Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2016 Grade 7 Algebra
Problem
Hint

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.
X has the greater value in equation ___

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:
  • (From the last measurement): Compute the number of bronze coins that equal a copper coin = ______
  • Substitute this in the second equation to get the number of bronze coins that equal a silver coin = _______
  • Substituting the above 2 expressions, ccmpute the number of bronze coins that equal 2 gold coins = ______ bronze coins
  • Divide this by 2 to get the number of bronze coins that equal 1 gold coin = ______ bronze coins
  • Multiply by 12 to get the number of bronze coins that equal 12 gold coins = _____ bronze coins.
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, write the inequality for Marcia's accumulated money after each week and solve for N when this exceeds $65

    N = ______
If this is not an integer, increment it to the next integer = _____

Problem
Hint

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? 		Time for 1st mile = ____ min.
Ignoring the 11⁄2 mile time, the increase in times for each mile are:
  1. Mile 1 to mile 2: ___ min = ____ minutes slower
  2. Mile 2 to mile 3: ___ min = ____ minutes slower than mile 1-2
Continuing this slow down rate:
  1. Mile 3 to mile 4: ___ min = ____ minutes slower than mile 2-3
  2. Mile 4 to mile 5: ___ min = ____ minutes slower than mile 3-4
So, adding these times to the mile 3 times gives us 6:49 + _____ minutes = ______

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 =
        ____ 25-cent stamps + ____ 15 cent stamps = $2.30
  2. Every time you decrease the number of 25-cent stamps by ___ you increase the number of 15-cent stamps by ___, because the least common multiple of 25 and 15 is ____.
  3. So, the progression is (starting from the maximum number of 25-cent stamps you computed above and decreasing the number of 25-cent stamps by ___ for each row):
    # 25-cent
    stamps    		 # 15-cent
    stamps    		Total $
    $2.30
    $2.30
    $2.30
    $2.30
    There are ____ to cover the exact postage.