Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 6 Algebra
Problem
Solution
1) Terri wants to buy some gum and comic books for a road trip. She is going to buy twice as many comic books as gum. Each package of gum costs $1.10 and each comic book costs $2.75. Terri lives in Oregon so there is no sales tax. What is the maximum number of comic books she can buy if she has $25 to spend? Let C = the number of comic books.
1. Then C x $2.75 is the amount spent on comic books
2. C/2 x $1.10 is the amount spent on gum
3. Write an equation that sums these two and makes them equal to the amount she has to spend and then solve for C.
    2.75 X C + 1.10 X C/2 = 25
    2.75C + .55C = 25
    3.3C = 25
    C = 7.5757.
    round down to 7.
Since this is odd, subtract 1 to get the answer
    6 comic books
2) Gerald buys apples and broccoli for his restaurant in whole numbers of pounds. Apples cost $2.33 per pound and broccoli cost $2.10 per pound. Gerald pays with a twenty dollar bill. If he gets $2.05 back and the difference between the number of pounds of broccoli and apples is less than 3, how many pounds of broccoli did he get? This is a guess-and-check problem.
The amount Gerald spent was $20 - the amount he got back =
    $20.00 - 2.05 = $17.95.
Note: If you divide this amount by 2.2 (roughly the cost of both a pound of apples and broccoli) you get the approximate number of pounds of both = 8.
Use this table:
Guess ## pounds
broccoli		# pounds
apples		    Total   
cost
153$17.49
235$17.95
Gerald bought 3 pounds of broccoli.

Problem
Solution
3) Figures #1, 2 and 3 of a sequence of growing dots figures is shown. If the sequence continues in the growing pattern how many dots would be in figure #9?
 1. The answer involves exponents
    Each figure contains N2 + 1 dots
    where N is the figure number
2. The next 6 figures (#4, - #9) contain
    17, 26 37, 50, 65 and 82 dots.
3. Figure #9 contains 82 dots.

4) Sidewalk Pizza Pockets sells pizza pockets. The table shows the prices for 3 pizza pockets, 6 pockets and 12 pockets. Based on the table how much should one pizza pocket cost to the nearest cent?
1. Divide the price for 3 pockets by 3 to get the price per pocket =
    $2/3 = $0.66667.
2. Round to the nearest cent = 67 cents.
5) Nicky says "Think of a number. Add 7 to it and multiply the result by 4. Then subtract 4, take a quarter of the result and subtract your number. I know the answer." Will Nicky always be right? If so, what is the answer? If not, show with examples why Nicky can't always know the answer. Use X for Nicky's number.
Write out the expression (it involves a lot of parentheses) and reduce it.
    (((X + 7) x 4 - 4))/4 - X
    (4X + 28 - 4)/4 - X
    (4X + 24)/4 - X
    X + 6 - X = 6