Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 5 Number Sense
Problem
Solution
1) On a number line, what fraction is halfway between
3/4 + 1/3 and 3/8 + 5/6?
 See Adding fractions with different denominators.
Note: Lots of common denominator problems here so be careful!
1. Add 3/4 + 1/3 = 9/12 + 4/12 = 13/12.
2. Add 3/8 + 5/6 = 18/48 + 40/48 = 58/48
3. Add these two fractions and multiply by 1/2:
First convert the 12ths to 48ths:
52/48 + 58/48 = 110/48.
Multiply by 2 and simplify:
    110/48 x 1/2 = 110/96 = 1 14/96 = 1 7/48
2) At Raymond's school, Tuesday was jeans day and everyone wore either blue jeans or black jeans. In his math class they collected data and concluded for every 7 children wearing blue jeans there were 4 wearing black jeans. If there were 451 students in the school on Tuesday, how many were wearing black jeans? 1. Their data was for every 11 students (7 blue jeans wearers + 4 black jeans wearers), so the fraction wearing black jeans was 4/11 of the student population.
2. Apply this percentage to the total student population.
Total black jeans wearers = 4/11 X 451 =
4 x 451 / 11 = 1804/11 = 164 students.
3) There are 30 students in Samir's class. They line up in the hall. Samir has a bag of jelly beans. He will hand them out according to the following rule. Each student gets a minimum of 6 jelly beans. However, if your place in line is a prime number then you get 6 jelly beans plus one jelly bean for every prime number spot so far, provided you can tell Samir the number of prime number spots, including yours, he has already visited. As an example, if you were standing in the spot that was the fourth prime number, you could receive ten beans if you were able to tell Samir that you were in the fourth prime number spot. What is the most number of jelly beans Samir could hand out to a student? 1. The prime numbers up to 30 are:
    2 3 5 7 11 13 17 19 23 29
2. There are 10 prime numbers below 30.
3. The first prime number (2) gets 1 extra jelly bean , the second (3) gets 2, and each additional one gets 1 more than the previous prime number, making the number of extra jelly beans the last student gets is 10.
4. Add this number to the 6 every student gets
    for a total of 16 jelly beans

Problem
Solution
4) Paulina and her Abuela (grandmother) sell eggs at the market. At the end of the day they have less than a half dozen eggs left. Abuela likes math humor. The next customer comes up and Abuela says "I have very few eggs left, but for 10 cents I will give you half of my remaining eggs and give you an extra half an egg without breaking any eggs if you can guess how many eggs I have." What is the most number of eggs Abuela could have? The key is that if she doesn't break any eggs, half of her remaining eggs must be odd so that when she adds half an egg it makes a whole number of total eggs. The most she could have is 5 eggs.
This is the highest number of eggs below 6 that is an odd number.

5) I am a whole number. I am a multiple of 4 and 6. I am divisible by 14. Subtract 100 from me and the result is a negative number. Who am I? 		Method 1: Compute the multiples
1. If you subtract 100 from the number and get a negative result, then the number is less than 100.
2. Since it is divisible by 14, write out all the multiples of 14 up to 100.
multiples of 14=>14 284256 708498
divisible by 4?NYNYNYY
divisible by 6?NNNNNYN
3. Find the one that is a multiple of both 4 and 6.
Number = 84.

Method 2: Find the least common multiple of 14,4 and 6
  1. The GCF of 14,4 and 6 is 2
  2. By the GCF-LCM equation:
        LCM(14,4,6) = (14 x 4 x 6) / 2 = 84