Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2008 Grade 8 Number Sense
Problem
Solution
1) Insert an additional set of parenthesis to make the following number sentence true.
8 - 6 - 3 x (4 - 10) ÷ 2 + 1 = 18
  1. The problem is that "+1" at the end of the expression. That means the rest of the expression must resolve to 17, an odd number.
  2. There is no way with a single set of parentheses to separate the 4 and the 10, so let's combine those:
    8 - 6 - 3 x -6 ÷ 2 + 1 = 18
  3. Try grouping the 8 and the 6:
    (8 - 6) - 3 x -6 ÷ 2 + 1 = 18
    = 2 -3x-6/2 + 1 = 12. No good.
  4. Try grouping the 6 and the 3:
    8 - (6 - 3) x -6 ÷ 2 + 1 = 18

    8 - 3x(-3) +1 = 8 + 9 + 1 = 18. That works!
  5. The answer is
    8 - (6 - 3) x -6 ÷ 2 + 1 = 18
2) What is the smallest whole number exponent, n, for which
    1 - (1⁄3) n is greater than 1 - (0.2)3 ?
  1. (0.2)3 = .008 which is 1⁄125
  2. So, (1⁄3) n must be less than 1⁄125, so
    3n must be greater than 125.
  3. Powers of 3: 3,9,27,81,243
  4. 35 is greater than 125 so n = 5
3) A builder orders 2 by 4 lumber for a project and always tags on an extra 5% rounded up just in case. If the total number of 2 by 4s is 625 how many of those are extra 2 by 4s that the builder may not need? Let A = the number of 2x4s necessary for the project, without the extras.
  1. A x 1.05 = 625
  2. A = 625/1.05 = 595
  3. 625 - 595 = 30 extra 2x4s

Problem
Solution
4) A sequence is created as follows:

What is the distance between the 9th and 10th term of the sequence when plotted on a number line?
Express your answer as a fraction.
  1. Evaluating the terms:
    1, .5, .75, .625
  2. Extending this list: 1, .5, .75, .625 .6875 .65625
  3. The difference between these terms is:
    .5, .25, .125, .0625, .03125
  4. Each difference is half the previous difference, so the expression for the differences is:
    D = (1⁄2) n
  5. Evaluating this expression for n = 9:
    D = (1⁄2) 9 = 1⁄512

5) A government program has a cost overrun of 0.3% on a total budget of 2 billion dollars. What will the overrun cost the tax payers? Express your answer in scientific notation.
  1. 2 billion = 2x109 dollars.
  2. .3% = .003 = 3x10-3
  3. 3x10-3 x 2x109 = 6x106 dollars