11 Section 1.3 Problem Solving Objective #1: Polya's four steps to problem solving. We will begin by introducing Polya's 4-Step Method for problem solving: Read the problem several times. The first time through, read it to get an overview. The second time through, read it phrase-by-phrase. The third time through, begin to write things down. Questions you need to ask yourself are: a) Do you understand all of the words in the problem? b) What are you asked to find or show? c) Can you rephrase the problem in your own words? d) Is there a picture or diagram that will help you solve the problem? e) Is there enough information to solve the problem? f) What information is not needed to solve the problem? Explore different ways to solve the problem and decide which strategy would be the best to use. Some possible of strategies are: Look for a pattern. Guess and work backwards. Guess and check. Eliminate possibilities. Consider special cases. Use direct reasoning Make an orderly list or table. Solve a simpler problem. Draw a picture or diagram. Relate it to a similar problem. Use common sense. Use a formula or an equation. Be patient and persistent. If the plan continues to not work, try another strategy. Does the answer satisfy the conditions of the problem? Does it make sense? Double-check your calculations. Also, think about what you did, what worked, and what didn't.
12 Objective #2: Evaluating the given information. The first step in problem solving is determining if there is enough information to solve a problem or if there is information that is not relevant to solving the problem. What necessary piece of information is missing that prevents solving the problem? Ex. 1 If a student saves every week, how long will it take to save enough money to buy a $700 TV? Given: A student wants to save $700. We must find how many weeks it took the student to save $700. The number of weeks it took to save the money is equal to $700 divided by amount saved per week. We are not given the amount that was saved per week, so we cannot solve the problem. Ex. 2 If it took Juan seven hours to drive to his friend s house, how fast was he driving per hour? Given: Juan drove for 7 hours. We must find how fast he was driving per hour. Juan's speed is equal to the distance driven divided by 7 hours. We are not given the distance Juan drove, so we cannot solve the problem. Which piece of information is not necessary to solve the problem? Ex. 3 A mail carrier receives an annual base salary of $49,200 plus time and a half for overtime. What is her weekly base salary? Given: Base Salary is $49,200 Time and a half paid for overtime We must find her weekly base salary.
Her weekly base salary is equal to her annual base salary divided by 52 weeks. The fact she gets paid time and a half is not important to solving the problem. Weekly base salary $49,200 52weeks $946.1538 $946.15 per week We can check our answer by multiplying $946.15 by 52: $946.15 52 49,199.80 $49,200. Objective #3: Solving problems. Use Polya's four-step method to solve the following problems: Ex. 4 a) Which is the better value? 1.5-pound box of cat food for $2.15 17.6-pound bag of cat food for $18.41 b) The supermarket displays the unit price for the 1.5-pound box in terms of cost per ounce, but displays the unit price for the 17.6-pound bag in terms of cost per pound. What are the unit prices, to the nearest cent, given by the supermarket? c) Based on our work in parts (a) and (b), does the better value always have the lower displayed unit price? a) Given: 1.5-pound box; $2.15 17.6-pound bag; $18.41 We need to determine which one is a better value. The better value will have the lower cost per pound. In order to be able to compare the two sizes, we need to find the cost per pound. We can do this by taking the cost and dividing it by the number of pounds. 1.5-pound box: $2.15 1.5 lb 1.4333 $1.43 per pound 17.6-pound bag: $18.41 1.0460 $1.05 per pound 17.6 lb Since the 17.6 pound bag has the lower cost per pound by $0.38 per pound, it is the better buy. 13
We can check our answer by multiplying the cost per pound by the number of pounds: Box: $1.43 1.5 2.145 $2.15 Bag: $1.05 17.6 18.48 $18.51 b) Given: 1.5-pound box; $2.15 17.6-pound bag; $18.41 We need to calculate the cost per ounce for the 1.5-pound box and the cost per pound for 17.6-pound bag. For the 1.5-pound box, we will first need to convert the pounds into ounces by multiplying 1.5 lb by 16 oz/lb. We will then need to divide that answer into $2.15. For the 17.6-pound, we will just use our work from part a. (1.5 lb)(16 oz/lb) 24 oz 1.5-pound box: $2.15 24 oz 0.0895 $0.09 per ounce 17.6-pound bag: $18.41 1.0460 $1.05 per pound 17.6 lb We can check our answer by multiplying the cost per ounce by the number of ounces: $0.09 24 2.16 $2.15 c) No. Even though we determined that the 17.6-pound bag was the better buy in part a, if the unit price displayed for the box is in cost per ounce, it will appear to be lower than the unit price for the bag. Ex. 5 A car sells for $8500. Instead of paying the total amount at the time of the purchase, Juanita pays $2000 down and $500 a month for 16 months. How much is saved by paying the total amount at the time of the purchase? Given: Car price: $8500 Down Payment: $2000 Monthly Payment: $500 Number of Months: 16 We need to determine how much is saved by purchasing the car upfront. 14
To find the total amount Juanita has to pay, we will multiply the monthly payment ($500) by the number of months (16) and then add the down payment ($2000). Then, to find how much is saved by paying upfront, we will subtract the price ($8500) from how much Juanita paid for the car. Total Monthly payments: 16($500) $8000 Plus the down payment: $8000 + $2000 $000 Subtract the price: $000 $8500 $1500. We can check our answer by adding how much extra she had to pay to the price. Then, subtract the down payment and divide the result by $500 to see if we get 16 months. $8500 + $1500 $000 $000 $2000 $8000 $8000/$500 16 months. Ex. 6 A storeowner orders 120 15.6" laptops that cost $265 each. The storeowner can sell each laptop for $399. The storeowner sold 112 laptops to customers. He had to return 8 laptops that were never sold and pay an $11 charge for each returned laptop (although the initial cost was refunded). What was the storeowner's profit? Given: Cost of a laptop: $265 Selling price of a laptop: $399 Number of laptops sold: 112 Number of laptops returned: 8 Cost to return a laptop: $11 The 120 laptops ordered and the fact that they are 15.6" laptops are not needed to solve the problem. We need to determine how much profit was made. First, find the profit per laptop sold by subtracting the cost ($265) from the selling price ($399). Multiply that answer by 112 laptops. Next, find the cost to return the unsold laptops by multiplying the cost ($11) by the number of laptops returned (8). Finally, subtract the two results to find the profit. 15
Profit per laptop sold: $399 $265 $134 Total profit from selling 112 laptops: $134 112 $15,008 Cost to return the 8 unsold laptops: $11(8) $88. Net Profit: $15,008 $88 $14,920 We can check our answer by calculating the revenue and the cost of the laptops and then subtracting the cost from the revenue. Revenue $399 112 $44,688 Cost $265 112 + $11 8 28680 + 88 $29,768 Profit Revenue Cost $44,688 $29,768 $14,920 which matches our result previously. Ex. 7 The average cost to attend a four-year private university was $34,000 per year. After 8 years, the average cost rose to $41,800 per year. Assuming the average cost increases steadily from year to year, what will the average cost to attend four-year private university be after another three years? Given: Initial cost: $34,000 Cost after 8 years: $41,800 Additional years: 3 We need to determine what the cost will be three years later. First, find how much the cost was rising per year over the eightyear period by subtracting $34,000 from $41,800 and dividing the result by 8 years. Next, find how much the cost increased over the next three years by multiplying the answer by 3. Finally, add that result to $41,800. Increase in cost: $41,800 $34,000 $7800 Increase per year: $7800 8 years $975 per year Increase over the next 3 years: $975(3) $2925 Average cost: $41,800 + $2925 $44,725 We can check our answer writing this situation as a linear equation. Our slope would be the increase per year ($975) and 16
the y-intercept would be the initial cost ($34,000): y mx + b 975x + 34000 Since the cost we are trying to find is 8 + 3 11 years after the initial cost, we need to evaluate the equation for x 11: y 975(11) + 34000 725 + 34000 $44,725 which matches our result previously. Ex. 8 A vending machine currently accepts nickels, dimes, and quarters only. Exact change is needed to make a purchase. How many ways can a person with six nickels, eight dimes, and five quarters make a $1.35 purchase from the machine? Given: nickels: 6 dimes: 8 quarters: 5 cost of the item: $1.35 exact change has to be used. We need to find all the combinations that add up to $1.35. Make an orderly table of all the possible combinations starting the coins that are worth more and work towards the coins that are worth less. Once the table is completed, count the number of rows. Make a table with headings quarters, dimes, nickels, and total. Row Quarters Dimes Nickels Total 1 5 1 0 $1.35 2 5 0 2 $1.35 3 4 3 1 $1.35 4 4 2 3 $1.35 5 4 1 5 $1.35 6 3 6 0 $1.35 7 3 5 2 $1.35 8 3 4 4 $1.35 9 3 3 6 $1.35 2 8 1 $1.35 11 2 7 3 $1.35 12 2 6 5 $1.35 13 1 8 6 $1.35 There are 13 different ways the purchase can be made. 17
18 Double-check that you have found all of the possible combinations and that they do total to $1.35 Ex. 9 As in sudoku, fill in the missing numbers in the 3-by-3 square so that it contains each of the 6 numbers from 5 through 13 exactly once. Also, 13 for this square, the rows, the columns, and the two diagonals must have different sums. 12 11 Given: The square above with the numbers 6,, 11, 12, & 13 already placed. We can only use the numbers 5 through 13 once. The remaining numbers are 5, 7, 8, and 9. We need to place the numbers in such a way that the sum of each row, of each column and of each diagonal is different. We need to do this by trial and error. We will randomly place the numbers 5, 7, 8, and 9 and check the sum of each row, of each column, and of each diagonal to see if they are all different. If not, we will try a different placement of numbers. 12 + 6 31 9 6 + 9 + 6 25 7 13 8 7 + 8 28 12 5 11 12 + 5 + 11 28 + 7 +12 9 +13 + 5 6 + 8 +11 29 27 25 + 11 34 12 + 6 31 9 6 + 9 + 6 25 8 13 5 8 + 5 26 12 7 11 12 + 7 + 11 30 + 8 +12 9 +13 + 7 6 + 5 +11 30 29 22 + 11 34
19 12 + 6 31 9 6 + 9 + 6 25 5 13 8 5 + 8 26 12 7 11 12 + 7 + 11 30 + 5 +12 9 +13 + 7 6 + 8 +11 27 29 25 + 11 34 12 + 6 31 7 6 + 7 + 6 23 8 13 5 8 + 5 26 12 9 11 12 + 9 + 11 32 + 8 +12 7 +13 + 9 6 + 5 +11 30 29 22 + 11 34 So, the correct answer is: 7 6 8 13 5 12 9 11 Double-check that all of the rows, columns, and diagonals have a different sum. Ex. A sales representative who lives in Helotes is required to visit to Atlanta, Boston, Charlotte, and Detroit. The one-way trip between each city is given below. Atlanta Boston Charlotte Detroit Helotes Atlanta $508 $208 $193 $213 Boston $508 $313 $368 $348 Charlotte $208 $313 $328 $173 Detroit $193 $368 $328 $3 Helotes $213 $348 $173 $3 Give the representative an order for visiting each city once and returning to Helotes for less than $1350.
Given: The chart above. Trip must start and end in Helotes and must visit each city once. For instance, one possible trip would be: Helotes Atlanta Boston Charlotte Detroit Helotes $213 + $508 + $313 + $328 + $3 $1672 We need to find a route that costs less than $1350. We can start by finding the cheapest route from Helotes to the another city. Then find the cheapest route from that city to the next city. We will continue this pattern until we hit all of the cities and then fly back to Helotes. To help us out, we will draw a diagram: Helotes 20 $3 $213 Detroit $368 $193 $208 Atlanta $328 $173 $508 $348 $313 Charlotte Boston Step #3: Devise a plan: Cheapest flight out of Helotes is to Charlotte ($173). From Charlotte, the cheapest flight is to Atlanta ($208). From Atlanta, the cheapest flight is to Detroit ($193). From Detroit, we have to fly to Boston ($368). From Boston, we have to fly back to Helotes ($348). Helotes Charlotte Atlanta Detroit Boston Helotes $173 + $208 + $193 + $368 + $348 $1290 which is below $1350. Verify the numbers that were used are the same as the ones in the chart. Also, estimate the answer to verify our calculations are correct: $200 + $200 + $200 + $400 + $300 $1300