Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 5 Probability & Statistics
Problem
Solution
1) A bag contains some marbles of the same size. Twelve of them are red. The rest are all white. The probability of drawing a white marble is 2/3.
How many white marbles are in the bag? 		1. If the probability of drawing a white marble is 2/3, then the probability of drawing a red marble is 1/3.
2. If 12 red marbles are 1/3 of the marbles, then 24 marbles are 2/3.
2) Ted spends some afternoons throwing darts. The length of the board is 30 cm and the width is 20 cm as shown in the scale drawing. The outer grey band has a width of 5 cm on each side, the next white band has a width of 2 cm on each side and the inner white square has side length 2 cm. His darts always hit the board, but assume at random. What is the probability he will hit the inner light grey band? Express your answer as a reduced fraction (i.e. in lowest terms) or as a decimal rounded to the hundredths place. 1. Compute the area of the large grey rectangle = 600 sq. cm.
2. Compute the area of the white rectangle, using the 5 cm width of the grey band = 20 x 10 = 200 sq. cm.
3. Compute the area of the inner light grey rectangle using the 5 cm and 2 cm widths of the outer rectangles = 16 x 6 = 96 sq. cm.
4. Subtract the area of the little square in the middle = 96 - 4 = 92 sq. cm.
5. Compute the probability of hitting the light grey band as the area of the light grey band divided by the area of the large grey rectangle = 92/600 = 46/300 = 23/150 = 0.15

Problem
Solution
3) A student got the following scores on her first five weekly math quizzes:
65, 82, 38, 86, and 74.
What is the least possible score she could get on the sixth quiz so that her median score would be 78? 		1. First, put the scores in numeric order:
    38 65 74 82 86
2. Since there will be 6 scores, the median will be the average of the last 2 numbers after you cross off the highest/lowest ... continuing until only 2 numbers are left. The current median before we add the 6th number is 74
3. The average of this 74 and 82 is 78.
4) Shooter's basketball team played in ten games.
The scores are charted as shown.
Which one is greater, the mean or the mode? By how much is it greater? 		1. Finding the median is easy! It's the horizontal line with the most diamonds on it! = 20
2. Compute the average (the mean) of the 10 scores = 21.5
3. The mean is greater than the median by 1.5.
5) A gum ball machine is filled with an equal number of white, blue, yellow, green, and purple gum balls. You cannot control which color you get from the machine and the gum balls cost 25 cents each. How much would you have to spend to be sure of getting 3 gum balls of the same color? You must assume there are at least 5 X 3 = 15 gum balls in the machine.
1. There are 5 different colors, so assume your first 5 gum balls are the wrong color.
2. This leaves 10 gum balls still in the machine.
3. Assume you buy 5 more gum balls and they, too, are the wrong color.
4. This leaves only 5 gum balls in the machine and that next gum ball purchase must be the right color, so the total number you would have to buy is 5 + 5 + 1 = 11 gum balls at 25 cents each = $2.75