Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 8 Algebra
Problem
Hint
1) Two teams enter a contest that has two parts to it. Part A problems are each worth the same amount, and Part B problems are each worth the same amount. The Geeks score 54 points. The Whizzes score 51 points. Their captains find out that the Geeks answered 10 problems in Part A correctly and 12 problems in Part B correctly. The Whizzes answered 11 problems in Part A correctly and 9 in Part B correctly. What would the Whizzes have scored if they had answered 2 more problems in Part B correctly for a total of 11 Part A problems and 11 Part B problems? It is important to note that although each question in part A is worth the same points and the same with point B, that the part A points are not worth the same as point B points. We need to find out what points were worth in each part. This is a 2 equations and 2 unknowns problem. Let x = the number of points for a part A problem and b the number of points for a part B problem. We need to know y to answer this problem.
  1. The Geeks equation:

  2. The Whizzes equation:

  3. Solve these 2 equations for y:





    y = _____
  4. The Whizzes would have scored ____ if they had answered 2 more part B problems.
2) Alex and Ian are shopping for new clothes. Alex says to Ian, "If you give me $5, we will have an equal amount of money and that would be fair." Ian responds, "That may be true, but if you give me $4, I will have twice as much money as you and that would be much better." How much money do Alex and Ian have together? This is another 2 equations with 2 unknowns problem. Let A = the amount of money Alex has and N the amount of money Ian has.
  1. The first equation is:

  2. The second equation is :

  3. Solve these 2 equations for A and N and add them together:




    A = ____ N = ____
  4. Together, they have _______

Problem
Hint
3) Mrs. Jyoti was keeping track of how many mystery books and how many adventure books her students were reading from the class collection. If this pattern in book reading continues, how many mystery books will be read in January and how many adventure books?
  1. The number of mystery books is mutiplied by ____ every month, so the number of mystery books read in January would be _____

  2. The number of adventure books is multiplied by ____ every month, so the number of adventure books read in January would be
    _________
4) Trina falls off a 300 foot cliff. Her height above the ground is modeled by the equation
T = 300 - 32t, t in seconds.
She's in luck and Superman flies after her. His height above the ground is modeled by the equation
S = 500 - 57t, t in seconds.
How high off the ground are they when he catches Trina? 		We first must find the time at which Trina and Superman meet, so we set the 2 equations equal to each other and solve for t:




t = ___ seconds
Now plug this time into either of the equations to get the height:



= _____ feet.
5) Jesse creates a pattern of equilateral triangles with rhombuses and triangles, the first three are shown. He sees that each rhombus is two triangles. How many rhombuses will he need if he wants to build the 20th triangle in the pattern?
 Let n = the number of the triangle.
  1. Counting just the small triangles, the sequence is:
    ___    ___     ___,
    which is _____________ (expression with n)
  2. So the 20th triangle contains ______ small triangles
  3. Counting the small triangles that are not part of a rhombus:
    ___     ___     ___
    which is ___________ (expression with n)
  4. So, the number of small triangles in the 20th triangle is ____
  5. Subtracting the small triangles and dividing by 2:
    ______ rhombuses.