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 3 steps every minute
  3. The progression is 75 + 72 + 69 + 66 + ...
  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.
        75 + 72 + 69 + 66 + 63 + 60 + 57 + 54 + 51 +
        48 + 45 + 42 + 39 + 36 + 33 + 30 = 840
  5. The number of minutes necessary to get to 832 steps is 16
The climb takes 16 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 3W + 2C = 32
  2. The second equation is 4W + 3C = 44
  3. Turning the first equation into an equation for C:
        C = (32 - 3W) / 2
  4. Plug this into the second equation:
        4W + 3(32 - 3W) / 2 = 44
        4W + (96 - 9W)/2 = 44
        8W + 96 - 9W = 88
        W = 8
  5. Plug this value for W into the first equation to get C:
        3x8 + 2C = 32
        2C = 8
        C = 4
  6. So, 2W + C = 2x8 + 4 = 20 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? 1. After 10 minutes: #cells: 32(3/4) x 2 = 48
2. After 20 minutes: 48(3/4) x 2 = 72
3. After 30 minutes: 72(3/4) x 2 = 108
4. After 40 minutes: 108(3/4) x 2 = 162 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. If the tens place digit is 3 times the hundredth's place digit, the number is one of 3 forms:
    • 3 X . X 1
    • 6 X . X 2 or
    • 9 X . X 3
  4. If the digits are in decending order then 3X.X1 is not possible because there aren't enough digits for the 2 in the middle
  5. If the sum is 27 then the 6X.X2 form is not possible because the max sum is 65.42 which is a sum of 15
  6. So, it is of the form 9X.X3. 9+3 = 12, so the missing 2 digits must add to 27 - 12 = 15. The only digits that fit that requirement are 8 and 7.
  7. The number is 98.73

5) How many integers satisfy (n - 4)(n + 6) ≤ 0?
  1. Any number greater than 4 results in a positive value for the expression.
  2. Any number less than -6 results in a positive value because a negative times a negative is a positive.
  3. So the range is -6 to +4 = 11 integers