Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 7 Algebra
Problem
Solution
1)There are 832 steps from the basement to the restaurant level of the Space Needle. You start out going up 25 steps every 20 seconds. After one minute you slow down to going up 24 steps every 20 seconds during the next minute, then 23 steps every 20 seconds in the next minute and so on. If this pattern continues, how long, to the nearest minute, will it take you to reach the restaurant?
  1. You start out at 75 steps per minute.
  2. After that you decrease by ___ steps every minute
  3. The progression is 75 + ___ + ___ + ___ + ...
  4. At this point add these numbers in your calculator until you get a sum greater or equal to 832 decreasing the number of steps by 3 at each minute. The number of minutes necessary to get to 832 steps is _____




  5. The climb takes _____ minutes.

2) Three watermelons and two cantaloupes weigh 32 pounds. Four watermelons and three cantaloupes weigh 44 pounds. All watermelons weigh the same and all cantaloupes weigh the same in this problem. What is the weight of two watermelons and one cantaloupe? 		Let W = the weight of a watermelon and C = the weight of a cantaloupe
This is 2 equations and 2 unknowns.
  1. Then the first equation is __________________

  2. The second equation is _____________________

  3. Turning the first equation into an equation for C:

        C = _________________
  4. Plug this into the second equation and solve for W:



        W = _____
  5. Plug this value for W into the first equation to get C:



        C = _____
  6. So, 2W + C = _____ pounds

Problem
Solution
3) A group of scientists is duplicating cells for a science experiment. They observe that during each 10-minute period, one-quarter of the cells present died and the rest of the cells all doubled. If there were 32 cells initially, how many cells were there after 40 minutes? If 1⁄4 of the cells died, then 3⁄4 lived, so, at each 10 minute interval multiply the number of cells by 3⁄4 and then by 2.



The number of cells left after 40 minutes was _____ cells.
4) A special four digit number has the following traits:
  • All of the digits are different
  • The digits in the tens place is 3 times the digit in the hundredths place.
  • The digits are in descending order, so the digit in the tens place is greater than the digit in the ones place, the digit in the ones place is greater than the digit in the tenths place and so on.
  • The sum of the digits is 27.
  1. Since there are "tenths" and "hundredths" in this number it is a decimal number.
  2. Since there is a "tens place", the number is of the form
        XX.XX
  3. Use the other clues to investigate possibilities for each digit.





    The number is ___________
5) How many integers satisfy (n - 4)(n + 6) ≤ 0?
  1. Look at the first expression (n - 4).
    Any number greater than ____ results in a positive value for the expression.
  2. Look at the second expression (n + 6).
    Any number less than _____ results in a positive value because a negative times a negative is a positive.
  3. So the range is ____ integers.