} } } {}, or, 16 is to 3 as 64 is to 12.

Example 8

98=x4098=x40 size 12{ { {9} over {8} } = { {x} over {"40"} } } {} Find the cross products.

9⋅40 = 8⋅x 360 = 8⋅x Divide 360 by 8. 360 8 = x 45 = x The unknown number is 45. 9⋅40 = 8⋅x 360 = 8⋅x Divide 360 by 8. 360 8 = x 45 = x The unknown number is 45.

Practice Set B

Find the unknown number in each proportion.

Exercise 5

x8=1232x8=1232 size 12{ { {x} over {8} } = { {"12"} over {"32"} } } {}

Solution

x=3x=3 size 12{x=3} {}

Exercise 6

7x=14107x=1410 size 12{ { {7} over {x} } = { {"14"} over {"10"} } } {}

Solution

x=5x=5 size 12{x=5} {}

Exercise 7

911=x55911=x55 size 12{ { {9} over {"11"} } = { {x} over {"55"} } } {}

Solution

x=45x=45 size 12{x="45"} {}

Exercise 8

16=8x16=8x size 12{ { {1} over {6} } = { {8} over {x} } } {}

Solution

x=48x=48 size 12{x="48"} {}

Proportions Involving Rates

Recall that a rate is a comparison, by division, of unlike denominate numbers. We must be careful when setting up proportions that involve rates. The form is impor­tant. For example, if a rate involves two types of units, say unit type 1 and unit type 2, we can write

unit type 1 over unit type 2 equals unit type 1 over unit type 2. The same units appear on the same side, in this case, the same units are part of the same fraction.

or

unit type 1 over unit type 2 equals unit type 1 over unit type 2. The same units appear on the same side, in this case, the same unit is in both denominators and the same unit is in both numerators.

Both cross products produce a statement of the type

unit type 1 ⋅ unit type 2 = unit type 1 ⋅ unit type 2 unit type 1 ⋅ unit type 2 = unit type 1 ⋅ unit type 2 size 12{ left ("unit type 1" right ) cdot left ("unit type 2" right )= left ("unit type 1" right ) cdot left ("unit type 2" right )} {}

which we take to mean the comparison

A comparison of types of units.

Examples of correctly expressed proportions are the following:

Two proportions. The first is miles over hr equals miles over hour, where the same unit is always in the denominator. The second is miles over miles equals hours over hours, where the same unit is in its own fraction.

However, if we write the same type of units on different sides, such as,

unit type 1 unit type 2 = unit type 2 unit type 1 unit type 1 unit type 2 = unit type 2 unit type 1 size 12{ { {"unit type 1"} over {"unit type 2"} } = { {"unit type 2"} over {"unit type 1"} } } {}

the cross product produces a statement of the form

A chart showing the comparison of different unit types.

We can see that this is an incorrect comparison by observing the following example: It is incorrect to write

2 hooks 3 poles = 6 poles 4 hooks 2 hooks 3 poles = 6 poles 4 hooks size 12{ { {"2 hooks"} over {"3 poles"} } = { {"6 poles"} over {"4 hooks"} } } {}

for two reason.

  1. The cross product is numerically wrong: 2⋅4≠3⋅62⋅4≠3⋅6 size 12{ left (2 cdot 4 <> 3 cdot 6 right )} {}.
  2. The cross product produces the statement “hooks are to hooks as poles are to poles,” which makes no sense.

Exercises

Exercise 9

A statement that two ratios or
are equal is called a
.

Solution

rates, proportion

For the following 9 problems, write each proportion in fractional form.

Exercise 10

3 is to 7 as 18 is to 42.

Exercise 11

1 is to 11 as 3 is to 33.

Solution

111=333111=333 size 12{ { {1} over {"11"} } = { {3} over {"33"} } } {}

Exercise 12

9 is to 14 as 27 is to 42.

Exercise 13

6 is to 90 as 3 is to 45.

Solution

690=345690=345 size 12{ { {6} over {"90"} } = { {3} over {"45"} } } {}

Exercise 14

5 liters is to 1 bottle as 20 liters is to 4 bottles.

Exercise 15

18 grams of cobalt is to 10 grams of silver as 36 grams of cobalt is to 20 grams of silver.

Solution

18  gr cobalt10  gr silver=36  gr cobalt20  gr silver18  gr cobalt10  gr silver=36  gr cobalt20  gr silver size 12{ { {"18"" gr cobalt"} over {"10"" gr silver"} } = { {"36"" gr cobalt"} over {"20"" gr silver"} } } {}

Exercise 16

4 cups of water is to 1 cup of sugar as 32 cups of water is to 8 cups of sugar.

Exercise 17

3 people absent is to 31 people present as 15 peo­ple absent is to 155 people present.

Solution

3  people absent31 people present=15  people absent155  people present3  people absent31 people present=15  people absent155  people present size 12{ { {3" people absent"} over {"31 people present"} } = { {"15"" people absent"} over {"155"" people present"} } } {}

Exercise 18

6 dollars is to 1 hour as 90 dollars is to 15 hours.

For the following 10 problems, write each proportion as a sentence.

Exercise 19

34=152034=1520 size 12{ { {3} over {4} } = { {"15"} over {"20"} } } {}

Solution

3 is to 4 as 15 is to 20

Exercise 20

18=54018=540 size 12{ { {1} over {8} } = { {5} over {"40"} } } {}

Exercise 21

3 joggers100  feet=6 joggers200   feet3 joggers100  feet=6 joggers200   feet size 12{ { {"3 joggers"} over {"100"" feet"} } = { {"6 joggers"} over {"200"" feet"} } } {}

Solution

3 joggers are to 100 feet as 6 joggers are to 200 feet

Exercise 22

12 marshmallows3  sticks=36 marshmallows9  sticks12 marshmallows3  sticks=36 marshmallows9  sticks size 12{ { {"12 marshmallows"} over {3" sticks"} } = { {"36 marshmallows"} over {9" sticks"} } } {}

Exercise 23

40 miles80 miles=2 gallons4 gallons40 miles80 miles=2 gallons4 gallons size 12{ { {"40 miles"} over {"80 miles"} } = { {"2 gallons"} over {"5 gallons"} } } {}

Solution

40 miles are to 80 miles as 2 gallons are to 4 gallons

Exercise 24

4 couches10  couches=2 houses5 houses4 couches10  couches=2 houses5 houses size 12{ { {"4 couches"} over {"10"" couches"} } = { {"2 houses"} over {"5 houses"} } } {}

Exercise 25

1 person1 job=8 people8 jobs1 person1 job=8 people8 jobs size 12{ { {"1 person"} over {"1 job"} } = { {"8 people"} over {"8 jobs"} } } {}

Solution

1 person is to 1 job as 8 people are to 8 jobs

Exercise 26

1 popsicle2  children=12 popsicle1 child1 popsicle2  children=12 popsicle1 child size 12{ { {"1 popsicle"} over {2" children"} } = { { { {1} over {2} } " popsicle"} over {"1 child"} } } {}

Exercise 27

2,000 pounds1  ton=60,000 pounds30  tons2,000 pounds1  ton=60,000 pounds30  tons size 12{ { {"2,000 pounds"} over {1" ton"} } = { {"60,000 pounds"} over {"30"" tons"} } } {}

Solution

2,000 pounds are to 1 ton as 60,000 pounds are to 30 tons

Exercise 28

1  table5 tables=2 people10 people1  table5 tables=2 people10 people size 12{ { {1" table"} over {"5 table"} } = { {"2 people"} over {"10 people"} } } {}

For the following 10 problems, solve each proportion.

Exercise 29

x5=615x5=615 size 12{ { {x} over {5} } = { {6} over {"15"} } } {}

Solution

x=2x=2 size 12{x=2} {}

Exercise 30

x10=2840x10=2840 size 12{ { {x} over {"10"} } = { {"28"} over {"40"} } } {}

Exercise 31

5x=10165x=1016 size 12{ { {5} over {x} } = { {"10"} over {"16"} } } {}

Solution

x=8x=8 size 12{x=8} {}

Exercise 32

13x=396013x=3960 size 12{ { {"13"} over {x} } = { {"39"} over {"60"} } } {}

Exercise 33

13=x2413=x24 size 12{ { {1} over {3} } = { {x} over {"24"} } } {}

Solution

x=8x=8 size 12{x=8} {}

Exercise 34

712=x60712=x60 size 12{ { {7} over {"12"} } = { {x} over {"60"} } } {}

Exercise 35

83=72x83=72x size 12{ { {8} over {3} } = { {"72"} over {x} } } {}

Solution

x=27x=27 size 12{x="27"} {}

Exercise 36

161=48x161=48x size 12{ { {"16"} over {1} } = { {"48"} over {x} } } {}

Exercise 37

x25=200125x25=200125 size 12{ { {x} over {"25"} } = { {"200"} over {"125"} } } {}

Solution

x=40x=40 size 12{x="40"} {}

Exercise 38

6530=x606530=x60 size 12{ { {"65"} over {"30"} } = { {x} over {"60"} } } {}

For the following 5 problems, express each sentence as a proportion then solve the proportion.

Exercise 39

5 hats are to 4 coats as xx size 12{x} {} hats are to 24 coats.

Solution

x=30x=30 size 12{x="30"} {}

Exercise 40

xx size 12{x} {} cushions are to 2 sofas as 24 cushions are to 16 sofas.

Exercise 41

1 spacecraft is to 7 astronauts as 5 spacecraft are to xx size 12{x} {} astronauts.

Solution

x=35x=35 size 12{x="35"} {}

Exercise 42

56 microchips are to x circuit boards as 168 microchips are to 3 circuit boards.

Exercise 43

18 calculators are to 90 calculators as xx size 12{x} {} students are to 150 students.

Solution

x=30x=30 size 12{x="30"} {}

Exercise 44

xx size 12{x} {} dollars are to $40,000 as 2 sacks are to 1 sack.

Indicate whether the proportion is true or false.

Exercise 45

316=1264316=1264 size 12{ { {3} over {"16"} } = { {"12"} over {"64"} } } {}

Solution

true

Exercise 46

215=1075215=1075 size 12{ { {2} over {"15"} } = { {"10"} over {"75"} } } {}

Exercise 47

19=33019=330 size 12{ { {1} over {9} } = { {3} over {"30"} } } {}

Solution

false

Exercise 48

6 knives7 forks=12 knives15  forks6 knives7 forks=12 knives15  forks size 12{ { {"6 knives"} over {"7 forks"} } = { {"12 knives"} over {"15"" forks"} } } {}

Exercise 49

33 miles1 gallon=99 miles3 gallons33 miles1 gallon=99 miles3 gallons size 12{ { {"33 miles"} over {"1 gallon"} } = { {"99 miles"} over {"3 gallons"} } } {}

Solution

true

Exercise 50

320 feet5 seconds=65 feet1 second320 feet5 seconds=65 feet1 second size 12{ { {"320 feet"} over {"5 seconds"} } = { {"65 feet"} over {"1 second"} } } {}

Exercise 51

35 students70 students=1 class2 classes35 students70 students=1 class2 classes size 12{ { {"35 students"} over {"70 students"} } = { {"1 class"} over {"2 classes"} } } {}

Solution

true

Exercise 52

9 ml chloride45 ml chloride=1 test tube7  test tubes9 ml chloride45 ml chloride=1 test tube7  test tubes size 12{ { {"9 ml chloride"} over {"45"" ml chloride"} } = { {"1 test tube"} over {7" test tubes"} } } {}

Exercises for Review

Exercise 53

((Reference)) Use the number 5 and 7 to illustrate the commutative property of addition.

Solution

5+7=127+5=125+7=127+5=12alignl { stack { size 12{5+7="12"} {} # size 12{7+5="12"} {} } } {}

Exercise 54

((Reference)) Use the numbers 5 and 7 to illustrate the commutative property of multiplication.

Exercise 55

((Reference)) Find the difference. 514−322514−322 size 12{ { {5} over {"14"} } - { {3} over {"22"} } } {}.

Solution

17771777 size 12{ { {"17"} over {"77"} } } {}

Exercise 56

((Reference)) Find the product. 8.06129⋅1,0008.06129⋅1,000 size 12{8 "." "06129" cdot 1,"000"} {}.

Exercise 57

((Reference)) Write the simplified fractional form of the rate “sixteen sentences to two paragraphs.”

Solution

8  sentences1  paragraph8  sentences1  paragraph size 12{ { {8" sentences"} over {1" paragraph"} } } {}

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