Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2007 Grade 8 Algebra
Problem
Solution
1) A bowl of candies sit on the counter. Jerry comes along and takes 1/4 of them. Willow comes along and takes 2/5 of the remaining candies in the bowl, leaving 27. How many were in the bowl to start with?
 This is a classic 'work backwards' problem.
  1. If Willow took 2⁄5 of the remaining candies and that left 27 candies, then those candies were 3⁄5 of the candies after Jerry took his 1⁄4 of the original candies.
  2. 3⁄5 X = 27, so X = 27x5/3 = 9x5 = 45 candies.
  3. Those 45 candies are 3⁄4 of the original candies, so
    3⁄4 X = 45
  4. X (the original number of candies) =
    45x4/3 = 60 candies
2) 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)

Problem
Solution
3) At Emerald Middle School there can be at most 26 students in a class. If the ratio of boys to girls in Danny's is class is 3:2, at most how many girls can be in his class?
  1. If you divide the class into fifths (3+2) then the girls are 2⁄5 of the class.
  2. If the class has at most 26 students, then 2⁄5 x 26 = 10.4, so the most girls that can be in the class is 10
4) An oil slick from a leaking tanker is roughly circular in shape with radius r. The radius increases in size by 1 foot every hour. Write an expression that would compute the area of the slick 3 hours from now if the radius now is r.
  1. Area = r2
  2. The radius at 3 hours is r + 3, so:
    Area = (r + 3)2
5) The lines given by y = 2x + 3 and
y = -x + k meet at the point (2, k-2). What is the value of k?
  1. Substituting x=2 and y=k-2 into the first equation gives us:
  2. k - 2 = 4 + 3
  3. k = 7 + 2 = 9