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Chapter 2.4 Rates, Ratios, and Proportions A ratio compares two quantities. Slide 5.1- 2 Parallel Example 1 Writing Ratios Tamara spent $13 on fish, $8 on salad and $7 on bread. Write each ratio as a fraction. a. Ratio of amount spent on salad to amount spent on bread. $8 8 The ratio of salad to bread is $7 Numerator Denominator (mentioned first) (mentioned second) 7 b. Ratio of fish to bread. $13 13 7 $7 Slide 5.1- 3 Parallel Example 2 Writing Ratios in Lowest Terms Write each ratio in lowest terms. a. 80 days to 20 days. Divide the numerator and denominator by 20. 80 80 20 4 20 20 20 1 b. 30 ounces of medicine to 140 ounces of medicine 30 10 3 30 140 140 10 14 Slide 5.1- 4 Parallel Example 3 Using Decimal Numbers in a Ratio The price of a bag of dog food increased from $22.95 to $25.50. Find the ratio of the increase in price to the original price. new price – original price = increase $25.50 $22.95 = $2.55 Find the ratio of the increase in price to the original price. 2.55 increase in price 22.95 original price Now write the ratio as a ratio of whole numbers. 255 255 1 2.55 2.55 100 255 22.95 22.95 100 2295 2295 255 9 Slide 5.1- 5 Parallel Example 4 Using Mixed Numbers in Ratios Write each ratio as a comparison of whole numbers in lowest terms. 1 a. 4 days to 4 days 4 Write the ratio and divide out common units. 4 days 4 41 4 1 days 4 4 Write as improper fractions. 4 4 4 17 4 4 16 1 17 1 4 1 17 17 4 41 4 Reciprocals Slide 5.1- 6 Parallel Example 4 Using Mixed Numbers in Ratios b. 4 3 pounds to 2 1 pounds 8 4 3 35 4 8 8 4 38 2 41 1 9 2 4 4 35 8 9 4 1 35 9 35 4 35 4 35 8 4 8 9 82 9 18 Slide 5.1- 7 Parallel Example 1 Writing Rates in Lowest Terms Write each rate as a fraction in lowest terms. a. 8 gallons of antifreeze for $40. 8 gallons 8 1 gallon 40 dollars 8 5 dollars b. 192 calories in 6 ounces of yogurt 192 calories 6 32 calories 6 ounces 6 1 ounce Slide 5.2- 8 Parallel Example 1 continued Writing Rates in Lowest Terms Write each rate as a fraction in lowest terms. c. 84 hamburgers on 7 grills. 84 hamburgers 7 12 hamburgers 7 grills 7 1 grill Slide 5.2- 9 When the denominator of a rate is 1, it is called a unit rate. For example, you earn $16.25 for 1 hour of work. This unit rate is written: $16.25 per hour Use per or a slash mark (/) when writing unit rates. Slide 5.2- 10 Parallel Example 2 Finding Unit Rates Find each unit rate. a. 445.5 miles on 16.5 gallons of gas 445.5 miles 16.5 gallons Divide to find the unit rate. 27 16.5 445.5 445.5 miles 16.5 27 miles 16.5 gallons 16.5 1 gallon The unit rate is 27 miles per gallon or 27 miles/gallon. Slide 5.2- 11 Parallel Example 2 continued Finding Unit Rates Find each unit rate. b. 413 feet in 14 seconds 413 feet 14 seconds Divide to find the unit rate. 29.5 14 413.0 The unit rate is 29.5 feet/second. Slide 5.2- 12 Parallel Example 3 Determining the Best Buy A local store charges the following prices for jars of jelly. $3.69 $3.09 $2.39 18 oz. 24 oz. 28 oz. The best buy is the container with the lowest cost per unit. All the jars are measured in ounces. Find the cost per ounce for each one by dividing the price of the jar by the number of ounces in it. Round to the nearest thousandth if necessary. Slide 5.2- 13 Parallel Example 3 continued Determining the Best Buy Size Cost per Unit (rounded) 18 ounces $2.39 $0.133 per ounce 18 ounces $3.09 $0.129 per ounce 24 ounces 24 ounces 28 ounces highest lowest $3.69 $0.132 per ounce 28 ounces The lowest cost per ounce is $0.129, so the 24-ounce jar is the best buy. Slide 5.2- 14 Parallel Example 4 Solving Best Buy Applications Juice is sold as a concentrated can as well as in a ready to serve carton. Which of the choices below is the best buy? 12 oz can makes 48 ounces of juice for $1.69 60 oz carton for $2.59 To determine the best buy, divide the cost by the number of ounces. Slide 5.2- 15 Parallel Example 4 continued Solving Best Buy Applications 12 oz can makes 48 ounces of juice for $1.69 60 oz carton for $2.59 Concentrate $1.69 48 ounces $0.0352 per ounce Carton $2.59 60 ounces $0.0432 per ounce Although, you must mix it yourself, the concentrated can of juice is the better buy. Slide 5.2- 16 Four numbers are used in a proportion. If any three of these numbers are known, the fourth can be found. Slide 5.4- 17 Parallel Example 1 Solving Proportions for Unknown Numbers Find the unknown number in each proportion. Round answers to the nearest hundredth when necessary. a. 30 48 x 40 Ratios can be written in lowest terms. You can do that before finding the cross products. 6 48 can be written in lowest terms as , 5 40 which gives the proportion 30 6 . x 5 Slide 5.4- 18 Parallel Example 1 continued Solving Proportions for Unknown Numbers 30 6 x 5 Step 1 Step 2 x 6 30 5 x 6 Find the cross products 30 5 Show that the cross products are equivalent. x 6 150 1 Step 3 x 6 150 6 6 1 x 25 Slide 5.4- 19 Parallel Example 1 continued Solving Proportions for Unknown Numbers 3 20 b. 7 x 3 20 7 x Step 1 3 x 140 Step 2 7 20 Find the cross products 3x Show that the cross products are equivalent. 1 Step 3 3 x 140 3 3 1 x 46.67 Rounded to the nearest hundredth. Slide 5.4- 20 Parallel Example 2 Solving Proportions with Mixed Numbers and Decimals Find the unknown number in each proportion. 6 32 x 8x a. 6 32 x 8 36 Find the cross products 8 36 2 6 36 3 2 Find 6 36. 3 12 2 20 36 240 240 6 36 3 3 1 1 1 Show the cross products are equivalent. 8 x 240 Divide both sides by 8. 8 x 240 8 8 x 30 Slide 5.4- 21 Parallel Example 2 continued 2 3 6 30 8 36 Solving Proportions with Mixed Numbers and Decimals 8 30 240 Equal 2 6 36 240 3 The cross products are equal, so 30 is the correct solution. Slide 5.4- 22 Parallel Example 2 continued Solving Proportions with Mixed Numbers and Decimals Find the unknown number in each proportion. 10.4 6.76 b. 12.4 x Show that cross products are equivalent. 10.4 ( x) (12.4)(6.76) 10.4 ( x) 83.824 10.4 x 83.824 10.4 Divide both sides by 10.4. 10.4 x 8.06 Slide 5.4- 23 Parallel Example 2 continued Solving Proportions with Mixed Numbers and Decimals 10.4 6.76 12.4 8.06 10.4 ∙ 8.06 = 83.824 Equal 12.4 ∙ 6.76 = 83.824 The cross products are equal, so 8.06 is the correct solution. Slide 5.4- 24 Similar Triangles • Similar Triangles whose angles have the same measure, but their sides have different lengths. • The triangles will look identical, but one will be smaller than the other. Slide 1- 25 • = = x y Slide 1- 26 Slide 1- 27 Slide 1- 28 Solution ℎ ℎ ℎ = ℎ ℎ ℎ Slide 1- 29 Hw section 2.4 1-20 Try 21 Slide 1- 30