Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 7 Algebra
Problem
Solution
1) Rudi and Ria go to the gym to do some weight lifting. Rudi goes first on the bench press, loading on three identical weights on each end of the bar. Ria can't lift as much and replaces each of the weights with weights that are 15 lbs lighter. If she is pressing 94 lbs total, what was Rudi pressing?
  1. Divide 94 by 6 to get the weight of one weight =
    94/6 = 152⁄3 pounds
  2. Add 15 pounds to get Rudi's individual weights =
    302⁄3 pounds
  3. Multiply by 6 to get Rudi's total weight =
    184 pounds
2) Megan wants to buy some bracelets and necklaces. Each bracelet is $8 and each necklace is $12. She buys a total of 12 items. How many of each did she buy if she has $100 to spend? Disregard sales tax. This is 2 equations with 2 unknowns.
Let N = the number of necklaces and B the number of bracelets.
  1. First equation: N + B = 12
  2. Second equation: 12N + 8B = 100
  3. Make the first equation an equation for N:
        N = 12 - B
  4. Substitute this in the second equation:
        12(12 - B) + 8B = 100
        144 - 12B + 8B = 100
        -4B = -44
        B = 11
        Therefore N = 12 - 11 = 1
  5. She bought 11 bracelets and 1 necklace.
3) I'm a mystery number. If you multiply me by 4 and then add 3 the result is the same number if you first add 4 to me and then multiply the result by 3. What number am I?
 Using X for the mystery number, the equation is:
    4X + 3 = 3(X + 4)
    4X + 3 = 3X + 12
    X = 12 - 3 = 9

Problem
Solution
4) Consider the pattern of arrangements of squares. The perimeter of the 1st arrangement is 4 units. What is the perimeter of the 10th arrangement of squares?
  1. Write out the sequence of perimeters:
        4 10 16
  2. Each perimeter adds 6 units to the previous one. This is a arithmetic sequence
  3. The equation for the nth term of an arithmetic sequence is:
        An = A1 + d(n-1) where:
        n = the term desired (= 10 for this problem)
        d is the number added to each term (= 6)
  4. A10 = 4 +6 (9) = 58
5) 14 monkeys are bouncing on the bed and one falls off every 2 minutes. Create a graph with appropriate scale that expresses this problem on the coordinate system provided on the answer sheet. The graph should show the x and y intercepts. What is a linear equation that passes through the integer points on the graph?
 See the graph to the left.
  1. 14 monkeys are on the bed, so 14 is the y-intercept
  2. Every 2 minutes one falls off, so the slope is
    (-1)/2 = -1⁄2
  3. so the x-intercept is when there are no monkeys =
    14x2 = 28 minutes.
  4. The equation of a line is
        y = mx + b
  5. So our equation is y = -1⁄2 x + 14