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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.

3 5 = 12 20 size 12{ { {3} over {5} } = { {"12"} over {"20"} } } {}

3 is to 5 as 12 is to 20

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10 items 5 dollars = 2 items 1 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

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8 is to 12 as 16 is to 24.

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

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

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

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Practice set a

Write or read each proportion.

3 8 = 6 16 size 12{ { {3} over {8} } = { {6} over {"16"} } } {}

3 is to 8 as 6 is to 16

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

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

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15 is to 4 as 75 is to 20.

15 4 = 75 20 size 12{ { {"15"} over {4} } = { {"75"} over {"20"} } } {}

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

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

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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 x size 12{x} {} , represent the unknown number. An example of such a proportion is

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

This proportion is read as " x 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)

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

x 4 = 20 16 means that   16 x = 4 20 4 x = 16 20 means that   4 20 = 16 x 5 4 = x 16 means that   5 16 = 4 x 5 4 = 20 x 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 x size 12{x} {} , whereas M size 12{M} {} was used in [link] .)

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 x size 12{x} {} represents the missing factor, then
x = ( product ) ÷ ( known factor ) size 12{x= \( "product" \) div \( "known factor" \) } {}

Sample set b

Find the unknown number in each proportion.

x 4 = 20 16 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.

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

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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