Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 8 Algebra
Problem
Solution
1) How many integers satisfy
(n - 4)(n + 6) ≤ 0?
  1. By the second expression (n + 6) If n is less than or equal to -6 then the inequality is satisfied.
  2. By the first expression (n - 4) if n is less than 5 then the inequality is satisfied.
  3. So the range of n is -6 ≤ n ≤ 4, so the number of integers that satisfy the the inequality is 11
2) Jack's science and math book together weigh 8 pounds. His social studies and science books together weight 11 pounds. His math and social studies books together weight 9 pounds. How much does the science book weigh by itself?

Let S = the weight of the science book, M the weight of the math book and SS = the weight of the social studies book.
  1. Then the 3 equations are:
    Equation 1: S + M = 8
    Equation 2: SS + S = 11
    Equation 3: M + SS = 9
  2. Turn the 3rd equation into an equation for M: M = 9 - SS
  3. Substitute this in the first equation:
    S + (9 - SS) = 8
  4. Turn this equation into an equation for S: S = SS -1
  5. Substitute this value for S into the second equation:
    SS + (SS - 1) = 11
    2 SS = 12; so SS = 6
  6. Therefore the science book, by itself, weighs SS - 1 = 5 pounds

Problem
Solution
3) What is the sum of the next two outputs in the table below?
  1. The values go up so fast there must be an exponent involved. Try squaring each number:
    12 = 1 Difference with output: 0
    22 = 4 Difference with output: 3
    32 = 9 Difference with output: 8
    42 = 16 Difference with output: 15
  2. Notice the differences are the square of the input -1, so add n2-1 to each term:
    12 + n2 - 1 = 1
    22 + n2 - 1 = 7
    32 + n2 - 1 = 17
    42 + n2 - 1 = 31
  3. So the expression for the output (using n as the input number) =
    n2 + n2 - 1 = 2n2 - 1
  4. The next 2 terms are:
    2 x 52 - 1 = 49 and 2 x 62 - 1 = 71,
    so 49 + 71 = 120
4) For what value of k do the lines
y = x⁄4, 3x + 4y = 32, and y = kx - 14
all intersect in exactly one point?
  1. Substitute the first value for y into the second equation:
    3x + 4(x⁄4) = 32
    3x + x = 32   so x = 8
  2. Therefore y = 8/4 = 2
  3. Substituting these values into the 3rd equation:
    2 = 8k - 14
    8k = 16     so k = 2
5) A stamp collector has a huge supply of old 4-cent stamps and 7-cent stamps. What is the greatest amount of postage that cannot be made up using these stamps? (For instance 25 cents can be made with one 4-cent stamp and three 7-cent stamps.)

# 4-cent stamps
↓ 		# 7-cent stamps
0 1 2 3 4 5 6
0 0 7 14 21 28 35 42
1 4 11 18 25 32 39 46
2 8 15 22 29 36 43 50
3 12 19 26 33 40 47 54
4 16 23 30 37 44 51 58
5 20 27 34 41 48 55 62
6 24 31 38 45 52 59 66
The table to the left is 4-cent and 7-cent stamp combinations.
  1. Notice that (for example) if you take the 0 4-cent and 2 7-cent stamps = 14 cents, to get to 15 you go down 2 rows and left 1 = 2 4-cent and 1 7-cent stamp = 15 cents.
    This is true for all numbers above a certain threshold. That's because 2 down (+8) and left 1 (-7) adds 1.
  2. By examining this table, you can see that the sums that are missing are: 5,6,9,10,13,and 17.
  3. The largest of these is 17