8th graders, click on the following link for practice problems for Tuesday's test on Systems:


7th graders, click on the following link for practice problems for Wednesday's test on Data and Probability:


Problem of the Week


A New Mathematical Operation


If a*bab– b, find (2*3)*4



A Quarter Pounder


A quarter-pound hamburger contains approximately 80 calories per ounce of meat, an average french-fry contains about 14 calories, a cola contains about 10 calories per ounce, and a bun contains 200 calories. Suppose you have a quarter-pound hamburger with a bun and six ounces of cola. How many french-fries can you eat and still keep your meal below 800 calories?


15. We have 80h+ 14f+ 10c+ 200 < 800. Using the information in the problem we can write 80(4) + 14f+ 10(6) + 200 < 800. This implies that 14f< 220 and f< 16. Therefore, you can eat 15 french fries.


Perfect Squares and Cubes...NOT


How many numbers from 1 to 1 million, inclusive, are not perfect squares or perfect cubes?


998,910. There are 1000 perfect squares between 1 and 1 million; these are the squares of the first 1000 integers. Similarly, there are 100 perfect cubes—the cubes of the numbers from 1 to 100. Subtract the squares and the cubes from 1 million to get 998,900. However, every number that is a perfect sixth power has been subtracted twice (the largest of these is 106= 1,000,000). Adding these back in gives 998,910.


999 Coins


Starting with a single pile of 999 coins, a person does the following in a series of steps: In step one, he splits the pile into two nonempty piles. Thereafter, at each step, he chooses a pile with 3 or more coins and splits this pile into two piles. What is the largest number of steps that is possible?


997. The number of steps is one less than the number of piles, and 998 is the largest number of piles, 997 with 1 coin and 1 with two coins.


House Prices


The average house price in Boomtown rose 30 percent each year for the last five years. If the average house price is currently $250,000, what was the average house price five years ago?


Approximately $67,332. If P was the average house price five years ago, then the current average price is 1.35P or 3.71293P. Thus, 250,000 = 3.71293P, so P=250,000/3.71293, which equals $67,332.27


Upright Integers


An integer is defined as upright if the sum of its first two digits equals its third digit. For example, 145 is an upright integer since 1 + 4 = 5. How many positive three-digit integers are upright?



45. From the definition, the first and second digits of an upright integer automatically determine the third digit, which is the sum of the first two digits. Consider those upright integers beginning with 1: 101, 112, 123, 134, 145, 156, 167, 178, and 189; there is a total of 9 such numbers. (Note that the second digit may not be 9; otherwise, the last “digit” would be 1 + 9 = 10.) Beginning with 2, the upright integers are 202, 213, 224, 235, 246, 257, 268, and 279; there is a total of 8 such numbers. We may continue this pattern of analysis to show that the numbers of upright integers beginning with a digit of 3, 4, 5, 6, 7, 8, or 9 are 7, 6, 5, 4, 3, 2, and 1, respectively. Therefore, there is a total of 9+8+7+6+5+4+3+2+1 = 45 three-digit upright integers.





A man has $10,000 to invest. He invests $4,000 at 5 percent and $3,500 at 4 percent. To have a yearly income of $500 from the investment, at what rate must he invest the remainder of the money?



About 65 minutes (to the nearest minute). The losing rates are $1000/hr., $500/hr., and $333.33/hr. The combined losing rate is $1833.33/hr. Thus,





What Is the Price?


Matthew and Matilda want to buy a set of DVDs. Matthew has $47 less than the purchase price, and Matilda has $2 less than the purchase price. If they pool their money, they still do not have enough to buy the DVDs. If the set costs a whole number of dollars, what is its price?




If Pis the price of the DVDs, Matthew has P- 47 dollars and Matilda has P- 2 dollars.

We know that (P- 47) + (P- 2) < P, since the amount of their combined savings is still less than the price of the DVDs.

Solving for P, we have P< 49; but P> 47, since Matthew had some money. Therefore, Pis a whole number such that 47 < P< 49, so the price of the DVDs is $48.


Tom and Bill


Tom is standing in a hole that is 4 feet deep. Bill asks him how much deeper he is going to dig the hole. Tom replies that he will dig 4 feet 2 inches deeper and that the top of his head will then be the same distance below ground level that it is now above ground level. How tall is Tom?


6 feet 1 inch. The top of his head will go down 4 feet 2 inches with the additional digging. Half that distance was above the hole before the additional digging, so he is 4 feet + 2 feet 1 inch, or 6 feet 1 inch, tall.



Wire Length

In the diagram, the rectangular wire grid contains 15 identical squares. The length of the grid is 10. What is the length of wire needed to construct the grid?


76. Since the length of the rectangular grid is 10, the side of each square in the grid is 10 ÷ 5 = 2. The height of the grid is therefore 6 (3 squares). There are four horizontal wires, each of length 10, and six vertical wires, each of length 6, for a total length of wire of 4 • 10 + 6 • 6 = 40 + 36 = 76.