Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 7 Algebra
Problem
Solution
1) If the difference of two numbers is 8 and their sum is 14, what is the difference of their squares? This is 2 equations with 2 unknowns Call the 2 numbers A and B.
  1. The first equation is B - A = 8
  2. The second equation is A + B = 14
  3. Turn the first equation into an expression for B:
        B = 8 + A
  4. Substitute the expression for B in the second equation and solve:
        A + (8 + A) = 14
        2A = 6
        A = 3
        B = 8 + A = 11
  5. The difference of their squares is 121 - 9 = 112
2) Corrie is 1/7 her dad's age. He is 28 years old. In how many years will Corrie be able to say "Dad, I'm half your age"?
  1. If Corrie is 1⁄7 her dad's age and he is 28, then she is 28/7 = 4.
  2. Let X be the number of years when she will be haif his age. The equation is:
        4 + X = (28 + X)/2
        8 + 2X = 28 + X
        X = 20 years.
3) Emilio wants to drown his sorrows by eating the full pan of brownies his mom left out on the counter to cool. In five minutes he eats 1/3 of the pan. In the next five minutes he eats 1/3 of the remaining pan. If this pattern continues how much of the pan of brownies will be left after 30 minutes? Express your answer as a fraction. The table provided may help you think it out.
 Here is the table showing the remaining fraction of the pan after 30 minutes:

Time 5 10 15 20 25 30
eats 1⁄3 2⁄3 x 1⁄3 =
2⁄9 		4⁄9 x 1⁄3 =
4⁄27 		8⁄27 x 1⁄3 =
8⁄81 		16⁄81 x 1⁄3
=
16⁄243 		32⁄243 x 1⁄3
=
32⁄729
left
in pan		 2⁄3 2⁄3 - 2⁄9
=
4⁄9 		4⁄9 - 4⁄27 =
8⁄27 		8⁄27 - 8⁄81 =
16⁄81 		16⁄81 - 16⁄243 =
32⁄243 		32⁄243 - 32⁄729 =
64⁄729

Problem
Solution
4) Tim is using a computer program to create a diamond cut out. What four equations can he use to create the diamond shown here?
 Finding the equation of a line from cartesian coordinates, the 4 equations are:
  1. Quadrant I: y-intercept = 4, slope = -4/2 = -2
        y = -2x + 4
  2. Quadrant II: y-intercept = 4, slope = 4/2 = +2
        y = 2x + 4
  3. Quadrant III: y-intercept = -4, slope = -4/2 = -2
        y = 2x -4
  4. Quadrant IV: y-intercept = -4, slope = 4/2 = +2
        y = 2x - 4
5) Juni has a perfect evergreen, meaning its silhouette is always a triangle, in her back yard. She planted it 7 years ago. She has kept a growth chart measuring its base and its height. This year she's having trouble measuring its height. If the tree follows the growth pattern in the chart, how tall should it be this year?
 You only need the last 2 years, 2008 and 2011 to establish the linear pattern. Using X for the height in 2011, the ratio is:
    55/66 = 75/X
    X = 75x66/55 = 90