Hands-on math!

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

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 , 4, 7 ...

        even sequence:
        5 , 8, 11 ...
  3. Determine d, the difference between each successive term of these sequences = 3 (for both of them)

  4. The 100th term of the original 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) d
    Solve for A50:

        A50 = 5 + (49) x 3 = 152
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. S = (1⁄6 + 1) x 5 /2
  3. Simplifying:
        S = 7⁄6 x 5 / 2 =
        S = 35⁄12 =
        211⁄12
Note: To find the sum of an arithmetic sequence amazingly does not require any of the terms except the first, the last and the total number of terms. So, what are the 3 inserted numbers? It doesn't matter. You don't need them!

Problem
Solution
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?

Cupcakes
Code:
Blue = every 3rd
Red = every 4th
Yellow = every 5th
White = no topping
Total 		= 27 cupcakes
= 13 cupcakes
=   8 cupcakes
= 32 cupcakes
    80 cupcakes
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 = (An - A1) + 1
                  d
  1. Count the ones with special toppings
    • Every 3rd cupcake: (Chocolate frosting)
          n = (79 - 1)/3 + 1 = 27
    • Every 4th cupcake:(sprinkles)
          n = (77 - 1)/4 + 1 = 20
    • Every 5th cupcake: (colored batter)
          n = (76 - 1)/5 + 1 = 16
    • Every 6th cupcake: (cherry)
          Since 6 is divisible by 3 all those that get cherries
          also get chocolate frosting so n = 0
    Total = 27 + 20 + 16 = 63 with special toppings.
  2. Remove the duplicates
    • Every 3rd and 4th:
          lcm(3,4) = 12: (Every 12th cupcake gets chocolate frosting and sprinkles)
          80/12 = 6.666 = 7 (cupcakes 1,13,25,37,49,61 and 73)
    • Every 3rd and 5th:
          lcm(3,5) = 15: (Every 15th cupcake gets chocolate frosting and colored batter)
          80/15 = 5.333 = 6 (cupcakes 1,16,31,46,61 and 76)
    • Every 4th and 5th:
          lcm(4,5) = 20: (Every 20th cupcake gets sprinkles and colored batter)
          80/20 = 4 (cupcakes 1,21,41 and 61)
    • But the first cupcake gets everything so we can't remove it multiple times! So we subtract 2 from the number of duplicates.
    Duplicates = 7 + 6 + 4 -2 = 15
Total that have any topping = 63 - 15 = 48
Therefore, the ones that have no topping are 80 - 48 = 32 cupcakes

Problem
Solution
4) What is the missing number in the figure below?
 Using N for the figure number:
  1. The expression for the lower left (LL) number is 4N + 1
        For N = 1     LL = 4N+1 = 5
        For N = 2     LL = 4N+1 = 9
        For N = 3     LL = 4N+1 = 13
  2. The expression for the lower right (LR) number is (4N + 1)(2N)
        For N = 1     LR = (4N+1)(2N) = 5x2 = 10
        For N = 2     LR = (4N+1)(2N) = 9x4 = 36
        For N = 3     (4N+1)(2N) = 13x6 = 78
  3. Just for completeness (although we don't need it), the expression for the top number (T) is:
        T = LR (2N):
        For N = 1     T = 10 (2) = 20
        For N = 2     T = 36 (4) = 108
        For N = 3     T = 78 (6) = 312

5) If 2x = 64 and xy = 216, what is yx ?
  1. If 2x = 64, then x = 6
  2. If 6y = 216, then y = 3
  3. Then 36 = 729