4.2 PROPORTIONS 70.00

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1 4.2 PROPORTIONS Jake has a part-time job walking dogs for an elderly neighbor. He is paid $20 for every ninety minutes he works. He has walked Boris, Buster, and Bonnie a total of 31 minutes this week. How much can Jake expect to be paid for his work 20 n $ Assess your readiness to complete this activity. Rate how well you understand: Not ready Almost ready Bring it on! the terminology and notation associated with proportions how to determine if a proportion is true how to solve a proportion for the unknown component how to validate the solution to a proportion Validating a proportion Solving a proportion answer rounded to the specified place validation of the answer 181

2 Chapter 4 Ratios and Proportions Technique Apply the Equality Test for Fractions and cross-multiply to determine if the proportion is true. Is this proportion true or false 7 miles 21. miles gallons 1. gallons Verify that the units are compared in the same order. miles per gallon in both ratios of the proportion Cross-multiply The proportion is false. Technique If it is obvious by inspection that you can multiply the numerator and denominator of one of the ratios by the same multiplier to make it equal to the other ratio, then the proportion is true. Is this proportion true or false It is apparent that multiplying the numerator by 3 will equal 1. Is the denominator 36 multiplied by 3 equal to Yes The proportion is true. 108

3 Activity 4.2 Proportions Technique If easily done, reduce both ratios to their simplest forms. If the reduced ratios are equal, then the proportion is true. Is this proportion true or false Reduce both sides: and 4 7 The proportion is false. If the relationship between the two ratios of a proportion is easily recognizable, that is, if one ratio is a multiple of the other, you can use the following technique to solve for the unknown quantity. Technique Step 1: Step 2: If one numerator is a multiple of the other numerator (or one denominator is a multiple of the other denominator), use the multiplier to determine the value of the variable. Validate that the solution is correct by substituting the answer into the original proportion. Apply the Equality Test for Fractions by cross-multiplying. Solve for n (a variable with an unknown value) in the proportion 03.. n Step Then 4 20 n Answer : n 20 Step 2 Validate:

4 Chapter 4 Ratios and Proportions If the relationship between the two ratios of a proportion is not easily recognizable, use the following methodology to solve for the variable. Example 1: 11 is to 1 as what number is to 37. Example 2: n 3 6 Try It! Steps in the Methodology Example 1 Example 2 Step 1 Set up the proportion with a variable. Step 2 Equate the cross products. Write the given proportion. If it is not already set up in proportion form, set up the proportion using a variable for the unknown component. Cross multiply and equate the crossproducts. Remember that the Equality Test for Fractions says that, if the ratios are equal, then their cross-products will be equal n n 1 Shortcut Reduce the known ratio fi rst (see Model 2) Step 3 Divide by the multiplier of the Divide both sides of the equation by the multiplier of the variable. This results in an equation that will provide the missing quantity of your proportion n Step 4 Solve for the variable. Calculate the value for the variable. Compute the numerator and divide the product by the denominator. Round to the specified place value, if necessary ) Step Present the answer. State your answer, the value for the unknown. n

5 Activity 4.2 Proportions Steps in the Methodology Example 1 Example 2 Step 6 Validate your answer. Validate your answer. In the original proportion, replace the variable with your answer and use one of the Techniques for Testing a Proportion. Note: When the answer is rounded, the cross-products will be close but not exactly equal Model 1 Solve: What number is to 3 as 1½ is to Step 1 Step 2 n n Step 3 Step 4 Step n Compute the numerator: Divide by the denominator: n Step 6 Validate:

6 Chapter 4 Ratios and Proportions Model 2 Shortcut: Reduce the Known Ratio First Solve for n: Step n n 7 2 Step 2 Use the shortcut and reduce: At this point, there is no need to cross-multiply. Use the Technique and skip Steps n Shortcut (optional): To simplify computations, reduce the known ratio before equating the cross-products n Step n Step 6 Validate: Make Your Own Model Either individually or as a team exercise, create a model demonstrating how to solve the most diffi cult problem you can think of. Answers will vary. Problem: Step 1 Step 4 Step 2 Step Step 3 Step 6 186

7 Activity 4.2 Proportions 1. Why must the comparison order of the units be verified before determining if a proportion is true The numerators should both refl ect the same unit and likewise the denominators of both ratios should refl ect the same unit. Therefore the comparison will both be the same, like dollars per share or miles per hour. 2. What are the ways you can validate that a proportion is true You can accomplish this by cross multiplication. If the answers are the same, then the proportion is true. You could also reduce or build both ratios to show that the fractions are equivalent. Corresponding parts can be reduced: numerator with denominator on the same side of the equal sign or numerator to numerator or denominator to denominator on opposite sides of the equal signs. 3. When should you use the Methodology versus the Technique for solving a proportion If the relationship between the two ratios is easily recognizable, then use a Technique, otherwise use the Methodology. 4. Why can you cross-multiply and equate the cross-products in Step 2 of the Methodology for Solving a Proportion This can be used because of the Equality Test for Fractions.. Why do you divide each side of the equation by the multiplier of the variable in Step 3 of the Methodology for Solving a Proportion Because the Special Properties of Division Involving One tell us that any number divided by itself equals one, and the Identity Property of Multiplication tells us that one times any number equals that same number. 6. How do you ensure that your answer for the unknown quantity (the variable) in a proportion is correct Replace the unknown with the answer that you found then cross-multiply to show that the cross products are equivalent. This will validate your answer. 187

8 Chapter 4 Ratios and Proportions 7. What aspect of the model you created is the most difficult to explain to someone else Explain why. Answers will vary. 1. Determine if the given proportions are true or false. Proportion Worked Solution True or False a) False b) 6.4 gallons 4 gallons acres 7 acres True c) 28 SUVs 16 SUVs 1 total vehicles 2 total vehicles True 188

9 Activity 4.2 Proportions Proportion Worked Solution True or False d) True 2. The directions on a well-known brand of parboiled rice say to combine the rice and water in the ratio of 1/2 cup rice to 2 1/4 cups water for four servings, and in the ratio of 1 1/2 cups rice to 3 1/3 cups water for six servings. Are the ratios in proportion to each other 3. For the following proportions, solve for the variable. Round to the nearest tenth, if necessary. Proportion Worked Solution Validation a) n b) 2 n

10 Chapter 4 Ratios and Proportions Proportion Worked Solution Validation c) 4 n d) n Determine if the following proportions are true or false feet 10 seconds feet 1 seconds TRUE FALSE 8 FALSE FALSE 7 cups sauce 3 servings 1 cups sauce 3 2 servings TRUE Solve for the variable in each of the following proportions. Round to the nearest tenth if necessary. Validate your solutions n ṇ 10 1 n ṅ or

11 Activity 4.2 Proportions In the second column, identify the error(s) you find in each of the following worked solutions. If the answer appears to be correct, validate it in the second column and label it Correct. If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer in the last column. Worked Solution What is Wrong Here Identify Errors or Validate Correct Process Validation 1) Is this a true proportion The decimal point was placed improperly in the product of 0.72 and 0. (Should have been 3 decimal places.) TRUE or Validation is not necessary for this problem. TRUE 2) Solve the proportion for the variable. 44 n You cannot cancel across the equal () sign. 11

12 Chapter 4 Ratios and Proportions Worked Solution What is Wrong Here Identify Errors or Validate Correct Process Validation 3) 11 is to as 33 is to what number The problem is set up incorrectly. The correct set up is: n 12