Assigned: Thursday, February 5, 1998
Due: Tuesday, February 10, 1998
Do 1, 4, 5, and 7.
Read all the problems before you attempt to solve them! You don't have to start with the first one!
Note. Most of these problems require numeric answers. Though many have answers that you can find by simply guessing and checking, most can also be solved using some simple algebraic statements.
If you are normally inclined to a strategy based on guessing, try algebraic solutions: use letters to stand for quantities and be careful in your definition of what quantities they stand for. Then, use the problem to help you write mathematical statements about the relationships between these quantities.
If you are normally inclined to a strategy based on algebraic solutions, you might want to see if there is a reasonable guessing strategy.
Warning Some problems may not be solvable by algebraic methods (and may not even by solvable).
0. Gender-based Group Learning
At Dartmouth, I participated in a regular discussion group on teaching and learning in the sciences. One interesting point that was raised (and that has some supporting research that I'll admit that I don't know well) is that while all learners do better when they work in groups, there are clear gender differences in what types of groups are best for which kinds of learners. Women generally do better in single-sex groups, men generally do better in mixed-sex groups.
Propose a hypothesis for why this is so and suggest a procedure that we might use to support or disprove your hypothesis.
1. A Lifetime in Math
[Based on the notes of Emily Moore.]
A woman passed one-sixth of her life in childhood, one twelfth in youth, and one-seventh more as a single woman. Five years after her marriage, she gave birth to a son who died four years before she did at half her final age. What was the woman's final age?
2. Three More Generations
My mother is twice as old as I am. The ratio between my mothers's age and my age is half the ratio between my age and my daughter's age. When I am as old as my mother, I will be eight times as old as my daughter is now. How old are we?
Is this similar to a problem we have recently encountered? How does it differ?
3. Selling Candy
[From the notes of Emily Moore. Extended by Sam Rebelsky]
A store stocks 10,000 candy bars. Of these, 1000 have neither nuts nor chocolate, 6500 have chocolate, and 7200 have nuts. How many have both chocolate and nuts? How many have nuts but no chocolate?
Is this question similar to any questions we've examined? Which ones is it most similar to?
4. The Fake Coin
[From an unknown source. Wording by Sam Rebelsky]
Melvin and Myra Miser have just received nine coins. They know that one is a fake and that it has a different weight than the other coins. A local scientist is willing to sell them time on his balance (which allows them to compare the weights of two groups of coins), but they must reserve and pay for the time in advance.
Given their miserly outlook, they want to reserve the shortest time they can. What is the minimum number of comparisons they have to do to guarantee that they've found the fake coin? How can they determine which coin is fake in that number of weighings?
How does your answer change if they have twelve coins? How does your answer change if they know that the fake coin is heavier than the other coins? Lighter?
I've found ways to do nine and twelve coins in three weighings and nine coins with a known difference in two weighings. Can you match my numbers? Can you beat them?
5. Campus Bikes
[Loosely based on a problem in the notes of Emily Moore.]
The new president of Grinnell has decided that Grinnell should keep a supply of Zoomsters on campus so that students, staff, and faculty can more easily get around campus. Experience shows that Zoomsters last for ten years and then fall apart, at which point the pieces are worth $100. Acme also sells Zoomcoat, which they claim gives the Zoomster a lifespan of five more years. However, Zoomcoat-treated bikes have no scrap value.
Acme has signed a contract committing to sell us Zoomsters for $1,100 and Zoomcoat for $300 for the foreseeable future. If Grinnell makes 5% interest on the endowment, should it purchase Zoomcoat?
If you ignore the interest, the problem should be much easier. You may want to try the easier problem first.
Is this problem similar to any other problem we've encountered? If so, which one(s) and how?
6. Differential Taxes
[Based loosely on a problem in the notes of Emily Moore.]
In the midst of a heated debate on fixed-rate taxation, a senator proposes a sliding scale tax rate, in which your tax rate is based on one five-thousandth of your income. For example, if you make $5,000, you pay 1% of your income in taxes, if you make $10,000, you pay 2% of your income; if you make $100,000, you pay 20%; if you make $500,000 or more, you pay 100% of your income.
Obviously, this isn't the most sensible policy (unless you believe in a form of social equity that limits the top amount someone can earn). Nonetheless, we will treat it as such. Under this policy, what is the optimal income?
7. The Bricklayers
[From the notes of Emily Moore. Modified slightly by Sam Rebelsky.]
A contractor estimated that one of her two bricklayers would take 9 hours to build a certain wall and the other 10 hours. However, she knew from experience that when they worked together, 10 fewer bricks were laid per hour. Since she was in a hurry, she put both workers on the job and found it took exactly 5 hours to build the wall. How many bricks did the wall contain?
8. The City Gates
[From the collective unconscious, modified by Sam Rebelsky.]
When I arrived at the city gates, I met someone with seven mates. Each mate had seven sacks. Each sack had seven cats. Each cat had seven kits. Kits, cats, sacks, and mates, how many were at the city gates?
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