Ratios and Rates: Proportions *

Transcription

1 OpenSta-CNX module: m3498 Ratios and Rates: Proportions * Wade Ellis Denny Burzynski This work is produced by OpenSta-CNX and licensed under the Creative Commons Attribution License 3.0 Abstract 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 nd 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 2 Ratios, Rates, and Proportions Ratio, Rate We have dened a ratio as a comparison, by division, of two pure numbers or two like denominate numbers. We have dened 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 eamples show how to read proportions. 2. Sample Set A Write or read each proportion. Eample 3 2 * Version.2: Aug 8, 0 8:49 pm

2 OpenSta-CNX module: m is to as 2 is to Eample 2 0 items 2 items dollars dollar 0 items is to dollars as 2 items is to dollar Eample 3 8 is to 2 as is to Eample 4 0 milligrams of vitamin C is to tablet as 300 milligrams of vitamin C is to 6 tablets Practice Set A Write or read each proportion. Eercise (Solution on p. 8.) Eercise 2 (Solution on p. 8.) 2 people window 0 people windows Eercise 3 (Solution on p. 8.) is to 4 as 7 is to. Eercise 4 (Solution on p. 8.) 2 plates are to tray as plates are to 0 trays. 3 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 specied numbers and one unknown number. It is customary to let a letter, such as, represent the unknown number. An eample of such a proportion is 4 This proportion is read as " is to 4 as is to." 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 eample, Notice that in a proportion that contains three specied 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. Specically, it is a missing factor statement. (See for a discussion of multiplication statements.) For eample,

3 OpenSta-CNX module: m means that 4 means that 4 4 means that 4 4 means that 4 Each of these statements is a multiplication statement. Specically, each is a missing factor statement. (The letter used here is, whereas M was used in.) 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 represents the missing factor, then (product) (known factor) 3. Sample Set B Find the unknown number in each proportion. Eample. Find the cross product Divide the product 80 by the known factor. 80 The unknown number is. This mean that 4, or is to 4 as is to. Eample 6. Find the cross product. 80 Divide the product 80 by the known factor The unknown number is 4. This means that 4, or, is to 4 as is to 6. Eample 7 Find the cross product Divide 92 by The unknown number is 2. The means that 3 64, or, is to 3 as 64 is to 2. 2 Eample 8 Find the cross products Divide 360 by The unknown number is 4.

4 OpenSta-CNX module: m Practice Set B Find the unknown number in each proportion. Eercise (Solution on p. 8.) Eercise 6 (Solution on p. 8.) Eercise 7 (Solution on p. 8.) 9 Eercise 8 (Solution on p. 8.) 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 eample, if a rate involves two types of units, say unit type and unit type 2, we can write or Both cross products produce a statement of the type (unit type ) (unit type 2) (unit type ) (unit type 2) which we take to mean the comparison Eamples of correctly epressed proportions are the following:

5 OpenSta-CNX module: m3498 However, if we write the same type of units on dierent sides, such as, unit type unit type 2 unit type 2 unit type the cross product produces a statement of the form We can see that this is an incorrect comparison by observing the following eample: It is incorrect to write 2 hooks 6 poles 3 poles 4 hooks for two reason.. The cross product is numerically wrong: ( ). 2. The cross product produces the statement hooks are to hooks as poles are to poles, which makes no sense. Eercises Eercise 9 (Solution on p. 8.) A statement that two ratios or are equal is called a. For the following 9 problems, write each proportion in fractional form. Eercise 0 3 is to 7 as 8 is to 42. Eercise (Solution on p. 8.) is to as 3 is to 33. Eercise 2 9 is to 4 as 27 is to 42. Eercise 3 (Solution on p. 8.) 6 is to 90 as 3 is to 4. Eercise 4 liters is to bottle as liters is to 4 bottles. Eercise (Solution on p. 8.) 8 grams of cobalt is to 0 grams of silver as 36 grams of cobalt is to grams of silver. Eercise 4 cups of water is to cup of sugar as 32 cups of water is to 8 cups of sugar. Eercise 7 (Solution on p. 8.) 3 people absent is to 3 people present as people absent is to people present. Eercise 8 6 dollars is to hour as 90 dollars is to hours. For the following 0 problems, write each proportion as a sentence. Eercise 9 (Solution on p. 8.) 3 4 Eercise 8 40 Eercise 2 (Solution on p. 8.) 3 joggers 00 feet 6 joggers 0 feet

6 OpenSta-CNX module: m Eercise 22 2 marshmallows 3 sticks 36 marshmallows 9 sticks Eercise 23 (Solution on p. 8.) 40 miles 2 gallons 80 miles 4 gallons Eercise 24 4 couches 0 couches houses Eercise 2 (Solution on p. 8.) person 8 people job 8 jobs Eercise 26 popsicle 2 2 popsicle children child Eercise 27 (Solution on p. 8.) 2,000 pounds ton 60,000 pounds 30 tons Eercise 28 table tables 0 people For the following 0 problems, solve each proportion. Eercise 29 (Solution on p. 8.) 6 Eercise Eercise 3 (Solution on p. 8.) 0 Eercise Eercise 33 (Solution on p. 8.) 3 24 Eercise Eercise 3 (Solution on p. 8.) Eercise Eercise 37 (Solution on p. 8.) 2 2 Eercise For the following problems, epress each sentence as a proportion then solve the proportion. Eercise 39 (Solution on p. 8.) hats are to 4 coats as hats are to 24 coats. Eercise 40 cushions are to 2 sofas as 24 cushions are to sofas. Eercise 4 (Solution on p. 8.) spacecraft is to 7 astronauts as spacecraft are to astronauts. Eercise 42 6 microchips are to circuit boards as 8 microchips are to 3 circuit boards.

7 OpenSta-CNX module: m Eercise 43 (Solution on p. 9.) 8 calculators are to 90 calculators as students are to 0 students. Eercise 44 dollars are to $40,000 as 2 sacks are to sack. Indicate whether the proportion is true or false. Eercise 4 (Solution on p. 9.) Eercise Eercise 47 (Solution on p. 9.) Eercise 48 6 knives 7 forks 2 knives forks Eercise 49 (Solution on p. 9.) 33 miles gallon Eercise 0 3 feet seconds 99 miles 3 gallons 6 feet second Eercise (Solution on p. 9.) 3 students class 70 students 2 classes Eercise 2 9 ml chloride test tube 4 ml chloride 7 test tubes. Eercises for Review Eercise 3 (Solution on p. 9.) () Use the number and 7 to illustrate the commutative property of addition. Eercise 4 () Use the numbers and 7 to illustrate the commutative property of multiplication. Eercise (Solution on p. 9.) () Find the dierence Eercise 6 () Find the product , 000. Eercise 7 (Solution on p. 9.) () Write the simplied fractional form of the rate siteen sentences to two paragraphs.

8 OpenSta-CNX module: m Solutions to Eercises in this Module Solution to Eercise (p. 2) 3 is to 8 as 6 is to Solution to Eercise (p. 2) 2 people are to window as 0 people are to windows Solution to Eercise (p. 2) 4 7 Solution to Eercise (p. 2) 2 plates tray plates 0 trays Solution to Eercise (p. 4) 3 Solution to Eercise (p. 4) Solution to Eercise (p. 4) 4 Solution to Eercise (p. 4) 48 Solution to Eercise (p. ) rates, proportion Solution to Eercise (p. ) 3 33 Solution to Eercise (p. ) Solution to Eercise (p. ) 8 gr cobalt 0 gr silver 36 gr cobalt gr silver Solution to Eercise (p. ) 3 people absent 3 people present people absent people present Solution to Eercise (p. ) 3 is to 4 as is to Solution to Eercise (p. ) 3 joggers are to 00 feet as 6 joggers are to 0 feet 40 miles are to 80 miles as 2 gallons are to 4 gallons person is to job as 8 people are to 8 jobs 2,000 pounds are to ton as 60,000 pounds are to 30 tons

9 OpenSta-CNX module: m Solution to Eercise (p. 7) 30 Solution to Eercise (p. 7) true Solution to Eercise (p. 7) false Solution to Eercise (p. 7) true Solution to Eercise (p. 7) true Solution to Eercise (p. 7) Solution to Eercise (p. 7) 7 77 Solution to Eercise (p. 7) 8 sentences paragraph