Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 7 Number Sense
Problem
Hint
1) A game comes with three types of cards:  Easy, Medium, and Hard. You can buy expansion packs to make your 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 _____ easy cards.
  2. Using the ratio of medium cards to hard cards, Evan has _____ hard cards.
  3. In total, Evan has _____ 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 = ______

  2. Second term = ______

  3. Third term = ______

  4. The pattern is ____________________________
  5. The 10th term is ______
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. Find the exponent of 10 that equals 10 million people = _____
  2. This is the number of cycles of calling, at 2 minutes per call needed to call 10 million people = ______ minutes.

Problem
Hint

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, Then delete any duplicates: 
        2:

        3:

        4:

        5:

        6:

        7:

        8:

        9:

  3. The total is ____ 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 ____ 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 8 and 25.
    • Factors of 8:
    • Factors of 25:

    • Factors of 8x25:


  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
    • Factors of 8:
    • Factors of 4:

    • Factors of 8x4:


  4. Use these 2 examples to find the greatest and least number of factors. Least = _____. Greatest = _____
  5. Therefore the difference between the greatest and the fewest number of factors is = _____