Presentation on theme: "An Introduction to Problem Solving"— Presentation transcript:
1 An Introduction to Problem Solving § 2.5An Introduction to Problem Solving
2 Strategy for Problem Solving General Strategy for Problem SolvingUNDERSTAND the problemRead and reread the problemChoose a variable to represent the unknownConstruct a drawing, whenever possiblePropose a solution and checkTRANSLATE the problem into an equationSOLVE the equationINTERPRET the resultCheck the proposed solution in the problemState your conclusion
3 Finding an Unknown Number Example:The product of twice a number and three is the same as the sum of five times the number and 12. Find the number.1.) UNDERSTANDRead and reread the problem. If we letx = the unknown number, then “twice a number” translates to 2x,“the product of twice a number and three” translates to 2x · 3,“five times the number” translates to 5x, and“the sum of five times the number and 12” translates to 5x + 12.Continued
4 Finding an Unknown Number Example continued:2.) TRANSLATEThe product ofthe sum of+is the same as=twice a number2x5 times the number5xand 33and 1212Continued
5 Finding an Unknown Number Example continued:3.) SOLVE2x · 3 = 5x + 126x = 5x Simplify left side.6x + (– 5x) = 5x + (– 5x) Add –5x to both sides.x = Simplify both sides.4.) INTERPRETCheck: Replace “number” in the original statement of the problem with 12. The product of twice 12 and 3 is 2(12)(3) = 72. The sum of five times 12 and 12 is 5(12) + 12 = 72. We get the same results for both portions.State: The number is 12.
6 Solving a Problem Example: A car rental agency advertised renting a Buick Century for $24.95 per day and $0.29 per mile. If you rent this car for 2 days, how many whole miles can you drive on a $100 budget?1.) UNDERSTANDRead and reread the problem. Let’s propose that we drive a total of 100 miles over the 2 days. Then we need to take twice the daily rate and add the fee for mileage to get 2(24.95) (100) = = This gives us an idea of how the cost is calculated, and also know that the number of miles will be greater than If we letx = the number of whole miles driven, then0.29x = the cost for mileage drivenContinued
7 Solving a Problem Example continued: 2.) TRANSLATE Daily costs 2(24.95)mileage costs0.29x100maximum budgetplus+is equal to=Continued
8 Solving a Problem Example continued: 3.) SOLVE 2(24.95) + 0.29x = 100 x = Simplify left side.Subtract from both sides.49.90 – x = 100 – 49.900.29x = Simplify both sides.Divide both sides by 0.29.x Simplify both sides.Continued
9 Solving a Problem Example continued: 4.) INTERPRET Check: Recall that the original statement of the problem asked for a “whole number” of miles. If we replace “number of miles” in the problem with 173, then (173) = , which is over our budget. However, (172) = 99.78, which is within the budget.State: The maximum number of whole number miles is 172.