Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 8 Algebra
Problem
Hint
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 _____ then the inequality is satisfied.
  2. By the first expression (n - 4) if n is less than ____ then the inequality is satisfied.
  3. So the range of n is ____ ≤ n ≤ ____, so the number of integers that satisfy the the inequality is ____
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 = ____
    Equation 2: SS + S = ____
    Equation 3: M + SS = ____

  2. Do some substitution of expressions from one equation into another until you can find an expression for S:

    S = _____
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 = ___ Difference with output: ___
    22 = ___ Difference with output: ___
    32 = ___ Difference with output: ___
    42 = ___ Difference with output: ___

  2. You should notice some pattern with the differences that will lead you to an equation for each output given the input n. Use this expression to compute the n=5 and n=6 terms and then add them.


    Expression for output given the input n = ____________
  3. Sum of terms 5 and 6 = ______

Problem
Hint
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 getting a value for x and then plug both those x and y values in the third equation to solve for k



    k = _____
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.) Here is the table of 4-cent and 7-cent stamp combinations:
# 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
7 28 35 42 49 56 63 70
By examining this table you should be able to find the largest amount of postage that cannot be made with 4- and 7-cent stamps


= _____ cents.