Lesson 2.1. Ahn and four friends decided to make macaroni & cheese. They were very hungry so Ahn decided to make two boxes instead of one.

Transcription

1 Write and Solve Proportions Lesson 2.1 EXPLORE! Ahn and four friends decided to make macaroni & cheese. They were very hungry so Ahn decided to make two boxes instead of one. The directions for one box are macaroni and Cheese Step 1: Find the amount of milk, butter and number of cheese packets Ahn will need to make both boxes of macaroni and cheese. Step 2: Write the simplified ratio of the amount of milk needed for one box to the amount of milk needed for two boxes. Step 3: Write the simplified ratio of the amount of butter needed for one box to the amount of butter needed for two boxes. Step 4: Write the simplified ratio of the number of cheese packets needed for one box to the number of cheese packets needed for two boxes. Step 5: What ratio did you write for Steps #2-4 Explain the meaning of the ratio. Step 6: Explain the meaning of a ratio of 1 : 3 in a similar situation. Step 7: The next day Ahn used 8 tablespoons of butter to make macaroni and cheese. a. How many boxes of macaroni and cheese did he make b. How much milk did he use In the Explore! Ahn first made two boxes of macaroni and cheese. The ratio of the amount of milk used in one box compared to two boxes was equal to the ratio of butter used in one box compared to two boxes. Since the ratios are equal, they can be written with an equals sign. 8 = 2_ 4 This is a proportion. A proportion is an equation stating that two ratios are equal. Proportions have equal cross products. Cross products are calculated by multiplying the numerator of one ratio by the denominator of the other ratio. Each cross product is placed on one side of an equals sign. For example: 8 = _ = = 16 Lesson 2.1 ~ Write and Solve Proportions 39

2 When one number in a proportion is missing, you can find it using equivalent fractions or cross products. Suppose Ahn tripled the amount of macaroni and cheese he wants to make. Each of his ingredients from one box to three boxes will be in a ratio of 1 : 3. Use a proportion to solve for the amount of milk Ahn will need. Write a proportion. Method 1 ~ Equivalent Fractions Figure out what to multiply 1 by to get 4. Multiply the denominator by this number. Solving Proportions _ 1 3 = _ 4 x 4 _ 1 3 = _ 4 x 4 Multiply to get the equivalent fraction. x = 12 Method 2 ~ Cross Products Write a proportion. _ 1 3 = _ 4 x Write the cross products. 1 x = 4 3 Simplify. 1x = 12 Solve for x by dividing both sides of the equation by 1. x = 12 Example 1 Solve each proportion using cross products. a. 2 3 = x 15 b. 6 a = _ Solutions a. Write the cross products. 3 x = 2 15 Simplify the equation. 3x = 30 Divide both sides by 3. x = 10 Check the answer. Substitute 10 for x. _ 2 3 = simplifies to 2_ 3, so the solution is correct. 15 b. Write the cross products. 1.5 a = 6 7 Simplify. 1.5a = 42 Divide both sides by 1.5. a = 28 Check the answer = _ (28) = = Lesson 2.1 ~ Write and Solve Proportions

3 Example 2 Determine whether each pair of ratios forms a proportion. a. _ 9 and 2 6 b. 15 and Solutions a. Write the proportion using =. 9 = _ 2 6 Check using cross products. 4 6 = No, 9 and 2_ 6DO NOT form a proportion. The cross products are not equal, so the ratios are not equivalent. b. Write the proportion using = = Find the cross products = = 450 Yes, 15 9 and 50 30DO form a proportion. Their cross products are equal. They are equivalent ratios. Example 3 Solution Write a proportion for the following phrase and solve it. $4 in 30 minutes; $b in 45 minutes If the first ratio is written as dollars in the numerator and minutes in the denominator, $4 30 minutes = $b 45 minutes the second must be written the same way. Write the cross products = 30 b Simplify. 180 = 30b Divide both sides of the equation by 30. _ = _ 30b 30 6 = b $6 in 45 minutes. Lesson 2.1 ~ Write and Solve Proportions 41

4 exercises Solve each proportion. 1. 2_ 3 = x 4. b 7. 9_ _ 3 = 45 x 3. a 12 = 2 = _ 8 = x y 10 = 9 = 21 x 8. a 17 = = _ x Determine whether each pair of ratios forms a proportion. Explain your reasoning _ 3 and and _ 4 and Write two different ratios that would form a proportion with the given ratio Use cross products to check the solution given for each proportion. Write yes or no for whether or not the solution works. Show all work necessary to justify your answer _ 4 = 15 x ; x = = 24 x ; x = _ 5 = a ; a = Write a proportion for each phrase and solve it miles in 2 hours; x miles in 6 hours minutes for 3 typed pages; 60 minutes for a typed pages pounds for $10.95; y pounds for $ feet in 4 seconds; 80 feet in x seconds 23. Larry paid $15 for 6 boxes of crackers. Nancy bought 4 boxes of the same crackers at another store. She paid $11. a. Find the ratio of the cost to the number of boxes of crackers for Larry s purchase. b. Find the ratio of the cost to the number of boxes of crackers for Nancy s purchase. c. Do the ratios form a proportion Explain how you know your answer is correct. 42 Lesson 2.1 ~ Write and Solve Proportions

5 24. Paulo and Sami solved the proportion = x 45. Their work is shown at right. a. Determine if their solutions are correct. b. Why does Paulo multiply by 3 in the numerator and denominator in Step 1 c. Sami wanted to use smaller numbers in her proportion. What did she do for Step 1 to create smaller numbers d. Which method do you like best Explain your reasoning. review Paulo s Work = x 1) = x 2) x = 36 Sami s Work _ = _ x 1) _ 4 5 = x 2) 5x = 180 3) x = 36 Find each unit rate balloons $ miles 1 1_ hours 28. _ 75 miles 5 gallons 20 ft 12 seconds 29. A store sells 2 pairs of shoes for $ Each pair of shoes is the same price. Find the price for each pair of shoes. 30. A car travels 204 miles using 12 gallons of gas. Find the miles per gallon the car used. Tic-Tac-Toe ~ M a p 1. Print or copy a local map that includes your home and school. Include the map with your work. 2. Record the scale of the map (map length : actual length). 3. Use a red pen to trace along the streets from your home to your school. Use string along the path in red to find the distance on the map from your home to your school. Record the length of the string. 4. Use proportions to find the actual distance from your home to your school. 5. Trace a path in blue between two places on your map and then trace a path in green between two different places on your map. (You can use any color, just different from the ones already used.) 6. Use string along each path in Part 5 to find the map distance. Record the map distance. 7. Use proportions to find the actual distance for each path. 8. Research how far it is from your home to the nearest zoo. How many inches would represent that distance on your map Lesson 2.1 ~ Write and Solve Proportions 43