Holt CA Course Solving Proportions 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 California Standards 4747 N 21

Holt CA Course Solving Proportions Vocabulary cross product The product of the numerator in one ratio and the denominator in the other when comparing two ratios.

Holt CA Course Solving Proportions 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 =

Holt CA Course Solving Proportions You can use the cross product rule to solve proportions with variables.

Holt CA Course Solving Proportions Use cross products to solve the proportion. Example 1: Solving Proportions Using Cross Products 15 · m = 9 · 5 15m = 45 15m 15 = m = 3 The cross products are equal. Multiply. Divide each side by 15. = m5m5 9 15

Holt CA Course Solving Proportions 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 x hr

Holt CA Course Solving Proportions 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? Example 2: Sports Application 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 Distance (mi) Walking Rate Time (hr) (0.25, 0.75)

Holt CA Course Solving Proportions 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? Example 2 Continued 0.75 · t = 0.25 · t = t 0.75 = t = 2.5 hr Distance Time Divide each side by = 7.5 mi t hr 0.75 mi 0.25 hr Set up a proportion in which each ratio compares distance to the time needed to walk that distance. The cross products are equal. Multiply. It will take Greg 2.5 hours to walk 7.5 miles.

Holt CA Course Solving Proportions 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? Example 2 Continued 0.75 · t = 0.25 · t = t 0.75 = t = 2.5 hr Walk to school Walk 7.5 miles Divide each side by = 0.25 hr t hr 0.75 mi 7.5 mi Method 2 Set up a proportion in which one ratio compares distance and one ratio compares time. The cross products are equal. Multiply.

Holt CA Course Solving Proportions Check It Out! Example 2 Continued 2.5 · t = 15 · t = t 2.5 = t = 51 min Marathon time Run 8.5 miles Divide each side by 2.5. = 8.5 mi t min 2.5 mi 15 min The cross products are equal. Multiply. 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.

Holt CA Course Solving Proportions 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? 1 Understand the Problem 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.

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

Holt CA Course Solving Proportions Solve = 26 x 3 · x = 4 · 26 3x = 104 3x33x3 = x = She needs inches for all 26 volumes. Write the proportion. The cross products are equal. Multiply. Divide each side by 3 to isolate the variable. Additional Example 3 Continued

Holt CA Course Solving Proportions Look Back = · 26 = · = 104 The cross products are equal, so is the answer. Additional Example 3 Continued

Holt CA Course Solving Proportions Check It Out! Example 1 Use cross products to solve the proportion. 7 · m = 6 · 14 7m = 84 7m77m7 = 84 7 m = 12 The cross products are equal. Multiply. Divide each side by 7 to isolate the variable = m 14

Holt CA Course Solving Proportions 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? Check It Out! Example 2 The labeled point on the graph shows that a Paige ran 2.5 miles in 0.15 hr. Let t represent the time in hours it will take Paige to run 8.5 miles Distance (mi) Running Rate Time (hr) (0.15, 2.5)

Holt CA Course Solving Proportions Check It Out! Example 2 Continued 2.5 · t =.15 · t = t 2.5 = t =.51 hr Distance Time Divide each side by 2.5. = 8.5 mi t hr 2.5 mi.15 hr Set up a proportion in which each ratio compares distance to the time needed to run that distance. The cross products are equal. Multiply. 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?

Holt CA Course Solving Proportions 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? 1 Understand the Problem 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.

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

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

Holt CA Course Solving Proportions Look Back = · 15 = · 25 = 150 The cross products are equal, so 15 is the answer. Check It Out! Example 3 Continued