Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 7 Number Sense
Problem
Hint

1) Find the largest whole number x, so x 8 > 8x
  1. Rearrange the expression as:
        x8   >   1
        8x
  2. Note that:
        88 = 1,
        88
  3. Use this to solve for the values of x around 8 to find the value that maximizes the expression

2) Place groups of parentheses to make the statement true
    6 - 2 x 3 + 2 ÷ 5 - 3 - 43 = 5
  1. First note that 43 = 64. There is no way the rest of the expression can be close to 64, therefore a set of parentheses must enclose the 4 with at least the 3 and possibly other numbers in this expression.
  2. Try just enclosing the 3 and 4 in parentheses and simplify.
  3. Note that the divide by 5 means that whatever is to the left of that divide must be a multiple of 5
  4. Keep working the problem from right to left until the expression simplifies to 5

Problem
Hint

3) The same relationship is used in each square. What is the missing number?
 Call the 4 numbers in the square a (top left), b (top right), c (bottom left) and d (bottom right)
  1. The first square shows that there must be a multiplication involved in order to result in d = -34, but which ones?
  2. Try different combinations of
    • axb ± c
    • axc ± b
    • bxc ± a
        ? = ______

4) Simplify each expression. Find what the 11th term in the sequence would be. Express your answer as a fraction in lowest terms.
  1. Simplify the first 3 terms and write them down.
  2. Extend the sequence to the 4th term and simplify it.
    (Be careful! It's easy to make a mistake here!)
  3. You should notice a pattern at this point

5) A box of red pens holds 13 pens. A box of black pens holds 21 pens. Chuck needs to order exactly 1,550 pens. For his order, he wants the number of red pens to be as close to the number of black pens as possible. How many boxes of black pens should he order? 		Let R = # boxes of red pens and B = # boxes of black pens. To restate the problem, we have 2 equations with 2 unknowns:
  • 13R + 21B = 1550 (exactly)
  • 13R = 21B (approximately)
Since we need exactly 1550 pens, we must have integer numbers for R and B that result in exactly 1550 pens.
  • Solve these 2 simultaneous equations for B as if the result is exact. B = ______. This is not an integer.
  • The easiest way is to create a table and increase integer B values until we find an integer value for R that results in exactly 1550 pens, like this:
    B R =
    (1550-21B)/13