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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, 21 5 8 size 12{"21" { {5} over {8} } } {} , 2 5 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

Compare 8 miles and 3 miles by subtraction.

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.

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Compare 12 and 5 by subtraction.

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

This means that 12 is 7 more than 5.

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

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

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Compare 36 and 4 by division.

36 ÷ 4 = 9 size 12{"36" div 4=9} {}

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

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Compare 8 miles and 2 miles by division.

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).

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Compare 30 miles and 2 gallons by division.

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.

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Practice set a

Make the following comparisons and interpret each one.

Compare 10 diskettes to 2 diskettes by

  1. subtraction:
  2. division:

  1. 8 diskettes; 10 diskettes is 8 diskettes more than 2 diskettes.
  2. 5; 10 diskettes is 5 times as many diskettes as 2 diskettes.

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Compare, if possible, 16 bananas and 2 bags by

  1. subtraction:
  2. division:

  1. Comparison by subtraction makes no sense.
  2. 16 bananas 2 bags = 8 bananas bag , 8 bananas per bag.

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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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