Chapter 7Chapter 7Proportional ReasoningProportional Reasoning

Section 7.2Section 7.2

Proportional Variation andProportional Variation and

Solving ProportionsSolving Proportions

IntrodutionIntrodution A bookstore runs a A bookstore runs a

clearance sale on its clearance sale on its paperback books, paperback books, advertising them as $3 advertising them as $3 each. each.

Complete the table Complete the table below in order to below in order to determine the cost (determine the cost (yy) ) when when xx books are books are purchased.purchased.

Determine the ratio Determine the ratio between between xx and and yy for for each pair of values. each pair of values. What do you notice?What do you notice?

xx yy

22

55

99

1414

Direct ProportionalityDirect Proportionality

Two quantities Two quantities vary proportionallyvary proportionally iff, as iff, as their corresponding values increase or their corresponding values increase or decrease, the ratios of the two quantities decrease, the ratios of the two quantities are always equivalent.are always equivalent.

Multiplicative Property of Quantities that Multiplicative Property of Quantities that Vary ProportionallyVary ProportionallyWhen quantities When quantities aa and and bb vary vary proportionally, a nonzero number proportionally, a nonzero number kk exists, for all corresponding values exists, for all corresponding values aa and and b,b, such that such that

aa = = kk, or , or aa = = b b • • k.k. bb

This type of proportional variation is This type of proportional variation is known as known as direct proportionalitydirect proportionality..

ExampleExample

If 2.8 inches of rain had fallen in If 2.8 inches of rain had fallen in 10 hours, how much would have 10 hours, how much would have accumulated at the end of 3 accumulated at the end of 3 hours?hours?

Proportions and Similar Proportions and Similar FiguresFigures

A A proportionproportion is an equation stating is an equation stating that two ratios are equivalent.that two ratios are equivalent.

Similar figuresSimilar figures are geometric are geometric shapes that have the same shape, shapes that have the same shape, but not necessarily the same size. but not necessarily the same size. Their corresponding angles are Their corresponding angles are congruent and congruent and theirtheir corresponding corresponding sides are proportionalsides are proportional..

ExamplesExamples

1.) On a blueprint, the dimensions of a 1.) On a blueprint, the dimensions of a room are 1 ½ inches by 2 ¾ inches. room are 1 ½ inches by 2 ¾ inches. If the scale is 1/16 in. to 1 foot, what If the scale is 1/16 in. to 1 foot, what are the actual dimensions of the are the actual dimensions of the room?room?

2.) At a certain time of day, a tree casts 2.) At a certain time of day, a tree casts a 25-foot shadow. At the same time, a 25-foot shadow. At the same time, a 6-foot tall man casts a 5-foot a 6-foot tall man casts a 5-foot shadow. Find the height of the tree.shadow. Find the height of the tree.

Properties of ProportionsProperties of Proportions

Cross-Product Property of Cross-Product Property of ProportionsProportions

Reciprocal Property of ProportionsReciprocal Property of Proportions

ExampleExample

A tennis magazine averages about A tennis magazine averages about 150 pages per issue. There are 150 pages per issue. There are seven ads for every 3 pages. How seven ads for every 3 pages. How many ads would you expect in a many ads would you expect in a typical issue?typical issue?

Set up a proportion to solve.Set up a proportion to solve.Re-write the proportion using the Re-write the proportion using the

Reciprocal Property.Reciprocal Property.Using the Cross-Product Property, Using the Cross-Product Property,

solve each proportion and verify the solve each proportion and verify the two proportions are equivalent.two proportions are equivalent.