Section 1.3 Problem Solving

What You Will Learn Upon completion of this section, you will be able to: Understand and use a general problem-solving procedure

Polya’s Procedure George Polya (1887-1985), a mathematician who was educated in Europe and taught at Stanford, developed a general procedure for solving problems.

Guidelines for Problem Solving Understand the Problem. Devise a Plan to Solve the Problem. Carry Out the Plan. Check the Results.

1. Understand the Problem. Read the problem carefully at least twice. Try to make a sketch of the problem. Label the information given. Make a list of the given facts that are pertinent to the problem. Determine if you have sufficient information to solve the problem.

2. Devise a Plan to Solve the Problem. Can you relate this problem to a previous problem that you’ve worked before? Can you express the problem in terms of an algebraic equation? Look for patterns or relationships. Can you express the problem more simply?

2. Devise a Plan to Solve the Problem. Can you substitute smaller or simpler numbers to make the problem more understandable? Will listing the information in a table help in solving the problem? Can you make an educated guess at the solution? You can work backward to determine the correct procedure.

3. Carrying Out the Plan. Use the plan you devised in step 2 to solve the problem.

4. Check the Results. Ask yourself, “Does the answer make sense?” and “Is it reasonable?” If the answer is not reasonable, recheck your method for solving the problem and your calculations. Can you check the solution using the original statement? Is there an alternative method to arrive at the same conclusion? Can the results of this problem be used to solve other problems?

Example 2: Shuttle Revenue Airport Express shuttle service provides service from San Antonio International Airport to downtown hotels, approximately 10 miles away. One particular shuttle makes 16 round trips per day, carrying 5 passengers per trip. The fare each way is $18. What are the receipts from one day’s operation for this particular shuttle?

Example 2: Shuttle Revenue Solution List all the information: Distance from airport to downtown hotels ≈ 10 miles * Number of round trips per day = 16 * Number of passengers per trip = 5 * Fare each way = $18 We need only the information with a *.

Example 2: Shuttle Revenue Solution Round trip costs 2 × $18 = $36 The total receipts for one day is the product of the number of round trips, the number of passengers, and the cost per round trip = 16 × 5 × $36 = $2880

Example 3: Retirement It is never too early to start planning for retirement. U.S. Census Bureau data indicate that at age 65 the average woman will live another 21 years and the average man will live another 19 years. The data also indicate that about 33% of the average person’s retirement income will come from Social Security.

Example 3: Retirement When discussing retirement planning, many investment firms and financial planners use the graph in the figure on the next slide which shows how long a typical retiree’s assets (or “nest egg”) will last based on the percentage of the assets withdrawn each year.

Example 3: Retirement

Example 3: Retirement a) If a typical retiree has retirement assets of $500,000, how much can he or she withdraw annually if he or she wishes the assets to last 21 years?

Example 3: Retirement

Example 3: Retirement Solution a) From the graph, we can see that for assets to last 21 years, about 6% of the assets can be withdrawn annually. The amount that can be withdrawn is Amount = 6% of assets Amount = 0.06(500,000) = $30,000 Thus, about $30,000 can be withdrawn annually.

Example 3: Retirement b) How much should a retiree have in assets if he or she wishes to withdraw $25,000 annually and wishes his or her assets to last 18 years?

Example 3: Retirement

Example 3: Retirement Solution b) From the graph, we can see that for assets to last 18 years, about 7% of the assets can be withdrawn annually. That amount has to equal $25,000: 7% of assets = $25,000 0.07 × assets = $25,000 Therefore, if the retiree has assets of about $357,142.86, he or she will be able to withdraw $25,000 annually and the assets will last 18 years.

Example 4: Determining a Tip The cost of Freddie’s meal before tax is $28.00. a) If a 6 ½% sales tax is added to his bill, determine the total cost of the meal including tax.

Example 4: Determining a Tip Solution Change 6 ½ to a decimal: 0.065 Sales tax = 6 ½ % of meal Sales tax = 0.065(28.00) = 1.82 The total bill = cost of meal + sales tax Total bill = 28.00 + 1.82 = $29.82 The bill including sales tax is $29.82

Example 4: Determining a Tip The cost of Freddie Rojo’s meal before tax is $28.00. b) If Freddie wants to leave a 10% tip on the pretax cost of the meal, how much should he leave?

Example 4: Determining a Tip Solution To find 10% of any number, we multiply the number by 0.10 10% of pretax cost = 0.10(28.00) = 2.80 A simple way to find 10% of any number is to move the decimal point in the number one place to the left: 28.00  2.80

Example 4: Determining a Tip The cost of Freddie Rojo’s meal before tax is $28.00. c) If he wants to leave a 15% tip on the pretax cost of the meal, how much should he leave?

Example 4: Determining a Tip Solution To find 15% of 28.00, multiply: 15% of 28.00 = 0.15(28.00) = 4.20 Or to find 15%, find 10% and add it to ½ that amount: 2.80 + = 2.80 + 1.40 = 4.20

Example 6: A Brine Solution Some recipes for smoking turkey recommend presoaking the turkey in a brine solution. To make enough brine solution to smoke a 12-pound turkey, use 16 tablespoons (tbsp) of salt in 2 gallons (gal) of water. a) If you wish to make 3.5 gallons of a brine solution, how much salt is needed?

Example 6: A Brine Solution Determine how much salt is needed to mix with 3.5 gallons of water. Use the information we know to set up a proportion to solve for the unknown quantity. Given ratio { Item to be found Other information given

Example 6: A Brine Solution Thus, 28 tbsp. of salt must be used to make 3.5 gal of a brine solution.

Example 6: A Brine Solution Some recipes for smoking turkey recommend presoaking the turkey in a brine solution. To make enough brine solution to smoke a 12-pound turkey, use 16 tablespoons (tbsp) of salt in 2 gallons (gal) of water. b) If you wish to make enough brine solution to soak a 20-pound turkey, how much salt is needed to make the brine solution?

Example 6: A Brine Solution Use the known information that a 12-lb turkey requires 16 tbsp of salt. Given ratio { Other information given Item to be found

Example 6: A Brine Solution Thus, about 26.67 tbsp. of salt are needed to make enough brine solution for a 20-lb turkey.