Course:MATH110/Archive/2010-2011/003/Homework/3

Homework 3, third part - Problem-Solving Skills

One of the best selling mathematics books of all time is a slim paperback called How to Solve It. Written by the Hungarian-American mathematician George Pólya (1887−1985), it is regarded as the first testament of mathematical problem solving. Because much of calculus involves problem-solving, the ideas in this book are extremely relevant.

At the heart of Pólya’s book is a four-step method designed to guide problem solving. The method can be summarized as Understand, Plan, Execute, Check. Here is Pólya’s four-step method with a few annotations included.

Step 1. Understand the problem.

The first step in problem solving is to determine where you are going. Be sure that you understand what the problem is asking. Read the problem carefully! If it helps, read it aloud. Record the quantities and conditions that are given for the problem. Identify the unknowns. Exactly what is to be determined? Draw a picture or diagram to help you organize the information and visualize the problem. If possible, restate the problem in different ways to clarify it.

Step 2. Plan a strategy for solving the problem.

Once you understand the problem, the next step is to decide how to go about solving it. This step is the most difficult; it requires creativity, organization, and experience. Try to think of a similar or related problem. Map out your strategy with a flow chart or diagram. Identify the appropriate analytical or computational tools needed for the solution

Step 3. Execute your strategy, and revise it if necessary.

After devising a strategy, the next step is to carry it out. Keep an organized record of your work, which will be helpful if revisions are needed. Double-check each step so that you do not propagate errors to the end of the solution.Assess your strategy as you work; if you find a flaw, return to Step 2 and revise your strategy.

Step 4. Check and interpret your result. It's tempting to stop after Step 3; however, the final step may be the most important. Be sure that your result makes sense; for example, check that it has the expected units and that numerical values are sensible. Recheck your calculations or find an independent way of checking the result. Check the consistency of the result by considering special or limiting cases. Write the solution clearly and concisely.

The following problems involve little mathematics. However, they may be challenging and they will allow you to exercise Pólya’s four-step method. Write out full solutions and discuss whether and how Pólya’s method was useful.

Problems

1. A bus traveled from the terminal to the airport at an average speed of 30 mi/hr and the trip took an hour and 20 min. The bus then traveled from the airport back to the terminal and again averaged 30 mi/hr. However, the return trip required 80 min. Explain.

2. A lady did not have her driver's license with her when she failed to stop at a stop sign and then went three blocks down a one-way street the wrong way. A policeman saw her, but he did not stop her. Explain.

3. One of three boxes contains apples, another box contains oranges, and another box contains a mixture of apples and oranges. The boxes are labeled APPLES, ORANGES and APPLES AND ORANGES, but each label is incorrect. Can you select one fruit from only one box and determine the correct labels? Explain.

4. I am the brother of the blind fiddler, but brothers I have none. How can this be?

5. Two quarters rest next to each other on a table. One coin is held fixed while the second coin is rolled around the edge of the first coin with no slipping. When the moving coin returns to its original position, how many times has it revolved?

6. Three kinds of apples are all mixed up in a basket. How many apples must you draw (without looking) from the basket to be sure of getting at least two of one kind?

7. Suppose you have 40 blue socks and 40 brown socks in a drawer. How many socks must you take from the drawer (without looking) to be sure of getting (i) a pair of the same color, and (ii) a pair with different colors?

8. Reuben says, “Two days ago I was 20 years old. Later next year I will be 23 years old.” Explain how this is possible.

9. A rope ladder hanging over the side of a boat has rungs one foot apart. Ten rungs are showing. If the tide rises five feet, how many rungs will be showing?

10. Suppose one-half of all people are chocolate eaters and one-half of all people are women. (i) Does it follow that one-fourth of all people are women chocolate eaters? (ii) Does it follow that one-half of all men are chocolate eaters? Explain.

11. A woman, her older brother, her son, and her daughter are chess players. The worst player’s twin, who is one of the four players, and the best player are of opposite sex. The worst player and the best player have the same age. If this is possible, who is the worst player?

12. A Manhattan fellow had a girlfriend in the Bronx and a girlfriend in Brooklyn. He decided which girlfriend to visit by arriving randomly at the train station and taking the first of the Bronx or Brooklyn trains that arrived. The trains to Brooklyn and the Bronx each arrived regularly every 10 minutes. Not long after he began his scheme the man's Bronx girlfriend left him because he rarely visited. Give a (logical) explanation.

13. If a clock takes 5 seconds to strike 5:00 (with 5 equally spaced chimes), how long does it take to strike 10:00 (with 10 equally spaced chimes)?

14. One day in the maternity ward, the name tags for four girl babies became mixed up. (i) In how many different ways could two of the babies be tagged correctly and two of the babies be tagged incorrectly? (ii) In how many different ways could three of the babies be tagged correctly and one baby be tagged incorrectly?

15. Alex says to you, “I'll bet you any amount of money that if I shuffle this deck of cards, there will always be as many red cards in the first half of the deck as there are black cards in the second half of the deck.” Should you accept his bet?

16. Suppose that each daughter in your family has the same number of brothers as she has sisters, and each son in your family has twice as many sisters as he has brothers. How many sons and daughters are in the family?

17. The zero point on a bathroom scale is set incorrectly, but otherwise the scale is accurate. It shows 60 kg when Dan stands on the scale, 50 kg when Sarah stands on the scale, but 105 kg when Dan and Sarah both stand on the scale. Does the scale read too high or too low? Explain.

18. Alice takes one-third of the pennies from a large jar. Then Bret takes one-third of the remaining pennies from the jar. Finally, Carla takes one-third of the remaining pennies from the jar, leaving 40 pennies in the jar. How many pennies were in the jar at the start?

19. One morning each member of Angela's family drank an eight-ounce cup of coffee and milk, with the (nonzero) amounts of coffee and milk varying from cup to cup. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. What is the least number of people in the family?

20. Of two clocks next to each other, one runs 5 min per hour fast and the other runs 5 min per hour slow. At midnight the clocks show the same time. At what time are they are one hour apart?

21. Sven placed exactly in the middle among all runners in a race. Dan was slower than Sven, in 10th place, and Lars was in 16th place. How many runners were in the race?

22. During a vacation, it rained on 13 days, but when it rained in the morning, the afternoon was sunny, and every rainy afternoon was preceded by a sunny morning. There were 11 sunny mornings and 12 sunny afternoons. How long was the vacation?

23. Suppose you overhear the following conversation: Paul: How old are your three children? Paula: The product of their ages is 36 and the sum of their ages is the same as today's date. Paul: That is not enough information. Paula: The oldest child also has red hair. If you were Paul could you determine the ages of Paula's children? Explain.

24. Two candles of equal length were lit at the same time. One candle took 6 hr to burn out and the other candle took 3 hr to burn out. After how much time was one candle exactly twice as long as the other candle?

25. Two candles of length $L$ and $L+1$ were lit at 6:00 and 4:30, respectively. At 8:30 they had the same length. The longer candle died at 10:30 and the shorter candle died at 10:00. Find $L$.