Math 4791/5791
Problem Set 1 - Word Problems - Fall 1998

Mathematical modeling could be described as the process of turning word problems into mathematics and choosing the appropriate tools for a solution. Some people might call it solving word problems! The following problems give you an opportunity to practice this process. These problems are not in order of dificulty and require many different tools. Use the tools appropriate to the task and do not work harder than necessary!

Using a fixed list, I will call on one student each class period to give an oral solution to a problem of his/her choice from the collection below. The chosen problem must not have been solved already during the semester. A complete, fully justified and convincing solution must be given that cannot exceed 5 minutes. Clearly you must be present to give a solution on your day! On the class period following your oral presentation, you must turn in a written solution. A small percentage of your grade will depend on your oral and written solutions.

1. Coffee and Milk. You transfer a teaspoon of coffee from a cup of coffee into a cup of milk. You then transfer a teaspoon of the mixture in the milk cup back into the cup of coffee. Compare the amount of coffee in the milk cup to the amount of milk in the coffee cup (is it more, less or the same?).
2. Clock Hands. At what time between 1:00 and 2:00 is the minute hand of a clock directly over the hour hand? At what time between n:00 and (n+1):00 does this occur?
3. Lost Time. Ralph can reach his destination on time if he averages 60 miles per hour. Halfway to the destination (in distance) he realizes that he has averaged only 30 miles per hour. How fast must he travel for the rest of the journey to arrive on time? A fraction p of the way to the destination he realizes he has averaged only v miles per hour. For what values of p and v can he still arrive on time?
4. Parades and Bikes. A cyclist begins at the tail of a parade that is four kilometers long and begins riding in the direction that the parade is moving. By the time the cyclist rides to the head of the parade and returns to the tail, the parade has traveled six kilometers. Assuming that the cyclist and the parade move at constant (but different) speeds, how far did the cyclist travel ?
5. Candles. Two candles of equal length are lit at the same time. One candle takes six hours and the other candle takes three hours to burn out. After how much time will one candle be exactly twice as long as the other?
6. Snowplow. With snow on the ground and falling at a constant rate, a snowplow begins plowing down a long straight road at noon. It plows twice as far in the first hour (between noon and 1:00) as in the second hour (between 1:00 and 2:00). When did the snow start falling? Assume that the speed of the plow is inversely proportional to the snow depth.
7. The Second Race. Jill and Jack ran a 100 meter race. When Jill crossed the finish line Jack had run only 95 meters. They decided to race again and Jill handicapped herself by starting 5 meters behing the starting line. If they both run at the same constant speed as they did in the first race, who wins the second race? (Do not use the distance-speed formula.)
8. Balls in Barrels. Ten large barrels are each filled with many golf balls which all look alike. However, the balls in one barrel weigh twice as much as the regular balls, which weight one ounce. With one weighing on a scale that gives absolute weight, how can you find the barrel with the counterfeit golf balls?
9. Walking and Riding. If a lady walks to work and rides home, it takes one and a half hours. When she rides both ways, it takes half an hour. How long would it take to walk the round trip?
10. Slingshots. Twenty people can produce 4 slingshots in 2 hours. How long will it take 15 people to make 30 slingshots? How many people are needed to make 40 slingshots in 2 hours? How many slingshots can be made by 5 people in 12 hours?
11. Solutions. A car radiator has a capacity of 21 liters and is filled with an 18 percent antifreeze solution. How many liters of the solution must be drained and then replaced by a 90 percent solution to leave a 42 percent solution in the radiator?
12. The Chelsea Pensioners (an old classic). If 70% of the pensioners have lost an eye, 75% have lost an ear, 80% have lost an arm, and 85% have lost a leg, what percentage of the pensioners (at least) have lost an eye, an ear, an arm and a leg?
13. Passing Boats. Two boats leave from opposite shores of a river at the same time and travel at constant but different speeds. They pass each other 700 yards from one shore, continue to the banks where they turn around. On their return trip the boats pass again 400 yards from the opposite shore. How wide is the river?
14. High School Mathematics Contest 1966. ``An escalator (moving staircase) of n uniform steps visible at all times descends at constant speed. Two boys, A and Z, walk down the escalator steadily as it moves, A negotiating twice as many escalator steps per minute as Z. A reaches the bottom after taking 27 steps while Z reaches the bottom after taking 18 steps''. Find n.
15. Buffon Needle Problem. Imagine a large horizontal board marked with parallel lines one inch apart. A one-inch-long needle is thrown randomly on the board. What is the probability that it lands on one of the parallel lines?
16. Colliding Ladybugs. Four ladybugs are at the corners of a ten-inch square. At the same moment each bug begins crawling toward its neighbor to the left at the same constant rate. How far do the bugs walk before they all meet?
17. Truth in Advertising. Three boxes are labeled APPLES, ORANGES and APPLES AND ORANGES. Each label is incorrect. Can you select one fruit from one box and determine the correct labels?
18. Working Together. Ann and Betty can do a job in 10 days; Ann and Carol can do the same job in 12 days; Betty and Carol can do the same job in 20 days. How long will it take Carol to do the job alone?
19. Parallel Pipes. Pipes A and B can fill a tank in two hours and three hours respectively. Pipe C can empty the full tank in five hours. If all pipes are opened at the same time when the tank is empty, how long will it take to fill the tank?
20. Stacking Dominoes. A set of dominoes have the same thickness and are 1 inch wide and 2 inches long. The dominoes are stacked on top of each other with their long edges aligned so that each domino overhangs the one beneath it. If there are n dominoes in the stack, what is the largest distance that the top domino can be made to overhang the bottom domino? How many dominos can be stacked altogether before the stack topples?
21. Commuting. A woman usually takes the 5:30 train home from work, arriving at the station at 6:00 where her husband meets her to drive her home. One day she left work early and took the 5:00 train, arrived at the station at 5:30 and began walking home. Her husband, leaving home at the usual time, met his wife along the way and brought her home ten minutes earlier than usual. How long did the woman walk?
22. Balls in a Can. Three tennis balls fit exactly (no room to spare) in a cylindrical can. Which is greater, the circumference or the height of the can?
23. Birthdays. Reuben says, "Two days ago I was 20 years old. Later next year I will be 23 years old." How is this possible?
24. Average Speed. A car went up a hill at 10 miles per hour and back down the same hill at 20 miles per hour. What was the average speed of the round trip?
25. Trains in Tunnels. A one-mile-long freight train goes through a one-mile-long tunnel at 15 miles per hour. How long does it take for the train to pass through the tunnel?
26. Switching Lockers. Five hundred students arrive for the first day of class and there are 500 numbered lockers at the school, all of which are closed. The first student switches every locker (if it's closed, it switches to open, and vice versa); the second student switches every other locker. The nth student switches every nth locker for . Describe the final state of the lockers.