Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 8 Algebra
Problem
Solution
1) Joni is watching her baby sister stack blocks and has discovered that there's an actual pattern! What is a formula that would allow you to compute how many blocks there are in the nth tower?
 Let n = the tower number.
  1. The number of blocks in each tower is
        1   4   7   10 ...
  2. The formula is 1 + 3(n - 1) or 3n - 2
2) Nancy took and exam which had 26 problems. Her mom told her she would pay her $0.99 for each problem she got right, but Nancy would have to pay her mom $0.61 for each problem she got wrong. If Nancy earned $8.14 for her work on the test, how many problems did she get right?

 This is 2 equations and 2 unknowns.
Let C = the number of problems Nancy got correct and W = the number she got wrong.
(Using cents instead of dollars for convenience)
  1. First equation: 99C - 61W = 814
  2. Second equation C + W = 26
  3. Turn the second equation into an equation for C:
    C = 26 - W
  4. Plug this into the first equation:
    99(26-W) - 61W = 814
    2574 - 99W - 61W = 814
    160W = 1760
    W = 11
    C = 26 - W = 15

Problem
Solution
3) I'm a mystery number. If you multiply me by 3 and then add 4 the result is the same number if you first add 3 to me and then multiply the result by 4. Write an equation to represent this problem. Then find the mystery number.

 Let N be the mystery number.
Then the equation is:
3N + 4 = 4(N + 3)
3N + 4 = 4N + 12

N = -8
4) A⁄7 + B⁄13 = 89⁄91
What is A + B if both fractions are positive?
  1. Convert both fractions to a common denominator = 7x13 = 91:
    A⁄7 = 13A⁄91
    B⁄13 = 7B⁄91
  2. 13A + 7B = 89, so
  3. 7B = 89 - 13A
    For A and B to be positive integers, 89 - 13A must be a multiple of 7 to make an integer B.
  4. Subtract multiples of 13 from 89 until you get a number divisible by 7:
    89 - 13 = 76 (doesn't work)
    89 - 26 = 63 <--- this works!
  5. So, A = 2, then
  6. 7B = 89 - 26 = 63, so B = 9
  7. Therefore 2 + 9 = 11
5) Sandy was scuba diving and exploring under an overhanging rock shelf. She lost track of time and realized just in time that she was almost out of air. Her location is as shown on the diagram. If she has 2.1 cubic feet of air left and she uses it at a rate of 0.7 cubic feet per minute, how fast does she have to swim along the shortest path to arrive safely on the surface? Express your answer rounding up to the nearest foot per minute to make sure she makes it.
 We must use the pythagorean theorem to find the diagonal distance (C) to the surface:

  1. C = √ 113 2 + 21 2)  = √13210 = 114.93 ft
  2. 2.1 cu. ft ÷ 0.7 cu. ft/min = 3 minutes of air
  3. She must swim at
    114.93 ft / 3 minutes = 38.3 ft/min = 39 ft/min