Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2016 Grade 7 Algebra
Problem
Problem
1) In which equation does X have the greater value?
    a) X + 21⁄2 = 51⁄4 or
    b) X - 1.7 = 2.05 or
    c) 6 / X = 2.1 		Solve each equation for X:
  1. ) X + 21⁄2 = 51⁄4
        X = 51⁄4 - 22⁄4 = 11⁄4 = 23⁄4
  2. ) X - 1.7 = 2.05
        X = 2.05 + 1.7 = 3.75 = 33⁄4
  3. ) 6 / X = 2.1
        X = 6 / 2.1 = 2.85 (approx)
X has the greater value in equation b
2) Jim has a balanced scale and several metallic coins. He finds that:
    2 gold coins = 3 silver coins and 1 copper coin
    2 silver coins = 1 copper coin and 1 bronze coin
    3 copper coins = 2 bronze coins

How many bronze coins will it take to balance 12 gold coins? 		Work this problem backwards:
  • 3 copper coins = 2 bronze coins, so
    1 copper coin = 2⁄3 of a bronze coin
  • 2 silver coins = 1 copper coin + 1 bronze coin, so
    1 silver coin = (1⁄2) x (2⁄3 of a bronze coin + 1 bronze coin) =
    1⁄2 x 12⁄3 bronze coins = 5⁄6 of a bronze coin
  • 2 gold coins = 3 silver coins and 1 copper coin, so
    2 gold coins = 3(5⁄6) bronze coins + 2⁄3 of a bronze coin =
    15⁄6 + 4⁄6 = 19⁄6 = 31⁄6 bronze coins
  • So 1 gold coin = 1⁄2 x (19⁄6) = 19⁄12 bronze coins
  • So 12 gold coins = 12 (19⁄12) = 19 bronze coins

Problem
Solution
3) Marcia is saving up to buy a book collection by her favorite author. The book collection costs $65. Marcia receives a $10 allowance every week, and she spends $8 every other week to see a movie with her friends. If she doesn't attend a movie on the week that she begins saving for the book collection, how many weeks will it be before she has saved enough to afford it? So, Marcia spends 8 dollars every other week, starting with the second and gets an allowance of $10 every week. Using N for the number of the week, the equation for the amount Marcia has left after each week is:
    10N -8(N/2) > 65
    10N -4N > 65
    6N > 65
    N > 10 5⁄6 weeks
Since Marcia gets her allowance only on integer number of weeks, she has enough for her book collection after week 11
4) Mikael is training for a fun run by jogging a five-mile loop around a lake every morning. After the first mile, he starts to slow his pace at a steadily decreasing rate. He records the following regarding his progress and time:

What time will he finish his jog? 		Ignoring the 11⁄2 mile time, the times (and increases in times) for each mile are:
  1. Mile 0 to mile 1: 8 min
  2. Mile 1 to mile 2: 11 min = 3 minutes slower
  3. Mile 2 to mile 3: 15 min = 4 minutes slower than mile 1-2
Continuing this slow down rate:
  1. Mile 3 to mile 4: 20 min = 5 minutes slower than mile 2-3
  2. Mile 4 to mile 5: 26 min = 6 minutes slower than mile 3-4
So, adding these times (20 and 26 minutes) to the mile 3 times gives us
    6:49 + 46 minutes = 6:95 = 7:35
5) You are mailing a large envelope that will require $2.30 in postage. Using only stamps valued at 15 and 25 cents, how many ways can you cover the exact postage?
  1. Using the maximum number of 25 cent stamps =     8 25-cent stamps + 2 15 cent stamps.
  2. Every time you decrease the number of 25-cent stamps by 3 (75 cents) you increase the number of 15-cent stamps by 5, because the least common multiple of 25 and 15 is 75.
  3. So, the progression is:
    # 25-cent
    stamps    		 # 15-cent
    stamps    		Total $
    8 2 $2.30
    5 7 $2.30
    2 12 $2.30
    There are 3 ways to cover the exact postage.