Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 7 Algebra
Problem
Solution
1) Danny bought a total of 20 game cards. The specialty cards cost $0.25 each and the rest cost $0.15 each. If Danny spent $4.20 to buy these cards how specialty cards did he buy? This is 2 equations with 2 unknowns
Use S for the number of specialty cards and and
C for the number of ordinary cards
  1. Set up the 2 equations:
        S + C = 20
        0.25 S + 0.15 C = 4.20
  2. Turn the first equation into an equation for C:
        C = 20 - S
  3. Substitute this expression for C into the second equation and solve for S:
        0.25 S + 0.15 (20 - S) = 4.20
        0.25 S + 3 - 0.15 S = 4.20
        0.10 S = 1.20
        S = 12. He bought 12 specialty cards
2) Tyler drove at a constant speed for 2 hours. He stopped for an hour to do some shopping and have a bite to eat. He then drove back home at a constant speed. Draw a possible graph on the axes provided with time as the input and total distance traveled in the car as the output.)

Problem
Solution
3) Before district play, the basketball team won 10 of their games, or 40% of their games. During district play, they won six more games and finished the season having won half their games. What percentage of their games did they win at districts? Round to the nearest percent.
  1. If the team won 10 games and that was 40% of their wins, then they played
        0.4 x = 10
        x = 10 / 0.4 = 25 games
  2. They won 10 + 6 = 16 games, which is half their games, so the total number of games played = 32
  3. The number of district games played =
    32 - 25 = 7
  4. Since they won 6 games at district out of 7, their winning percentage is (100 x 6) / 7 = 85.7% = 86%
4) I am a mystery number. If you multiply me by 2 and take away 5 the result is the same number when you first add 2 to me and then multiply the result by 5. What number am I? Let n be the number.
The equation is:
    2n - 5 = (n + 2)5
    2n - 5 = 5n + 10
    -3n = 15
    n = -5
5) The table shows some data about the number of students enrolled at a junior high. A line fits the data well for the data shown. If the line is accurate, what is an equation you can use to compute the enrollment for 2015 and other years? Use t as the number of years since 2000, and S for student enrollment.
  1. The equation for a line is y = m x + b where:
    m = the slope of the line and
    b = the y-intercept
  2. For our equation, the year is the x axis and the student enrollment is y. We will call the year 2000 x = 0.
  3. The slope of this line (using the first 2 points) is (change in y) / (change in x) =
        m = (1245 - 1200) / 1 = 45
  4. Plug this into the equation for the line to get the y-intercept (use the first point):
        b = y - m x
        b = 1200 - 45 (0) = 1200
  5. The equation, using S for student enrollment and t for the number of years since 2000 =
        S = 45t + 1200