2 5 Solving Proportions Preview Warm Up California

2 -5 Solving Proportions Preview Warm Up California Standards Lesson Presentation

2 -5 Solving Proportions Warm Up Solve each equation. 1. 48 2. 5 m = 18 3. 6 Multiply. 3. 7 10 4. Change each percent to a decimal. 5. 73% 0. 73 6. 112% 7. 0. 6% 0. 006 8. 1% 0. 01 1. 12 Change each fraction to a decimal. 9. 0. 5 10. 0. 3

2 -5 Solving Proportions California Standards 15. 0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.

2 -5 Solving Proportions Vocabulary ratio rate cross products scale drawing proportion unit rate percent scale model

2 -5 Solving Proportions A ratio is a comparison of two quantities. The ratio of a to b can be written as a: b or , where b ≠ 0. A statement that two ratios are equal, such as is called a proportion.

2 -5 Solving Proportions Additional Example 1: Using Ratios The ratio of the number of bones in a human’s ears to the number of bones in the skull is 3: 11. There are 22 bones in the skull. How many bones are in the ears? Write a ratio comparing bones in ears to bones in skull. Write a proportion. Let x be the number of bones in ears. Since x is divided by 22, multiply both sides of the equation by 22. There are 6 bones in the ears.

2 -5 Solving Proportions Check It Out! Example 1 The ratio of red marbles to green marbles is 6: 5. There are 18 red marbles. How many green marbles are there? green red 5 6 Write a ratio comparing green to red marbles. Write a proportion. Let x be the number green marbles. Since x is divided by 18, multiply both sides by 18. 15 = x There are 15 green marbles.

2 -5 Solving Proportions A common application of proportions is rates. A rate is a ratio of two quantities with different units, such as Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as or 17 mi/gal. You can convert any rate to a unit rate.

2 -5 Solving Proportions Additional Example 2: Finding Unit Rates Ralf Laue of Germany flipped a pancake 416 times in 120 seconds to set the world record. Find the unit rate. Round your answer to the nearest hundredth. Write a proportion to find an equivalent ratio with a second quantity of 1. 3. 47 ≈ x Divide on the left side to find x. The unit rate is approximately 3. 47 pancake flips per second.

2 -5 Solving Proportions Check It Out! Example 2 a Find the unit rate. Round to the nearest hundredth if necessary. Cory earns $52. 50 in 7 hours. Write a proportion to find an equivalent ratio with a second quantity of 1. 7. 50 = x Divide on the left side to find x. The unit rate is $7. 50 per hour.

2 -5 Solving Proportions Check It Out! Example 2 b Find the unit rate. Round to the nearest hundredth if necessary. A machine seals 138 envelopes in 23 minutes. Write a proportion to find an equivalent ratio with a second quantity of 1. 6=x Divide on the left side to find x. The unit rate is 6 envelopes seals per minute.

2 -5 Solving Proportions 3. 6 Cross Products We will continue from here tomorrow!

2 -5 Solving Proportions In the proportion the products a d and b c are called cross products. You can solve a proportion for a missing value by using the Cross Products Property

2 -5 Solving Proportions Additional Example 3 A: Solving Proportions Solve the proportion. Use cross products. 3(m) = 9(5) 3 m = 45 Divide both sides by 3. m = 15

2 -5 Solving Proportions Additional Example 3 B: Solving Proportions Solve the proportion. Use cross products. 6(7) = 2(y – 3) 42 = 2 y – 6 +6 +6 48 = 2 y Add 6 to both sides. Divide both sides by 2. 24 = y

2 -5 Solving Proportions Check It Out! Example 3 a Solve the proportion. Check your answer. Use cross products. – 5(8) = 2(y) – 40 = 2 y Divide both sides by 2. – 20 = y

2 -5 Solving Proportions Check It Out! Example 3 a Continued Solve the proportion. Check your answer. Check Substitute – 20 for y. – 2. 5

2 -5 Solving Proportions Check It Out! Example 3 b Solve the proportion. Check your answer. Use cross product. 4(g + 3) = 5(7) 4 g + 12 = 35 – 12 4 g = 23 Subtract 12 from both sides. Divide both sides by 4. g = 5. 75

2 -5 Solving Proportions Check It Out! Example 3 b Continued Solve the proportion. Check your answer. Check Substitute 5. 75 for b. 1. 75

2 -5 Solving Proportions Another common application of proportions is percents. A percent is a ratio that compares a number to 100. For example, 25% = You can use the proportion find unknown values. to

2 -5 Solving Proportions Additional Example 4 A: Percent Problems Find 30% of 80. Method 1 Use a proportion. Use the percent proportion. Let x represent the part. 100 x = 2400 x = 24 30% of 80 is 24. Find the cross product. Since x is multiplied by 100, divide both sides to undo the multiplication.

2 -5 Solving Proportions Additional Example 4 B: Percent Problems 230 is what percent of 200? Method 2 Use an equation. 230 = x 200 230 = 200 x 1. 15 = x Write an equation. Let x represent the percent. Since x is multiplied by 200, divide both sides by 200 to undo the multiplication. The answer is a decimal. Write the decimal as a percent. This answer is reasonable; 230 is more than 100% of 200. 230 is 115% of 200. 115% = x

2 -5 Solving Proportions Additional Example 4 C: Percent Problems 20 is 0. 4% of what number? Method 1 Use a proportion. Use the percent proportion. Let x represent the whole. 2000 = 0. 4 x Cross multiply. Since x is multiplied by 0. 4, divide both sides by 0. 4. 5000 = x 20 is 0. 4% of 5000.

2 -5 Solving Proportions Check It Out! Example 4 a Find 20% of 60. Method 1 Use a proportion. Use the percent proportion. Let x represent the part. 100 x = 1200 x = 12 20% of 60 is 12. Find the cross product. Since x is multiplied by 100, divide both sides to undo the multiplication.

2 -5 Solving Proportions Check It Out! Example 4 b 48 is 15% of what number? Method 1 Use a proportion. Use the percent proportion. Let x represent the whole. 4800 = 15 x x = 320 48 is 15% of 320. Find the cross product. Since x is multiplied by 15, divide both sides by 15 to undo the multiplication.

2 -5 Solving Proportions are used to create scale drawings and scale models. A scale is a ratio between two sets of measurements, such as 1 in. : 5 mi. A scale drawing, or scale model, uses a scale to represent an object as smaller or larger than the actual object. A map is an example of a scale drawing.

2 -5 Solving Proportions Additional Example 5 A: Scale Drawings and Scale Models A contractor has a blueprint for a house drawn to the scale 1 in. : 3 ft. A wall on the blueprint is 6. 5 inches long. How long is the actual wall? Write the scale as a fraction. Let x be the actual length. Use cross products to solve. x 1= 3(6. 5) x = 19. 5 The actual length is 19. 5 feet.

2 -5 Solving Proportions Additional Example 5 B: Scale Drawings and Scale Models A contractor has a blueprint for a house drawn to the scale 1 in. : 3 ft. A wall in the house is 12 feet long. How long is the wall on the blueprint? Write the scale as a fraction. Let x be the blueprint length. Use cross products to solve. x 3 = 1(12) x=4 The blueprint length is 4 inches.

2 -5 Solving Proportions Reading Math A scale written without units, such as 32: 1, means that 32 units of any measure corresponds to 1 unit of that same measure.

2 -5 Solving Proportions Check It Out! Example 5 a The actual distance between North Chicago and Waukegan is 4 mi. What is the distance between these two locations on the map? Write the scale as a fraction. Let x be the map distance. 18 x = 4 x ≈ 0. 2 Use cross products to solve. The distance on the map is about 0. 2 in.

2 -5 Solving Proportions Check It Out! Example 5 b A scale model of a human heart is 16 ft long. The scale is 32: 1 How many inches long is the actual heart that the model represents? Write the scale as a fraction. Let x be the actual distance. 32 x = 16 Use cross products to solve. x = 0. 5 The actual heart is 0. 5 feet or 6 inches.

2 -5 Solving Proportions Lesson Quiz: Part l 1. In a school, the ratio of boys to girls is 4: 3. There are 216 boys. How many girls are there? 162 Find each unit rate. Round to the nearest hundredth if necessary. 2. Nuts cost $10. 75 for 3 pounds. $3. 58/lb 3. Sue washes 25 cars in 5 hours. 5 cars/h Solve each proportion. 4. 6 5. 16

2 -5 Solving Proportions Lesson Quiz: Part ll 6. Find 20% of 80. 16 7. What percent of 160 is 20? 12. 5% 8. 35% of what number is 40? 114. 3 9. A scale model of a car is 9 in. long. The scale is 1: 18. How many inches long is the actual car the model represents? 162 in.