Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 6 Probability & Statistics
Problem
Solution
1) Mercedes is looking at the calendar for the month of May 2012 and sees May 5th is a Saturday. If she draws a box around four dates that all touch at a corner, what is the probability the box contains two dates that are multiples of 6? (May has 31 days.) 1. Here is the calendar (Saturday in the last column)
2. The dates that are multiples of 6 are:
    6, 12, 18, 24 and 30
3. Note that for the first week (starting on Tuesday)
  there are 4 4-date boxes that you can make with week #2.
4. For each week, count the number of 4-date boxes that can be made with the following week (week #1 = 4 boxes). Note that the last week you can do this is the week starting the 20th.
The total is 20 4-date boxes.
5. Now go back and find 4-date boxes that contain 2 dates in your list above. There are 3 of them.
    11 12     16 17       17 18
    18 19     23 24       24 25
6. So the probability of a box containing 2 dates that are multiples of 6 is 3/20
2) Lucy is playing a cup game with her brother. She has 10 cups turned upside down in front of her. She has hidden 5 pieces of candy, one under each of 5 cups. To get all the candy, her brother has to uncover three pieces of candy in the first three tries. What is the probability that he will win all the candy? 1. What is the probability that he will uncover a piece of candy on the first try?
    5/10 = 1/2
2. Assuming he got one, what is the probability that he will uncover a piece of candy on the second try? 4/9
3. Assuming once again that he got that second one, what's the probability that the 3rd cup will uncover a piece? 3/8
4. Multiply these together to get the total probability =
    1/2 x 4/9 x 3/8 = 12/144 = 6/72 = 3/36 = 1/12

Problem
Solution

3) LaMarcus does a survey of 57 students about the pets they own. There are 870 students in his school. If the survey is an accurate reflection of pet ownership at his school, what percentage of students are cat owners?
 The figure shown is a Venn diagram.
1. Count up the total number of cat owners as all the ones who appear in the circle marked "cats". This includes the ones shared by dog and hamster owners.
Total cat owners = 10 + 5 + 1 + 2 = 18.
2. Divide this by the number of students in the survey =
    = 18/57 = 31.58 = 32%

4) Tad needs to maintain an 85 or higher average in math class for his parents to let him play on two sports teams. His current average based on four test scores is 94. He doesn't have much time to study for the next test. What is the lowest he can score on the next test to maintain an 85 or above average?		 Let T4 = the total of 4 test scores that average to 94.
    T4 = 94 x 4 = 376
Let S = his score on a 5th test.
1. Then the equation is:
    (376 + S) / 5 = 85.
2. Solve this for S.
    376 + S = 425
    S = 49
5) Doreen likes to play Words with Friends, a version of scrabble, on her phone. She has a Z, two vowels and two consonants left, and the letters are all different. Her strategy in this round is to try combinations using her Z for 10 points, a vowel, and one other of her letters in that order someplace on the board and see if they work. How many different options does she have for the three letter combinations without considering where they'll go on the board? Doreen has 5 letters: a Z, 2 vowels and 2 consonants.
1. The Z goes in the first position.
2. The second position is a vowel and she has 2 choices for that.
3. Any of the 3 remaining letters can go in the 3rd position.
4. The total number of options is 1 x 2 x 3 = 6