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An Introduction to Problem Solving 2.5 An Introduction to Problem Solving

Strategy for Problem Solving General Strategy for Problem Solving 1. UNDERSTAND the problem. Read and reread the problem. Choose a variable to represent the unknown. Construct a drawing. Propose a solution and check. 2. TRANSLATE the problem into an equation. 3. SOLVE the equation. 4. INTERPRET the result: Check proposed solution in problem. State your conclusion.

Example Twice a number plus 3 is the same as the number minus 6. Objective A

Finding an Unknown Number Example The product of twice a number and three is the same as the difference of five times the number and ¾. Find the number. 1. Understand Read and reread the problem. If we let x = 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 difference of five times the number and ¾” translates to 5x – ¾.

continued 2. Translate The product of · the difference of – is the same as = twice a number 2x 5 times the number 5x and 3 3 and ¾ ¾

continued 3. Solve 2x · 3 = 5x – ¾ 6x = 5x – ¾ 4. Interpret Check: Replace “number” in the original statement of the problem with –¾. The product of twice –¾ and 3 is 2(–¾)(3) = –4.5. The difference of five times –¾ and ¾ is 5(–¾) –¾ = – 4.5. We get the same results for both portions. State: The number is –¾.

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? x = the number of whole miles driven, then 0.29x = the cost for mileage driven 2(24.95) + 0.29x = 100

continued 2(24.95) + 0.29x = 100 49.90 + 0.29x = 100 49.90 – 49.90 + 0.29x = 100 – 49.90 0.29x = 50.10 x  172.75

continued 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 49.90 + 0.29(173) = 100.07, which is over our budget. However, 49.90 + 0.29(172) = 99.78, which is within the budget. State: The maximum number of whole number miles is 172.

Example The measure of the second angle of a triangle is twice the measure of the smallest angle. The measure of the third angle of the triangle is three times the measure of the smallest angle. Find the measures of the angles. Draw a diagram. Let x = degree measure of smallest angle 2x = degree measure of second angle 3x = degree measure of third angle

continued Recall that the sum of the measures of the angles of a triangle equals 180. measure of first angle measure of second angle measure of third angle equals 180 x 2x + 3x + = 180

continued x + 2x + 3x = 180 6x = 180 x = 30 Check: If x = 30, then 2x = 2(30) = 60 and 3x = 3(30) = 90 The sum of the angles is 30 + 60 + 90 = 180. State: The smallest angle is 30º, the second angle is 60º, and the third angle is 90º.

Example The sum of three consecutive even integers is 252. Find the integers. x = the first even integer x + 2 = next even integer x + 4 = next even integer Translate: x + x + 2 + x + 4 = 252

continued The sum of three consecutive even integers is 252. Find the integers. x + x + 2 + x + 4 = 252 3x + 6 = 252 3x = 246 The integers are 82, 84 and 86.