Ratios and Proportions, Percent of Change Objective: Compare ratios. Solve proportions. Find the percent of change. Solve problems involving percent of change.

Ratios and Proportions A ratio is a comparison of two numbers by division. The ratio of x to y can be expressed in the following ways. x to yx:y x / y An equation stating that two ratios are equal is called a proportion.

Example 1 Determine whether 7 / 8 and 49 / 56 are equivalent ratios. Write yes or no. Justify your answer. Yes they are equivalent because 49 / 56 reduces down to 7 / 8.

Proportions There are special names for the terms in a proportion. 1.5 and 1.2 are called the means. They are the middle terms of the proportion. 0.2 and 9.0 are called the extremes. They are the first and last terms of the proportion. 0.2 : 1.5 = 1.2 : 9.0

Means-Extremes Property of Proportion In a proportion, the product of the extremes is equal to the product of the means. If a / b = c / d and b, d ≠ 0, then ad = bc. Since 2 / 4 = 1 / 2, 2(2) = 4(1) or 4 = 4. Another way to determine whether two ratios form a proportion is to use cross products. If the cross products are equal, then the ratios form a proportion. This is the same as multiplying the means, and multiplying the extremes.

Example 2 Use cross products to determine whether each pair of ratios forms a proportion. a., 0.25(2) = 1.25(0.6) ? 0.5 ≠ 0.75 The ratios do not form a proportion.

Example 2 Use cross products to determine whether each pair of ratios forms a proportion. b., 2(20) = 16(2.5) ? 40 = 40 The ratios do form a proportion.

Solve Proportions To solve proportions use cross products.

Example 3 Solve each proportion. If necessary, round to the nearest hundredth. a.. b.. n(8) = 3(12) 8n = 36 8 n = 4.5 (x + 4)(4) = 3(12) 4x + 16 = x = 20 4 x = 5

Rate The ratio of two measurements having different units of measure is called a rate. For example, a price of $9.99 per 10 songs is a rate. A rate tells how many of one item is being compared to 1 of another item is called a unit rate.

Example 4 The ratio of a gear on a bicycle is 8:5. This means that for every 8 turns of the pedals, the wheel turns 5 times. Suppose the bicycle wheel turns about 2435 times during a trip. How many times would you have to crank the pedals during the trip? – Compare pedal turns to wheel turns 8(2435) = p(5) 19,480 = 5p = p You will need 3896 pedal turns.

Rate A rate called a scale is used to make a scale model of something too large or too small to be convenient at actual size.

Example 5 In a road atlas, the scale for the map of Connecticut is 5 inches = 41 miles. What is the distance in miles represented by 2 ½ inches on the map? – Compare inches and miles. 5(m) = 2.5(41) 5m = m = 20.5 The distance is 20 ½ miles.

Percent of Change Percent of change is the ratio of the change in an amount to the original amount expressed as a percent. If the new number is greater than the original number, the percent of change is a percent of increase. If the new number is less than the original number, the percent of change is a percent of decrease.

Example 6 Determine whether each percent of change is a percent of increase or a percent of decrease. Then find the percent of change. a.original: 32 new: – 32 = 8 8(100) = r(32) 800 = 32r = r Increase of 25%.

Example 6 Determine whether each percent of change is a percent of increase or a percent of decrease. Then find the percent of change. b.original: 20 new: 4 20 – 4 = 16 16(100) = r(20) 1600 = 20r = r Decrease of 80%.

Example 7 The price a used-book store pays to buy a book is $5. The store sells the book for 28% above the price that it pays for the book. What is the selling price of the book? – 28% is a percent of increase. – Therefore s – 5 represents the amount of change. (s – 5)(100) = 28(5) 100s – 500 = s = s = 6.4 The selling price is $6.40.

Solve Problems Two applications of percent of change are sales tax and discounts. Sales tax is an example of a percent of increase. Discount is an example of a percent of decrease.

Example 8 A meal for two at a restaurant costs $ If the sales tax is 5%, what is the total price of the meal? – 5% = = $1.64 is the tax = $34.39

Example 9 A dog toy is on sale for 20% off the original price. If the original price of the toy is $3.80, what is the discounted price? – 20% = = 0.76 $0.76 is the discount – 0.76 = $3.04

Check Your Progress Choose the best answer for the following. – A baseball cap is on sale for 15% off the original price. If the original price of the cap is $19.99, what is the discounted price? A.$9.99 B.$4.99 C.$16.99 D.$ % = = $3.00 is the discount – 3.00 =

Check Your Progress Choose the best answer for the following. Determine whether 5 / 6 and 40 / 49 are equivalent ratios. A.They are not equivalent ratios. B.They are equivalent ratios. C.Cannot be determined. does not reduce.

Check Your Progress Choose the best answer for the following. – Use cross products to determine whether the pair of ratios below forms a proportion. A.The ratios do form a proportion. B.The ratios do not form a proportion. C.Cannot be determined. 0.5(1.17) = 0.45(1.3) ? = 0.585

Check Your Progress Choose the best answer for the following. – Use cross products to determine whether the pair of ratios below forms a proportion. A.The ratios do form a proportion. B.The ratios do not form a proportion. C.Cannot be determined. 5(15) = 12(6) ? 75 ≠ 72

Check Your Progress Choose the best answer for the following. – Solve the proportion A.10 B.63 C.6.3 D.70 r(10) = 7(9) 10r = 63 10

Check Your Progress Choose the best answer for the following. – Solve the proportion A.6 B.10 C.-10 D.16 (x – 6)(8) = 5(16) 8x = x – 48 =

Check Your Progress Choose the best answer for the following. – Trent goes on a 30-mile bike ride every Saturday. He rides the distance in 4 hours. At this rate, how far can he ride in 6 hours? A.7.5 mi B.20 mi C.40 mi D.45 mi Compare miles and hours. 30(6) = m(4) 180 = 4m 4

Check Your Progress Choose the best answer for the following. – The scale on a map of the United States is 1 3 / 8 inches = 750 miles. The distance, on the map, between Los Angeles and Washington, D.C., is about 3 7 / 8 inches. What is the distance in miles between the two locations? A.About 750 miles B.About 1500 miles C.About 2000 miles D.About 2114 miles

Check Your Progress Choose the best answer for the following. – Determine whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change. original: 20 new: 18 A.increase of 10% B.decrease of 10% C.increase of 90% D.decrease of 90% 20 – 18 = 2 2(100) = r(20) 200 = 20r 20

Check Your Progress Choose the best answer for the following. – Determine whether the percent of change is a percent of increase or a percent of decrease. Then find the percent of change. original: 12 new: 48 A.increase of 300% B.decrease of 300% C.increase of 25% D.decrease of 25% 48 – 12 = 36 36(100) = r(12) 3600 = 12r 12

Check Your Progress Choose the best answer for the following. – At one store the price of a pair of jeans is $ At another store the same pair of jeans has a price that is 22% higher. What is the price of jeans at the second store? A.$38.00 B.$31.72 C.$25.00 D.$27.72 (p – 26)(100) = 22(26) 100p – 2600 = p =

Check Your Progress Choose the best answer for the following. – A portable CD player costs $ If the sales tax is 6.75%, what is the total price of the CD player? A.$64.27 B.$ C.$76.74 D.$ % = = $4.72 is the tax =