Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 5 Probability & Statistics
Problem
Solution
1) Among the 78 fifth graders at Presley Elementary, 51 have at least one sister and 63 have at least one brother. How many have at least one sister and one brother? 1. The total number of students that have at least one sibling is
    51 + 63 = 114
2. Since this is more than 78, the excess number must have at least one brother and one sister.
    Excess = 114 - 78 = 36.
2) A factor of 60 is chosen at random. What is the probability that this factor has factors of 2 and 5? Express your answer as a fraction. 1. Here are all the factors of 60:
  2 3 4 5 6 10 15 20 30
2. The number of factors of 60 is 9
3. Find the ones that have factors of both 2 and 5:
    10 20 30
4. The number of (2,5) factors of 60 = 3
5. Divide this number by the number of total factors to get the probability =
    3/9 = 1/3
3) The PTA is raising money with an auction and part of the auction is a raffle which allows you to draw from a jar of jelly beans. Each color bean is worth a different amount. There are 5 colors and they are in the following proportions: 7 red to 10 blue to 5 yellow to 5 green to 6 orange. There are 165 jelly beans in the jar. Red is worth $50, blue is worth $50, yellow is worth $100, green is worth $1 and orange is worth $20. If you are the first to draw, what is the probability that you will draw a bean worth at least $20? Express your answer as a fraction. 1. The only color you need to worry about is green because it is the only color worth less than $20.
2. 5 out of 33 beans are green.
3. The proportion that are not green is (33 - 5)/33 = 28/33.

Problem
Solution
4) Toni's teacher gives a 20 point quiz every Friday. On her first five quizzes Toni has scores: 12, 10, 11, 20, and 12. What are the possible quiz scores for the next quiz that would keep the median the same but raise Toni's average? Quiz scores are whole numbers. 1. It helps to rewrite the scores in scoring order:
  10, 11, 12, 12, 20
2. The current median is 12
3. The current average is 65/5 = 13
4. The added score must be greater than the current average to raise Toni's average.
5. The added score must be 14 to 20 .

5) You are given the scrabble letter tiles that spell TRAVEL. How many three letter arrangements can you make with your tiles that have the pattern: vowel, consonant, vowel? The arrangements need not spell a real word though many will. (Note: since they are tiles you can use a letter only once.) 		1. There are 2 vowels and 4 consonants in TRAVEL.
2. There are 2 ways to choose the first vowel.
3. There is 1 way to choose the last vowel after you choose the first vowel.
4. There are 4 ways to choose the middle consonant.
5. Muliply these 3 ways to get the answer: 8 ways