Chapter 7 Proportional Reasoning Section 7.2 Proportional Variation and Solving Proportions

Introdution A bookstore runs a clearance sale on its paperback books, advertising them as $3 each. A bookstore runs a clearance sale on its paperback books, advertising them as $3 each. Complete the table below in order to determine the cost (y) when x books are purchased. Complete the table below in order to determine the cost (y) when x books are purchased. Determine the ratio between x and y for each pair of values. What do you notice? Determine the ratio between x and y for each pair of values. What do you notice? xy

Direct Proportionality Two quantities vary proportionally iff, as their corresponding values increase or decrease, the ratios of the two quantities are always equivalent. Two quantities vary proportionally iff, as their corresponding values increase or decrease, the ratios of the two quantities are always equivalent. Multiplicative Property of Quantities that Vary Proportionally Multiplicative Property of Quantities that Vary Proportionally When quantities a and b vary proportionally, a nonzero number k exists, for all corresponding values a and b, such that a = k, or a = b k. a = k, or a = b k. b This type of proportional variation is known as direct proportionality. This type of proportional variation is known as direct proportionality.

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

Proportions and Similar Figures A proportion is an equation stating that two ratios are equivalent. A proportion is an equation stating that two ratios are equivalent. Similar figures are geometric shapes that have the same shape, but not necessarily the same size. Their corresponding angles are congruent and their corresponding sides are proportional. Similar figures are geometric shapes that have the same shape, but not necessarily the same size. Their corresponding angles are congruent and their corresponding sides are proportional.

Examples 1.) On a blueprint, the dimensions of a room are 1 ½ inches by 2 ¾ inches. If the scale is 1/16 in. to 1 foot, what are the actual dimensions of the room? 2.) At a certain time of day, a tree casts a 25-foot shadow. At the same time, a 6- foot tall man casts a 5-foot shadow. Find the height of the tree.

Properties of Proportions Cross-Product Property of Proportions Cross-Product Property of Proportions Reciprocal Property of Proportions Reciprocal Property of Proportions

Example A tennis magazine averages about 150 pages per issue. There are seven ads for every 3 pages. How many ads would you expect in a typical issue? Set up a proportion to solve. Set up a proportion to solve. Re-write the proportion using the Reciprocal Property. Re-write the proportion using the Reciprocal Property. Using the Cross-Product Property, solve each proportion and verify the two proportions are equivalent. Using the Cross-Product Property, solve each proportion and verify the two proportions are equivalent.