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Warm Up

Preview. Warm Up. California Standards. Lesson Presentation. Warm Up. Determine whether the ratios are proportional. 5 8. 15 24. ,. 1. yes. 16 25. 12 15. ,. 2. no. 15 10. 20 16. ,. 3. no. 14 18. 42 54. ,. yes. 4. California Standards.

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Warm Up

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  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up Determine whether the ratios are proportional. 5 8 15 24 , 1. yes 16 25 12 15 , 2. no 15 10 20 16 , 3. no 14 18 42 54 , yes 4.

  3. California Standards NS1.3 Use proportions to solve problems (e.g., determine the value of N if = , find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Also covered:AF2.2, AF2.3 4 7 N 21

  4. Vocabulary cross product

  5. 6 15 2 5 = For two ratios, the product of the numerator in one ratio and the denominator in the other is a cross product. If the two ratios form a proportion, then the cross products are equal. 5 · 6 = 30 2 · 15 = 30

  6. It is important to set up proportions correctly. Each ratio must compare corresponding quantities in the same order. 16 mi 4 hr 8 mi x hr 16 mi 8 mi 4 hr xhr = =

  7. You can use the cross product rule to solve proportions with variables.

  8. 9 15 m 5 = Additional Example 1: Solving Proportions Using Cross Products Use cross products to solve the proportion. 15 · m = 9 · 5 The cross products are equal. 15m = 45 Multiply. 15m 15 = 45 15 Divide each side by 15. m = 3

  9. Walking Rate 0.75 0.5 Distance (mi) 0.25 0 0.1 0.2 0.3 0.4 0.5 Additional Example 2: Sports Application The graph shows the time and distance Greg walked on his way to school. At this rate, how long would it take Greg to walk 7.5 miles? The labeled point on the graph shows that Greg walked 0.75 miles in 0.25 hr. Let t represent the time in hours it will take Greg to walk 7.5 miles. (0.25, 0.75) Time (hr)

  10. 0.75 mi 0.25 hr 7.5 mi thr = Additional Example 2 Continued The graph shows the time and distance Greg walked on his way to school. At this rate, how long would it take Greg to walk 7.5 miles? Method 1 Set up a proportion in which each ratio compares distance to the time needed to walk that distance. Distance Time The cross products are equal. 0.75 · t = 0.25 · 7.5 0.75t = 1.875 Multiply. 0.75t 0.75 1.875 0.75 Divide each side by 0.75. = t = 2.5 hr

  11. 0.75 mi 7.5 mi 0.25 hr thr = Additional Example 2 Continued The graph shows the time and distance Greg walked on his way to school. At this rate, how long would it take Greg to walk 7.5 miles? Method 2 Set up a proportion in which one ratio compares distance and one ratio compares time. Walk to school Walk 7.5 miles The cross products are equal. 0.75 · t = 0.25 · 7.5 0.75t = 1.875 Multiply. 0.75t 0.75 1.875 0.75 Divide each side by 0.75. = t = 2.5 hr

  12. 1 Understand the Problem Additional Example 3:Problem Solving Application If 3 volumes of Jennifer’s encyclopedia takes up 4 inches of space on her shelf, how much space will she need for all 26 volumes? Rewrite the question as a statement. • Find the space needed for 26 volumes of the encyclopedia. List the important information: • 3 volumes of the encyclopedia take up 4 inches of space.

  13. Make a Plan 2 Additional Example 3 Continued Set up a proportion using the given information. Let x represent the inches of space needed. volumes inches 3 volumes 4 inches 26 volumes x =

  14. 3 Solve Additional Example 3 Continued 3 4 26 x Write the proportion. = 3 · x = 4 · 26 The cross products are equal. 3x = 104 Multiply. Divide each side by 3 to isolate the variable. 3x 3 104 3 = 2 3 x = 34 2 3 She needs 34 inches for all 26 volumes.

  15. 4 Additional Example 3 Continued Look Back 4 · 26 = 104 26 3 4 = 2 3 34 2 3 = 104 3 · 34 2 3 is the The cross products are equal, so 34 answer.

  16. 6 7 m 14 = Check It Out! Example 1 Use cross products to solve the proportion. 7 · m = 6 · 14 The cross products are equal. 7m = 84 Multiply. 7m 7 84 7 Divide each side by 7 to isolate the variable. = m = 12

  17. Running Rate 3.0 2.0 Distance (mi) 1.0 0 10 20 30 40 50 Check It Out! Example 2 The graph shows the time and distance Paige ran in one marathon. At this rate, how long would it take Paige to run 8.5 miles? The labeled point on the graph shows that a Paige ran 2.5 miles in 15 minutes. Let t represent the time in hours it will take Paige to run 8.5 miles. (15, 2.5) Time (min)

  18. 2.5 mi 15 min 8.5 mi tmin = Check It Out! Example 2 Continued The graph shows the time and distance Paige ran in one marathon. At this rate, how long would it take Paige to run 8.5 miles? Method 1 Set up a proportion in which each ratio compares distance to the time needed to run that distance. Distance Time The cross products are equal. 2.5 · t = 15 · 8.5 2.5t = 127.5 Multiply. 2.5t 2.5 127.5 2.5 = Divide each side by 2.5. t = 51 min

  19. 2.5 mi 15 min 8.5 mi tmin = Check It Out! Example 2 Continued The graph shows the time and distance Paige ran in one marathon. At this rate, how long would it take Paige to run 8.5 miles? Method 2 Set up a proportion in which one ratio compares distance and one ratio compares time. Marathon time Run 8.5 miles The cross products are equal. 2.5 · t = 15 · 8.5 2.5t = 127.5 Multiply. 2.5t 2.5 127.5 2.5 = Divide each side by 2.5. t = 51 min

  20. 1 Understand the Problem Check It Out! Example 3 John filled his new radiator with 6 pints of coolant, which is the 10 inch mark. How many pints of coolant would be needed to fill the radiator to the 25 inch level? Rewrite the question as a statement. • Find the number of pints of coolant required to raise the level to the 25 inch level. List the important information: • 6 pints is the 10 inch mark.

  21. Make a Plan 2 Check It Out! Example 3 Continued Set up a proportion using the given information. Let p represent the pints of coolant. 6 pints 10 inches p 25 inches pints inches =

  22. 3 Solve Check It Out! Example 3 Continued 6 10 p 25 Write the proportion. = 10 · p = 6 · 25 The cross products are equal. 10p = 150 Multiply. Divide each side by 10 to isolate the variable. 10p 10 150 10 = p = 15 15 pints of coolant will fill the radiator to the 25 inch level.

  23. 4 Check It Out! Example 3 Continued Look Back 10 · 15 = 150 15 6 10 = 25 6 · 25 = 150 The cross products are equal, so 15 is the answer.

  24. x 9 19 57 = 2. Lesson Quiz Use cross products to solve each proportion. 45 t 25 20 1. = t = 36 x = 3 2 3 r 36 n 10 28 8 = n = 35 3. r = 24 4. = 5. Carmen bought 3 pounds of bananas for $1.08. June paid $ 1.80 for her purchase of bananas. If they paid the same price per pound, how many pounds did June buy? 5 pounds

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