Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 7 Algebra
Problem
Solution
1) Billy wants to purchase a new bike which costs $185. Billy mows lawns in his neighborhood to earn money. For each lawn Billy mows he makes $10. After every 3 lawns he mows, Billy must refill his mower's gas tank. It costs $8.50 to refill the tank. How many whole lawns will Billy have to mow to make enough money to buy the new bike?
  1. For every 3 lawns, Billy earns $30 and spends $8.50 for a profit of $21.50
  2. Dividing the price of his bike by this gives:
        $185/$21.50 = 8.6 or 8 3-lawn cycles.
  3. At the end of these 8 3-lawn cycles he has made a profit of $21.50 x 8 = $172, leaving him needing $13 to get his bike.
  4. To make that last $13 he needs to mow 2 more whole lawns, for a total of 8 x 3 + 2 = 26 lawns
2) Consider the sequence of triangles shown. If the patterns in the side lengths continue, what is the area of the 5th triangle in the sequence in square units?
 Call the base of the triangle b and it's height h.

The bases, b, are n + 1 in length, where n is the number of the triangle.
  1. The triangle heights are 2 to the base power,
        h = 2b = 2(n+1)
  2. Area of the triangle is therefore
        A = (n + 1)(2 (n+1))/2
  3. So the area of the 5th triangle is
        A = (6 x 26) / 2 = 3 x 64 = 192 sq. units

Problem
Solution
3) Marie lives 5 miles from school and bikes at an easy rate of 6 miles per hour. Her sister Kate takes the same route but is a bike racer. She rides at a rate of 25 miles per hour. If they arrive at school at the same time, how many minutes earlier than Kate does Marie need to leave?
  1. Marie's time =
    5 mi / 6 mi/hr = 5/6 hour = 5x60 / 6 = 50 minutes
  2. Kate's time =
    5 mi / 25 mi/hr = 1/5 hour = 12 minutes.
  3. Marie leaves 50 - 12 = 38 minutes earlier

4) Consider the sequence of figures made up of little squares shown. Divide the total number of little squares in Figure 10 by 11. What is that quotient?
 Careful! Tricky wording! They are asking for the number of squares in the 10th figure divided by the number 11, not the number of squares in the 11th figure!
Use n for the figure number.
  1. The number of squares in the figures are 8, 12, 16
    This is an arithmetic sequence where each figure adds 4 squares.
  2. The formula for the nth figure is
    An = A1 + d (n - 1)
  3. Putting in our values:
    A10 = 8 + 4 (9) = 44
  4. 44/11 = 4
5) Molly's Limo service provides the graphic below to display its airport service prices. I live 20 miles from the airport. What will it cost me if I use this service?
 You must find the equation of the line in y = m x + b form where m is the slope and b is the y-intercept.
  1. The y-intercept is obviously $5
  2. The slope, m, is the change in cost per mile.
    Use the 0 and 2 mile points:
    m = (20 - 5)/2 = 15/2 = $7.5
  3. The equation for Molly's Limo cost is:
        Cost = $7.5 x miles + $5
  4. For a 20 mile trip it will cost:
        Cost = 7.5 x 20 + 5 = 150 + 5 = $155

Editor's comment: Boy! That's an expensive limo!