Hands-on math!

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

1) Find the 100th term in this sequence:
    1, 5, 4, 8, 7, 11, etc
  1. This is an arithmetic sequence combining 2 sequences.
  2. Rewrite this as 2 sequences:
        (1) the sequence of only the odd terms and
        (2) the sequence of only the even terms:

        Odd sequence:
        1 , ____, ____, ____, ____ , ...

        even sequence:
        5 , ____, ____, ____, ____ , ...
  3. Determine the difference between each successive term of these sequences = ____

  4. The 100th term of the orginal sequence is the 50th term of the above even sequence, so, using the formula for the nth term of an arithmetic sequence:
        An = A1 + (n-1)
    Solve for A50:

  5. A50 = _________

2) If you insert three numbers between 1/6 and 1 to form an arithmetic sequence (i.e., values evenly spaced), what would the sum of this sequence?
(Express your answer as a mixed number)
  1. The sum of an arithmetic sequence is:
        S = (A1 + An) n / 2
  2. Using the information about this sequence, solve for S:

        S = _____

Problem
Hint

3) Liz is making cupcakes for a charity event. She decides that every 3rd cupcake will have chocolate frosting, every 4th cupcake will have sprinkles, every 5th cupcake will use colored batter, and every 6th cupcake will be topped with a cherry. If Liz makes 80 cupcakes, how many will have none of these additions? 		The approach here is to count all the ones that get special toppings and then remove all the dupicated ones due to those that get multiple toppings. This involves finding the least common multiples (lcm) of each of the combinations.
We will use the equation for finding the Nth term of an arithmetic sequence to do the counting. That equation is:
    An = A1 + (n - 1) d
    Turning this around into an expression for n:

    n = ____________________
  1. Count the ones with special toppings
    • Every 3rd cupcake: (Chocolate frosting)
          n = _______________
    • Every 4th cupcake:(sprinkles)
          n = _______________
    • Every 5th cupcake: (colored batter)
          n = _______________
    • Every 6th cupcake: (cherry)
          Since 6 is divisible by 3 all those that get cherries
          also get chocolate frosting so n = _____
    Total = _____ with special toppings
  2. Remove the duplicates
    • Every 3rd and 4th:
          lcm(3,4) = ___: (Every ___th cupcake gets chocolate frosting and sprinkles)
          = _____ cupcakes
    • Every 3rd and 5th:
          lcm(3,5) = ___: (Every ___th cupcake gets chocolate frosting and colored batter)
          = _____ cupcakes
    • Every 4th and 5th:
          lcm(4,5) = ____: (Every ___th cupcake gets sprinkles and colored batter)
          = ____ cupcakes
    • But the first cupcake gets everything so we can't remove it multiple times! So we subtract ___ from the number of duplicates.

Duplicates = _____ cupcakes that have more than 1 topping.

Total that have any topping = ____

Therefore, the ones that have no topping are ____ cupcakes

Problem
Hint

4) What is the missing number in the figure below?
 Using N for the figure number:
  1. Look at the lower left (LL) numbers:
        5, 9 13
    The expression for the lower left (LL) number in terms of N is
        LL = __________
  2. Look at the lower right numbers:
        10, 36, ?
    The expression for the lower right (LR) number in terms of the lower left (LL) number is

        LR = ___________
  3. Using this expression for LR solve for the last number ?:

        ? = _____________
Note: You don't need the top number.

5) If 2x = 64 and xy = 216, what is yx ?
  1. If 2x = 64, then x = ___

  2. If xy = 216, then y = ___

  3. Then yx = ____