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Problem Solving

Problem Solving

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Problem Solving

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  1. Problem Solving

  2. Problem Solving • Read the problem. Look for key phrases. Assign a variable to the main unknown. • Plan and think. Write an equation. Make a table. Draw a picture.

  3. Problem Solving • Solve. • Check. Did you answer the question? Does your answer seem reasonable?

  4. Learning the Language • The length is 5 feet longer than the width. • l = 5 + w

  5. Learning the Language • One plane travels 50 miles faster than the other plane. • F = 50 + S

  6. Learning the Language • The hiking rate was 3 mph slower returning than going. • R = G – 3

  7. Learning the Language • There are half as many purple marbles as green marbles. • P = 1/2G

  8. Learning the Language • There are 6 more blue marbles than pink marbles. • B = 6 + P

  9. Consecutive Integer Problems • Consecutive Integers: • 1st = x • 2nd = x + 1 • 3rd = x + 2

  10. Consecutive Integer Problems • Consecutive Odd or Even Integers: • 1st = x • 2nd = x + 2 • 3rd = x + 4

  11. Consecutive Numbers • Find three consecutive odd integers whose sum is 1,197. • x + x + 2 + x + 4 = 1197 • 3x + 6 = 1197 • 3x = 1191 • x = 397 • The three numbers are 397, 399, and 401.

  12. What’s your angle? • Two angles are complementary (add up to 90). The measure of one is 1/2 the measure of the other. What is the measure of each angle?

  13. What’s your angle? • x = one angle • 1/2x = other angle • x + 1/2x = 90 • 3/2x = 90 • 2/3 • 3/2x = 90 • 2/3 • x = 60 • The angles are 60 and 30.

  14. Working hard. . . • Caleb uses 92 feet of fencing to enclose a field whose length is 16 feet longer than its width. What are the dimensions of the field?

  15. Working hard. . . • w: width & l: length = w + 16 • P = 2l + 2w • 92 = 2(w + 16) + 2w • 92 = 2w + 32 + 2w • 60 = 4w; 15 = w • l = w + 16 • l = 15 + 16; l = 31 • The field is 31 feet by 15 feet.

  16. Distance Problems

  17. Distance Formula • Rate x Time = Distance

  18. Solving Distance Problems • Make a table to organize information about the rate and time. • Draw a picture.

  19. Solving Distance Problems • Distance column is always filled in by multiplying rate by time. • Ask – What do you know about the distance?

  20. Example • Martin and Sean planned to meet at a campground. Martin traveled 50 mph and Sean traveled 75 mph. If they left at the same time from Sean’s house, but Sean arrived ½ hour before Martin, how long did Martin and Sean travel?

  21. Martin • Sean r • t = d 50 x 50x • x –1/2 75(x – 1/2)

  22. 50x = 75(x – 1/2) • 50x = 75x – 75/2 • 75/2 = 25x • x = 1.5 • Martin: 1.5 hours • Sean: 1 hour

  23. Example • Brad and Peter traveled toward each other on the 100-mile road between Sioux City and Omaha. Brad left 30 minutes before Peter, but both traveled at 63 mph. When they met, how many miles had each traveled?

  24. Rate • What is Brad’s rate? • Rate = 63 mph

  25. Time • If Peter’s time is “x”, what is Brad’s time? • x – ½ • x + ½ • 1/2x • 2x • None of these

  26. Distance • What is the total distance? • Distance = 100 miles

  27. Time • What is Peter’s time (rounded to the nearest thousandth)? • Time = 0.544 hr

  28. Distance • What is Brad’s distance (rounded to the nearest hundredth)? • Distance = 65.77 miles

  29. Example • Tim and Matt went hiking, and the trip up the mountain took 4 hours. The trip down the same trail only took 3 hours. If the rate going down was 1 mph faster than going up, what was the length of the trail?

  30. Rate • If the rate going up was “x”, what was the rate going down? • x + 1 • x – 1 • x • None of these

  31. Rate • Solve for x. • x = 3

  32. Distance • What is the length of the trail, one way? • Distance = 12 miles LQ8

  33. Example • Two airplanes leave at the same time from airports that are 1300 miles apart and travel toward each other. If one plane travels 120 mph faster than the other, and they pass each other in 2 hours, how fast is each traveling?

  34. Equation • What is the equation? • 2x = 2x + 240 • 2x + 2x + 240 = 1300 • 2x + 240 = 2x + 1300 • 2x + 240 = 1300 – 2x • None of these

  35. Rate • Solve for x. • x = 265 mph

  36. Section 2.5 • pp. 60-61

  37. Problem 1 r • t = d Clark Blair 150 + 165 = x x = 315 miles 50 3150 55 3165

  38. Page 60 • 2. x = blue marbles • 1/2x = red marbles • 1/2x + 8 = yellow marbles • 3. x = number • 4. x = width • x + 21 = length

  39. Page 60 • 5. x = smaller number • x + 81 = larger number • 6. x = A • 4x = B

  40. Page 60 • 7. x = 1st integer • x + 2 = 2nd integer • x + 4 = 3rd integer

  41. Problem 8 1st 2nd 2x + 2x + 252 = 880 x = 157 1st plane: 157 mph 2nd plane: 283 mph r • t = d x 22x x + 126 2 2(x+126)

  42. Problem 9 Go Ret. 3x = 2x + 2 x = 2 Going: 2 mph Returning: 3 mph r • t = d x 33x x + 1 2 2(x + 1)

  43. Problem 13 r • t = d Mr. F. Mr. C. 56x + 64x = 658 x = 5.48 hours 56 x56x 64 x64x

  44. Problem 14 r • t = d Go Ret. 8x = 7x + 7 – 4 x = 3 Total time: 7 hours 8 x8x 7 x + 1 7(x + 1)