Preview Warm Up California Standards Lesson Presentation

Warm Up Solve each equation. 1. Multiply. 3. 5. 7. 48 2. 5m = 18 3.6 5. 7. 48 2. 5m = 18 3.6 4. 10 7 Change each percent to a decimal. 73% 0.73 6. 112% 0.6% 0.006 8. 1% 0.01 1.12 Change each fraction to a decimal. 9. 0.5 10. 0.3

California Standards 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.

Vocabulary ratio scale rate scale model cross products scale drawing proportion unit rate percent

A ratio is a comparison of two quantities 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.

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.

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.

A common application of proportions is rates 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.

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.

Check It Out! Example 2a 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.

Check It Out! Example 2b 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.

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

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

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

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

Check It Out! Example 3a Continued Solve the proportion. Check your answer. Check Substitute –20 for y.  –2.5 –2.5

Check It Out! Example 3b Solve the proportion. Check your answer. Use cross product. 4(g + 3) = 5(7) 4g + 12 = 35 –12 –12 4g = 23 Subtract 12 from both sides. Divide both sides by 4. g = 5.75

Check It Out! Example 3b Continued Solve the proportion. Check your answer. Check Substitute 5.75 for b. 1.75 1.75 

Another common application of proportions is percents 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 to find unknown values.

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

Additional Example 4B: Percent Problems 230 is what percent of 200? Method 2 Use an equation. 230 = x  200 Write an equation. Let x represent the percent. 230 = 200x Since x is multiplied by 200, divide both sides by 200 to undo the multiplication. 1.15 = x The answer is a decimal. Write the decimal as a percent. This answer is reasonable; 230 is more than 100% of 200. 115% = x 230 is 115% of 200.

Additional Example 4C: Percent Problems 20 is 0.4% of what number? Method 1 Use a proportion. Use the percent proportion. Let x represent the whole. 2000 = 0.4x Cross multiply. Since x is multiplied by 0.4, divide both sides by 0.4. 5000 = x 20 is 0.4% of 5000.

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

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

Proportions are used to create scale drawings and scale models 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.

Additional Example 5A: 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.

Additional Example 5B: 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.

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

Check It Out! Example 5a 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. 18x = 4 Use cross products to solve. x ≈ 0.2 The distance on the map is about 0.2 in.

Check It Out! Example 5b 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. Use cross products to solve. 32x = 16 x = 0.5 The actual heart is 0.5 feet or 6 inches.

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

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.