Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2007 Grade 6 Algebra
Problem
Solution
1) Define the operation V as p V q =
(p + q)/(2p + q).
What is 6 V 4 in lowest terms?		 1. In the expression for V, p = 6 and q = 4.
2. Put these values into the equation for V and solve =
    (6 + 4)/(2x6 + 4) =
    10/16 = 5/8
2) Rory knows he is 8 inches shorter than his sister. He's figured out that his height in inches is 7/8 her height in inches. How tall is he in inches? 1. If he is 7/8ths of his sister's height then what remains is the fraction 1/8.
2. This fraction of his sister's height is 8 inches.
3. Rory's sister's height is
    Let h be his sister's height.
    Then h/8 = 8 inches.
    h = 8 x 8 = 64 inches.
    He is 8 inches shorter than this which is 56 inches.
3) Mrs. Pack notices that in her stamp drawer she still has unfinished rolls of old 8 cent stamps and 15 cent stamps. She needs to mail a package for which the postage is $0.78. What's the fewest number of 8 cent stamps and 15 cent stamps she can use to get the exact postage amount? 1. Divide 78 cents by 15 to get the number of 15 cent stamps = 5
2. Compute the remainder by subtracting the value of this many 15 cent stamps from 78 cents = 78 - 15x5 = 3 cents.
3. If this remainder is not a multiple of 8, reduce the number of 15 stamps by 1 and repeat until you get a remainder that is divisible by 8
4. Mrs. Pack needs 6 8 cent stamps and 2 15 cent stamps.

Problem
Solution
4) A TV station uses the following method for determining commercial time for any of its special 2 hour broadcasts. It divides the broadcast into 20 minute segments. In the first segment no commercials are aired. Then for each subsequent 20 minute segment the amount of time given to the special is reduced by 25% from the previous 20 minute segment. The remainder of the 20 minutes is given to commercials. In the first hour how much time is spent on commercials to the nearest second? Fill out this table:
Time segment
(mins)		% special% commercialCommercial
time (mins)
0-20100%0% 0
21-4075% 25%5 min
41-6056.25% 43.75%8.75 min
Total commercial time = 13.75 minutes. = 13 min 45 sec.
5) In the following pattern of rectangles, how many squares are needed to make the nth rectangle? Express your answer in terms of n.
 1. For each n (1 - 3) write out the number of cubes on the bottom row:
              n = 1     2     3
      # cubes  2     4     6
2. The expression for this is 2n in terms of n.
3. Notice that the number of layers is equal to n.
    and that you already have 1 layer accounted for.
4. The expression for the nth term is
    2n + 2n(n - 1)
    Combine terms: 2n + 2n2 - 2n
    Expression = 2n2
    Note: the expression 2n + 2n(n - 1), although not reduced should be considered correct also.