Math 2000 - Solution 1

Fall 2000

Part C

1. The key to this problem is reading it very carefully. The question is: How many apples must you draw from the basket to be sure of getting two apples of one kind? Suppose the three kinds of apples are red, green, and yellow (you could say red delicious, granny smith, and yellow delicious). On the first draw, suppose you select a red apple. Thinking in terms of the “worst case scenario,” suppose you select a green apple on the second draw. If you did select a red apple on the second draw, then you would have two of the same kind, but you can’t be sure this will happen. And suppose that on the third draw, you select a yellow apple. So far you still don’t have two apples of the same kind. But what must happen on the fourth draw? You must select an apple that matches one of the apples you have drawn. Thus, it takes four draws to be sure of having two apples of the same kind.

2. The convoluted Congressman has used a triple negative. So let’s unravel the negatives (where I have underline the negative terms in the analysis). If the Congressman supported anti-abortion demonstrations, we might conclude that he opposes abortion. If the Congressman supported a ban on anti-abortion demonstrations, we might conclude that he supports abortion. Therefore, because he opposes a ban on anti-abortion demonstrations, we might conclude that he opposes abortion. I realize that there are other interpretations to this problem; for example, the Congressman may just oppose demonstrations of all kinds. But I have looked at this as a logical problem.

3. We will talk more about converting fractions to decimals and percentages. If you use your calculator to divide 2 by 5, you will see that 2/5 = 0.40. And the decimal 0.40 is the same at 40%.

4. There may have been some ambiguity in this question. I intended ten billion, four hundred and six million, eight hundred and twenty three to be one number; but I can see how it might have been read otherwise. As one number it is 10,406,000,823.

5a. The expression (450 ¸ (6 + 23)) ´ 14 should be evaluated in steps starting with the innermost parentheses:

(450 ¸ (6 + 23)) ´ 14 = (450 ¸ 29) ´ 14 = 15.52 ´ 14 = 217.24.

I have rounded the numbers to two decimal places.

5b. 3⁴ ´ 5² = (3 × 3 × 3 × ) × (5 × 5) = 81 × 25 = 2025.

6. The statement 1.06 quarts = 1 liter means it takes 1.06 quarts of water to fill a one-liter bottle. Thus, it takes more than one quart to fill a one-liter bottle, which means one quart must be smaller than one liter. Perhaps more evident is the case of 1 yard = 3 feet, which means one foot is smaller than one yard.

7. Recall that 60% means six-tenths or 0.6. IT follows that the number of married people is

0.6 × 120 = 72.

Therefore, the number of unmarried people (which is what was asked) is 120 - 72 = 48 people. This solution assumes that every person is either married or unmarried, but not both or neither!

8. One way to compare to two numbers is to divide one by the other. To compare 1 billion = 1,000,000,000 to 1 thousand = 1,000, we can divide:

Thus, 1 billion is 1 million times larger than 1 thousand. To compare one one-thousandth =0.001 to one tenth = 0.1, we again divide:

We see that than or, equivalently, one one-thousandth is 100 times smaller than one tenth.

9. Here is one way to visualize this problem. Imagine a pie cut into six nice equal pieces. Half of the pieces is three pieces and one-third of the three pieces is one piece. One piece out of six, is one-sixth or 1/6 of the pie. Alternatively, you could recall that “of” usually implies multiplication. So one-third of one-half is

If you are not too comfortable working with fractions, we can use you calculator to see that

The squiggly equal sign means “approximately equal to.” Sometimes fractions cannot be represented exactly by decimals.

10. I realize that there is more than one way to interpret the question. Let’s assume that the words “a warehouse contains bicycles, tricycles, and cars” means that there is at least one of each vehicle. (One student assumed that because each term is plural, there must be at least two of each vehicle.) Noting that bicycles, tricycles, and cars have two, three, and four wheels, respectively, we can use a little trial and error. If there is at least one of each vehicle, then we have 2 + 3 + 4 = 9 wheels. So we need to find ways to get nine more wheels. One solution is to have two of each vehicle. We could also have one bicycle, four tricycles, and one car (check 2 + 12 + 4 =18) and four bicycles, two tricycles, and one car (check 8 + 6 + 4 =18). I think that’s all!

11. We can make a Venn diagram with a circle for softball and a circle for soccer. Because everyone in the story is a nurse, we can’t use a circle for nurses. The region common to both circles is for nurses who play both soccer and softball; we are told there are 50 such nurses. Of the 75 nurses who play soccer, 50 also play softball; therefore, 75 - 50 = 25 nurses play soccer only. Of the 95 nurses who play softball, 50 also play soccer; therefore, 95 - 50 = 45 nurses play softball only. With these figures, we can make the Venn diagram shown below. Notice that each nurse appears in only one region. Adding up the number of nurse in the diagram, we have accounted for 120 nurses (and we don’t even know how many nurses play neither sport. We are told that there are 100 nurses in the entire group, so there must be an error in the study.