Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2008 Grade 7 Number Sense
Problem
Solution
1) In the diagram the sum 1/3 + 1/6 + 1/12 + 1/24 + 1/48 + 1/96 represents the area of the shaded sections. If the area of the entire rectangle is 1, what is the sum of the areas of the two unshaded regions?
  1. The large unshaded region is equal to the first = 1⁄3
  2. That makes the sum of the shaded regions and the little unshaded square 2⁄3
  3. Convert all the fractions to 96ths and add:
    32⁄96 + 16⁄96 + 8⁄96 + 4⁄96 + 2⁄96 +1⁄96 = 63⁄96
  4. So the area of the little unshaded region is:
    2⁄3 - 63⁄96 = 1⁄96
  5. The sum of the unshaded regions =
    1⁄96 + 1⁄3 = 1⁄96 + 32⁄96 = 33⁄96 = 11⁄32
2) For a baseball team, the ratio of the team's wins to its losses is 3:2. If the team continues at this pace, how many games will it win out of 110 games?
  1. The ratio 3:2 makes the win fraction 3⁄5 because the ratio divides the number of games into fifths (3+2).
  2. 3⁄5 x 110 = 3 x 22 = 66 wins
3) In the sequence below, what is the value of the 20th term?
  1   1   1  
2 1 - 2 -1
3 1 - 2 + 3 2
4 1 - 2 + 3 - 4 -2
  1. Another column has been added to the table to the left to do the sums indicated.
  2. As can be seen, for each even numbered term the sum is -n/2
  3. Therefore the 20th term is -10

Problem
Solution
4) Mr. Tenpercent always marks up his retail price by 10% and then offers a 10% discount when he has trouble selling an item. If retail price for a particular item is $1000.00 what's his final price when he applies the discount?
  1. A markup of 10% is a multiply of the principal by 1.1, while a discount of 10% is a multiply of the principal by .9
  2. ($1000 x 1.1) x .9 = $1000 x .99 = $990
5) Insert one additional set of parentheses to make the following number sentence true.
8 - 6 - 3 x (4 - 10) ÷ 2 + 1 = 18


  1. The expression (4-10) = -6, so it must be multiplied by another negative number to get a positive number. Here is the sentence after that simplification:
    8 - 6 - 3 x - 6 ÷ 2 + 1 = 18
  2. So it must be multiplied by either -3 or -(6 -3).
    Let's try the first option (-3):
    8 - 6 +18 ÷ 2 + 1 = 18
  3. There is no way to make the expression evaluate to 18 with that, so let's try the second option -(6 -3):
    8 - (6 - 3) x - 6 ÷ 2 + 1 = 18
    8 - 3 x (- 6) ÷ 2 + 1 = 18
    8 + 18 ÷ 2 + 1 = 18
    8 + 9 + 1 = 18 and this is correct!
  4. So the answer is
    8 - (6 - 3) x (4 - 10) ÷ 2 + 1 = 18