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Section 1.3

Math in Our World. Section 1.3. Problem Solving. Learning Objectives. State the four steps in the basic problem solving procedure. Solve problems by using a diagram. Solve problems by using trial and error. Solve problems involving money. Solve problems by using calculation.

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Section 1.3

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  1. Math in Our World Section 1.3 Problem Solving

  2. Learning Objectives • State the four steps in the basic problem solving procedure. • Solve problems by using a diagram. • Solve problems by using trial and error. • Solve problems involving money. • Solve problems by using calculation.

  3. Polya’s Four-Step Problem-Solving Procedure • Step 1 Understand the problem. Read the problem slowly, jotting down the key ideas • Step 2 Devise a plan to solve the problem. Draw a diagram, find a formula, look for patterns • Step 3 Carry out the plan to solve the problem. Solve the problem, follow the numbers, create an equation • Step 4 Check the answer. Does your answer make sense? Did you solve for the requested unknown?

  4. EXAMPLE 1 Solving a Problem by Using a Diagram A gardener is asked to plant eight tomato plants that are 18 inches tall in a straight line with 2 feet between each plant. How much space is needed between the first plant and the last one? Be careful—what seems like an obvious solution is not always correct! You might be tempted to just multiply 8 by 2, but instead we will use Polya’s method.

  5. EXAMPLE 1 Solving a Problem by Using a Diagram SOLUTION • Step 1 Understand the problem. In this case, the key information given is that there will be eight plants in a line, with 2 feet between each. We are asked to find the total distance from the first to the last. • Step 2 Devise a plan to solve the problem. When a situation is described that you can draw a picture of, it’s often helpful to do so. • Step 3 Carry out the plan to solve the problem. The figure would look like this: Now we can use the picture to add up the distances: 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14 feet • Step 4 Check the answer. There are eight plants, but only seven spaces of 2 feet between them. So 7 x 2 = 14 feet is right.

  6. EXAMPLE 2 Solving a Problem Using Trial and Error Suppose that you have 10 coins consisting of quarters and dimes. If you have a total of $1.90, find the number of each type of coin.

  7. EXAMPLE 2 Solving a Problem Using Trial and Error • SOLUTION • Step 1 Understand the problem. We’re told that we have a total of 10 coins and that some are dimes (worth $0.10 each) and the rest are quarters (worth $0.25 each). The total value is $1.90. The problem is to find how many quarters and dimes together are worth $1.90. • Step 2 Devise a plan to solve the problem. One strategy that can be used is to make an organized list of possible combinations of 10 total quarters and dimes and see if the sum is $1.90. For example, you may try one quarter and nine dimes. This gives 1 x $0.25 + 9 x $0.10 = $1.15.

  8. EXAMPLE 2 Solving a Problem Using Trial and Error • SOLUTION • Step 3 Carry out the plan. Since one quarter and nine dimes is wrong, try two quarters and eight dimes. • This doesn’t work either. • So continue… • Answer: six quarters and four dimes. • Step 4 Check the answer. In this case, our answer can be checked by working out the amounts: 6 x $0.25 + 4 x $0.10 = $1.90.

  9. EXAMPLE 3 Solving a Problem Involving Salary So you’ve graduated from college and you’re ready for that first real job. In fact, you have two offers! One pays an hourly wage of $19.20 per hour, with a 40-hour work week. You work for 50 weeks and get 2 weeks’ paid vacation. The second offer is a salaried position, offering $41,000 per year. Which job will pay more?

  10. EXAMPLE 3 Solving a Problem Involving Salary SOLUTION • Step 1 Understand the problem. The important information is… • The hourly job pays $19.20 per hour for 40 hours each week • You will be paid for 52 weeks per year. • We are asked to decide if that will work out to be more or less than $41,000 per year. (The fact that you get 2 weeks’ paid vacation is irrelevant to the problem.) • Step 2 Devise a plan to solve the problem. We can use multiplication to figure out how much you would be paid each week and then multiply again to get the yearly amount. Then we can compare to the salaried position.

  11. EXAMPLE 3 Solving a Problem Involving Salary SOLUTION • Step 3 Carry out the plan to solve the problem. Multiply the hourly wage by 40 hours; $19.20 x 40 = $768 This shows that the weekly earnings will be $768. Now we multiply by 52 weeks: $768 x 52 = $39,936 This gives an annual income of $39,936. The salaried position, at $41,000 per year, pays more. • Step 4 Check the answer. We can figure out the hourly wage of the job that pays $41,000 per year. We divide by 52 to get a weekly salary of $788.46. Then we divide by 40 to get an hourly wage of $19.71. Again, this job pays more.

  12. EXAMPLE 4 Solving a Problem by Using Calculation A 150-pound person walking briskly for 1 mile can burn about 100 calories. How many miles per day would the person have to walk to lose 1 pound in one week? It is necessary to burn 3,500 calories to lose 1 pound.

  13. EXAMPLE 4 Solving a Problem by Using Calculation SOLUTION • Step 1 Understand the problem. • A 150-pound person burns 100 calories per 1 mile • The person needs to burn 3,500 calories in 7 days to lose 1 pound • The problem asks how many miles per day the person has to walk to lose 1 pound in 1 week. • Step 2 Devise a plan to solve the problem. We will calculate how many calories need to be burned per day and then divide by 100 to see how many miles need to be walked.

  14. EXAMPLE 4 Solving a Problem by Using Calculation SOLUTION • Step 3 Carry out the plan. Since 3,500 calories need to be expended in 7 days, divide to find out how many are needed to be burned per day. 3,500 ÷ 7 = 500 Then divide 500 by 100 to get 5 miles. It is necessary to walk briskly 5 miles per day to lose 1 pound in a week. • Step 4 Check the answer. Multiply 5 miles per day by 100 calories per mile by 7 days to get 3,500 calories.

  15. Classwork p. 32-34: 5, 10, 13, 15, 19, 22, 28, 30, 33, 37, 41, 46

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