Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 6 Algebra
Problem
Solution
1) On January 1st, Danny has $5 and Alyssa has $131. Each month Danny saves $12 more than he spends while Alyssa spends $6 more than she saves. At the beginning of what month do they have an equal amount of money? Using N for the number of months past January, set up an equation for "$5 plus 12 times the number of months equals $131 minus $6 times the number of months past January", and solve it.
    5 +12N = 131 - 6N
    12N + 6N = 131 - 5
    18N = 126
    N = 7 and that month is August
2) Swiper, Dora, and Bob collect Pokemon cards. Dora has 3/5 more cards than Swiper. Bob has 2/3 more cards than Dora. If altogether they have 79 cards, how many does Swiper have? Let S = the number of cards Swiper has,
    D = Dora's cards and B = Bob's cards.
1. Dora has 3/5 more cards than Swiper,
    The expression for this is D = (1 + 3/5)S; D = 8S/5
2. Bob has 2/3 more cards than Dora,
    The expression for this is B = (1 + 2/3)D; B = 5D/3
3. Set up an equation that adds these together to equal 79 and solve.
    D + B + S = 79
    8S/5 + 5D/3 + S = 79
    13S/5 + 5D/3 = 79
    Since D = 8S/5, put this back in:
    13S/5 + 5(8S/5)/3 = 79
    13S/5 + 8S/3 = 79
    39S/15 + 40S/15 = 79
    79S/15 = 79
    S = 15
3) Lena is keeping track of time as she walks a trail in the park. This is the chart she created with her data when she got home. What time was it when she passed the 2 3/4 mile marker?
1. Subtract the mile 2 time from the mile 0 time and divide by 2 to get the time/mile = 32/2 = 16 min/mile.
2. Multiply this by 3/4 and add to the mile 2 time =
    3/4 x 16 = 12 minutes + 1:57 = 2:09

Problem
Solution
4) 14 monkeys are bouncing on the bed and one falls off every 2 minutes. Create a 2-coordinate graph with appropriate scale that would allow you to track this data that expresses this problem on the coordinate system provided on the answer sheet. Label the point on the graph that shows how many monkeys are still on the bed after 12 minutes.
Method 1: Plot the points on the graph
1. After 12 minutes 6 monkeys have fallen off the bed.
2. Subtract this from 14 to get the number of monkeys still on the bed after 12 minutes
    = 8 monkeys.
See plot to the left.

Method 2: Use the equation of a line:
  1. Using x for time and y for monkeys on the bed, create 2 points:
      (0,14) starting position
      (2,13) after 1 monkey falls off
  2. Slope = (13 - 14) / (2 - 0) = -1⁄2
  3. Y intercept = 14 (the starting number of monkeys!)
  4. Equation: y = -x/2 +14
  5. Solve for x = 12:
        y = 14 - 12/2 = 14 - 6 = 8 monkeys
Method 3: Use logic:
  1. If 1 monkey falls off the bed every 2 minutes, then in 12 minutes 6 monkeys will have fallen off the bed.
  2. 14 - 6 = 8 monkeys
5) Darryl spends $17.45 on 5 pounds of trail mix from the bulk food aisle in preparation for a hike with his buddies. He now thinks 5 pounds is not enough. How much more can he buy for $10.00 to the nearest tenth of a pound? 1. Compute the cost per pound of the trail mix =
    $17.45/5 = $3.49/pound
2. Divide $10 by this and round to the nearest tenth of a pound =
    $10/$3.49 = 2.865 pounds.
    Rounded = 2.9 pounds