Complete all work on a separate piece of paper.
- Duplicate this chart on your paper. Then, determine if each number is evenly divisible by 2, 3, 4, 5, 6, 9, or 10. Write YES or NO in the appropriate place on the chart. (10 points)
For questions 2 and 3, write the questions and the answers and show all steps. Full credit will only be given if all work is shown.
- Find a shortcut to get these answers. (25 points)
1 + 2 + 3 + 4 + 5 + ..... + 98 + 99 + 100 = __________
1 + 3 + 5 + 7 + 9 + ..... + 95 + 97 + 99 = __________
- 1 + 2 - 3 + 4 - 5 + 6 - ..... + 98 - 99 + 100 = __________
- 1 - 2 + 3 + 4 - 5 - 6 + 7 + 8 - ..... - 97 - 98 + 99 + 100 = __________
- 2 + 4 - 6 + 8 - 10 + 12 - ..... + 196 - 198 +200 = __________
- Find the sum of each infinite series. (15 points)
27 + 9 + 3 + 1 + 1/3 + .....
7/5 +6/5 +5/5 + 4/5 + 3/5 + .....
1 + 3/4 + 9/16 + 27/64 + 81/256 + .....
All word problems should be answered using complete sentences, and paragraphs if necessary. Draw a picture if you would find it helpful. You do not have to rewrite the questions.
Your answers should be convincing! How do you know your answer is right? Some questions will require much explanation; others, not much at all.
- Alex, Beverly, and Calvin live on the same straight road. Alex lives 10 miles from Beverly and Calvin lives 2 miles from Beverly. How far does Alex live from Calvin? (5 points)
- If posts are spaced 10 feet apart, how many posts are needed for 100 feet of straight line fence? (5 points)
- Six glasses are standing in a row. The first three are empty and the last three are full of water. By moving only one glass, it is possible to change this arrangement so that no empty glass is next to another empty one and no full glass is next to another full one. How can this be done? (5 points)
- A farmer has to get a fox, a goose, and a bag of corn across a river in a boat that is large enough only for him and one of these three items. If he leaves the fox alone with the goose, the fox will eat the goose. If he leaves the goose alone with the corn, the goose will eat the corn. How does he get all the items across the river? (5 points)
- A Strange Journey
A person walks 10 miles south, then immediately 10 miles east, and then immediately 10 miles north. After doing this the person is back where he started. How is this possible? (5 points)
- Connect The Dots Duplicate this arrangement of dots on your paper. Then, without lifting y our pen or pencil, draw four straight lines through the dots so that every dot has a line going through it. Do not retrace a line previously drawn. (5 points)
- The Counterfeit Coin
You have nine coins, but you are told that one of the coins is counterfeit and weighs a little more than an authentic coin. How can you determine the counterfeit with just two weighings on a two-pan balance scale? (10 points)
- The Inheritance
In the will of a recently deceased old couple, it states that their land should be given to their four children. However, there is no mention of how the land is to be divided, except that it should be split by three north-south dividing lines. What remains unknown is where the dividing lines should be placed and how the four strips are to be assigned among the heirs.
Since no agreement can be reached, they have come to you as an unbiased arbiter to help resolve the situation. Can you find a way to divide and assign the land in a way that will make all four feel that they got a fair deal?
Note: This problem is not as simple as merely dividing up the land equitably. You have to do more than split it in a way you think is fair. The land has to be divided in such a way that each recipient believes that he or she got a fair deal. Keep in mind that each person will have slightly different criteria for what makes a piece of land valuable. (10 points)
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