Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 8 Algebra
Problem
Solution
1) Emil cooks 64 hotdogs for a barbeque. He uses 5 packages of hotdogs plus 4 hotdogs left over from a meal earlier in the week. A friend wonders how many hotdogs were in a package. If x is the number of packages and h is the number of hotdogs, write an equation that could be used to answer the wondering. From the first sentence, 64 hotdogs came from 5 packages + 4 extra dogs, so, if the number of packages is x, then :
64 = 5x + 4

So, the number of dogs/package, using this expression, is:
5x = 60
x = 12
2) Each camp counselor at Camp Wallaby walked 6 miles for health and fitness activity. Each camper walked 2 miles. The camp leader paid $0.50 into a Fun Day account for every mile walked. They raised $48. If the ratio of camp counselor to camper is 1 to 13, how many campers are at Camp Wallaby?
 This is 2 equations with 2 unknowns. Let X = number of counselors and Y = number of campers.
  1. First equation: 13X = Y
  2. Second equation: Y + 3X = 48
  3. Substitute the first expression for Y into the second equation:
        13X + 3X = 48
        X = 48/16 = 3
  4. Substitute this in the first equation to get the number of campers:
        Y = 13x3 = 39

Problem
Solution
3) Two teams enter a contest that has two parts to it. In part A all problems are worth the same number of points, and in Part B problems all problems are worth the same number of points. The Bulldogs score 71 points. The Rams score 72 points. The captain of the Bulldogs finds out that her team answered 9 problems in Part A correctly and 5 problems in Part B correctly. The Rams answered 11 problems in Part A correctly and 4 problems in Part B correctly. What are the Part A problems worth? (If you're stuck, find a way of writing 1 point in terms of A and B.) Another problem with 2 equations and 2 unknowns Let A be the value of part A problems and B the value of part B problems.
  1. Equation 1: The Bulldogs: 9A + 5B = 71
  2. Equation 2: The Rams:    11A + 4B = 72
  3. Turn equation 1 into an equation for A:
        A = (71 - 5B) / 9
  4. Plug this into equation 2:
        11/9(71 - 5B) + 4B = 72
        11(71 - 5B) + 36B = 648
        781 - 55B + 36B = 648
        133 = 19B
        B = 7
  5. Plug this back in equation 1:
        9A + 5x7 = 71
        9A = 36; A = 4
Editor's note: By the way, we were not stuck!
4) A biologist studies an insect colony with an initial population of 50 insects. She notices that every three weeks after a new generation is born the population has doubled. What was the difference between the population size after the fourth generation and the population size after the sixth generation?
  1. Population after the 4th generation = 50(24) =50x16 = 800
  2. Population after the 6th generation = 50(26) = 50x64 = 3200
  3. The difference is 3200 - 800 = 2400
5) The table shows some data about the number of students enrolled at a junior high. A line fits the data well for the data shown. If the line is accurate, what is the enrollment for 2015?
  1. Use the first 2 data points to determine the increase in students per year:
    (1245 - 1200)/1 year = 45 students/year.
  2. Apply this to 2015:
    1650 + (2015 - 2010) x 45 = 1650 + 225 = 1875