Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2007 Grade 7 Algebra
Problem
Solution
1) 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 20 minute 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 = 5 + 8.75 =
13.75 minutes. = 13 min 45 sec.

2) 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.

Problem
Solution
3) Rosita wants to take a DVD movie with her on the family vacation. She compares prices. Happy Viewing will give her the DVD for 3 days at $3.00 and then charges $1.50 per day for a late return. Movies-At-Home will give her the DVD for 5 days at $5.00 and charges $1.75 per day for a late return. For how many days can she keep the movie if the total cost to her is the same at both stores? This is the break even day.
 Let D = the number of days the two move prices break even.
  1. The expression for Happy Viewing is: 3 + 1.5(D - 3)
  2. The expression for Movies_At_Home is : 5 + 1.75(D - 5)
  3. Set these 2 expressions equal to each other and solve for D:
        3 + 1.5(D - 3) = 5 + 1.75(D - 5)
        3 + 1.5D - 4.5 = 5 + 1.75D - 8.75
        1.5D - 1.75D = 5 - 8.75 +4.5 - 3
        -.25D = -2.25
        D = -2.25 / -.25 = 9 days
4) Lisa is designing her own house. She draws a side view sketch of the stairs she wants to use to connect the downstairs with the upstairs. She notices there's a line that connects all the edges of the steps as shown in the diagram. If the coordinates giving the edge of the second step are ( 2ft, 3/2ft), what are the coordinates of the edge of the top step? You must find the equation for a line given the coordinates of the second step and the fact that the line passes through the origin.
  1. The equation is of the form y = mx + b where:
    m is the slope and b is the y-intercept
  2. The y-intercept (b) is 0.
  3. The slope (m) = 3⁄2 ÷ 2 = 3⁄4 The equation is y = 3⁄4 x + 0
  4. The top step is at x = 14, so
        y = 14 x 3⁄4 = 21⁄2 = 10.5 feet
  5. So the coordinates of the top step are (14,10.5)

5) At Emerald Middle School there can be at most 26 students in a class. If Danny's class has twice as many boys as girls what's the biggest his class can be?
  1. If Danny's class has twice as many boys as girls then the boys are 2⁄3 of the class.
  2. 2⁄3 of 26 is 171⁄3 which can't be because you can't have 1⁄3 of a student!
  3. The number of students must be a multiple of 3 and the next lowest number of students that fit this is 24