Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 8 Number Sense
Problem
Solution
1) In the expression below place minus signs "-" and parentheses on the left hand side of the "=" to create a true equation.
Note: 2(4 - 1) = 6 and - (2) = -2, (2-1) = 1.

3   4   5   6 = 3
  1. The problem is that 6. We can't use "+" so we have to make it positive: -(-6) = +6
  2. Now, if the remainder of the expression can be made to equal to -3, then the total expression will evaluate to +3
  3. That can be accomplished by subtracting 5 from 4 and multiplying it by 3, so the final expression is
  4. 3(4-5) -(-6) = 3
2) If 23 x 37 x 104 = 6, then what is k?
        (36)k x 54
  1. 23 x 37 x 104/54 = 279936
  2. (36)k = 279936 / 6 = 46656
  3. 363 = 46656, so
  4. k = 3
3) Ibrahim wants to buy an X-Box elite console that costs $380. His dad says he'll buy it for him with his credit card but Ibrahim has to pay him back. Ibrahim can pay him $10 on the first day of every month. Ibrahim's dad will also charge him 18% annual interest on the unpaid balance at the end of every 12 month cycle (year) so that he can get an idea of how credit cards work. If Ibrahim agrees, how much will he end up paying his dad for the console?
  1. If Ibrahim pays his dad $10 every month, it will take him 380/120 = 3 years and 2 months to pay him back
  2. The interest he will pay is:
    1. End of year 1: (380 - 120) x .18 = $46.80
    2. End of year 2: (380 - 2x120 + $46.80) x .18 = $33.62
    3. End of year 3: ($20 + $46.80 + $33.62) x .18 = $18.08
  3. The total he pays is $380 + $46.80 + $33.62 + $18.08 = $478.50

Problem
Solution
4) Athena uses two packs of 500 sheets of paper to make the estimate that one sheet of paper is 4 x 10-3 inches thick. A paper mill produces 5.22 x 106 sheets of paper per day. If all the sheets produced in a day were stacked up, how many feet high would the stack reach? Express your answer in feet without using scientific notation.
  1. 5.22x106 x 4x10-3 = 20.88x103 inches.
  2. 20.88x103 / 12 = 1.74x103 = 1740 feet.

5) The math teacher plays a game of Guess My Number with her class for the last 10 minutes of the period. The students narrow it down to a number between 100 and 500. With 2 minutes left she tells them the square root of the number is a whole number greater than 15 and the square of the square root of the number is a whole number with a 9 as the units digit. What's the number?
  1. √500 = 22.36, so the square root of the number is between 16 and 22.
  2. Squares of the numbers between 16 and 22:
    256 289 361 400 441 and 484
  3. The number is 289