Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 7 Number Sense
Problem
Solution
1) A game comes with three types of cards:  Easy, Medium, and Hard. You can buy expansion packs to make you collection bigger. Every pack comes in the ratio 5:  4:  3, so for every 5 Easy cards there are 4 Medium cards and 3 Hard cards. Evan buys some expansion packs and now has a collection of cards that is complete and has 24 medium cards. How many cards are in his collection?
  1. Since Evan has 24 medium cards, using the ratio of easy to medium cards, he has
    24/4 = Easy/5
    Easy cards = 24x5/4 = 6x5 = 30 cards
  2. Using the ratio of medium cards to hard cards, Evan has
    24/4 = Hard/3
    Hard = 24x3/4 = 6x3 = 18 cards
  3. In total, Evan has 24 + 30 + 18 = 72 cards in his collection.
2) When you simplify the terms in the following sequence to reduced fractions an interesting pattern occurs. Find out what it is and use it to determine the value of the tenth term as a reduced fraction.
  1. First term = 1⁄3
  2. Second term = 1 - 1/1⁄3 = 1 - 3 = -2
  3. Third term = 1 - 1/(-2) = 1 1⁄2
  4. Fourth term = 1 - 1/3⁄2 = 1 - 2⁄3 = 1⁄3
  5. The terms repeat every 3rd term.
  6. 10/3 = 3, remainder 1, so the tenth term repeats the first:  1⁄3
3) Junie hits on a phenomenal idea and sends text messages to 10 friends. These 10 friends simultaneously text 10 more friends. They then send the idea on by simultaneously texting 10 more friends. Assuming each text message always reaches a new person and each round of texting 10 friends takes 2 minutes, how long before 10 million people have Junie's idea texted to them?
  1. 10 million = 107 = 7 cycles of 10
  2. Each of the 7 cycles takes 2 minutes, so the time it takes to reach 10 million people is 2 x 7 = 14 minutes

Problem
Solution
4) How many fractions when reduced to lowest terms have denominators smaller than 10 and are between 0.29 and 0.81 on the number line?
  1. The denominators must be between 2 and 9 to keep their values between 0.29 and 0.81
  2. Count the ones with each denominator and reduce: 
        2:  1⁄2 = 1 fraction
        3:  1⁄3 and 2⁄3 = 2
        4:  2⁄4 and 3⁄4 = 1⁄2 and 3⁄4 = 1 new fraction
        5:  2⁄5, 3⁄5 and 4⁄5 = 3
        6:  2⁄6, 3⁄6 and 4⁄6 = 1⁄3, 1⁄2 and 2⁄3 = no new ones
        7:  3⁄7, 4⁄7 and 5⁄7 = 3
        8:  3⁄8, 4⁄8, 5⁄8 and 6⁄8 = 3⁄8, 1⁄2, 5⁄8 and 3⁄4 = 2 new ones
        9:  3⁄9, 4⁄9, 5⁄9, 6⁄9 and 7⁄9 = 1⁄3, 4⁄9, 5⁄9, 2⁄3 and 7⁄9 = 3 new ones
  3. The total is 15 fractions
5) Take a number with exactly 4 factors and another with exactly 3 factors. Multiply the two numbers. What is the difference between the greatest number of factors the product could have and the fewest number of factors the product could have?
  1. When 2 numbers are multiplied, the factors of the first, multiplied by all the factors of the second become factors of their product. So, a number with 4 factors multiplied by a number with 3 factors has 4x3 = 12 possible factors, assuming none of them (other than 1) are duplicated.
  2. The greatest number of factors is when the two numbers have no factors, other than 1, in common, like 6 and 25:
          6: Has 4 factors: 1,2,3,6, and
        25: Has 3 factors: 1,5,25
        6 x 25 = 150: Has 12 factors: 1,2,3,5,6,10,15,25,30,50,75,150
        Conclusion: The maximum number of factors = 4 x 3 = 12 factors, none of which are duplicated.
  3. The least number of factors is when the second number's factors are the same as 3 of the factors of the first number, like 8 and 4:
        8: Has 4 factors: 1,2,4,8 and
        4: Has 3 factors: 1,2,4
        8 x 4 = 32: has 6 factors 1,2,4,8,16,32
  4. Why doesn't the second example have 4x3 = 12 factors?
    Because half of them are duplicated!
  5. Therefore the difference between the greatest and the fewest number of factors is 12 - 6 = 6