3.6 Cross Products

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.