Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 7 Probability and Statistics
Problem
Solution
1) A gold coin is hidden in a slice of triangular cake that Joan and her best friend have just purchased. They are both coin collectors and want to add the coin to their collections. If they cut the slice as shown and Joan gets the bottom half, what is the probability the coin is in her piece? Assume the coin is entirely in one of the pieces and not touching the cut line.
  1. the whole cake and the smaller piece are similar (because when you cut a triangle parallel to any of its sides, the smaller triangle and the whole triangle are similar and their sides are all in the same ratio to each other) which means the base of the smaller triangle is 1⁄2 the base of the whole cake = 11⁄2 inches.
  2. The area of the whole cake is 3x6/2 = 9 sq. in.
  3. The area of the smaller piece is
    (11⁄2) x 3 / 2 = 9⁄4 sq. in.
  4. Subtracting: 9 - 9⁄4 = 27⁄4
  5. The probability of the bottom half having the coin is the ratio of these 2 areas: 27⁄4 ÷ 9 = 27⁄36 = 3⁄4
2) A number cube has 5 odd numbers on it and one even number. The cube is a fair cube. The cube is tossed and the number showing on top is odd. What's the probability that the number on the bottom is even? There are 5 remaining numbers on the cube and only 1 of them is even, so the probability that the bottom number is even is 1⁄5 or 20%

3) The lowest number in a set of five numbers is 10. The range is 60 and the mean is 40. What is the biggest the median could be?
  1. The highest number is 70 (10 + the range).
  2. In order to maximize the median, the second number must be equal to the lowest number, like this:
    (M = the median, N = the other number):
    10 10 M N 70
  3. Now, in order to maximize M, N must be as low as possible to maintain the mean of 40, so that value of N is equal to M:
    10 10 M M 70
  4. For the mean to be 40, the 2 Ms must add to 110. Therefore, M must be 55

Problem
Solution
4) Bo, Darius, Kareena, Mary, and Paco all go with Mrs. Smart for math enrichment during math time. When they leave she lines them up by calling out their names, but she does this randomly. What's the probability that today Darius will be first and Paco will be last? Express your answer as a fraction.
  1. The probability that Darius will be first is 1⁄5
  2. If Darius is first, then there are 4 remaining students that are candidates for the last position so that probability is
    1⁄4
  3. The probability of both happening is:
    1⁄5 x 1⁄4 = 1⁄20
5) Temperatures were recorded for 100 days equally spread throughout the year in the small town of Buffon. The data in the bar chart below was uses to create a pie chart with the five categories: 0-20 degrees F, 21-40 degrees F, 41 - 60 degrees F, 61 - 80 degrees F, 81 - 100 degrees F. What is the central angle of the section that represents the 0-20 degree F range? Give your answer to the angle to the nearest degree. Note: a single bar represents a range of 5 degrees, so the first bar represents temperatures from 1 degree F to 5 degrees F.
  1. The 0-20 degree part is the sum of the first 4 bars = 17 days
  2. the 0-20 part is 17/100 = 17% of the pie chart =
    .17 x 360 = 61.2 degrees =
    61 degrees.
    Note: You don't really need all those bars over 20 degrees.