Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2016 Grade 8 Number Sense
Problem
Hint
1) Rewrite 0.72 as a fraction.

 0.72 is a repeating decimal.
  1. The number (we'll call it x) is 0.72727272 . . .
  2. Multiply it times 100 to get the repeated part to the left of the decimal:
    100x = ___________________ . . .
  3. Subtract the original number from this number:


    = __________________
  4. This value is 100x - x = 99x

  5. The fraction is ________________
2) A doctor makes a box-and-whisker plot(see below) to show the number of patients she sees each day. Identify upper and lower quartiles.
  1. The upper quartile is ___________

  2. The lower quartile is ___________

Problem
Hint
3) How many numbers between 100 and 1,000 have exactly 3 factors?
  1. All numbers have 2 factors: 1 and the number itself. Prime numbers have these factors only.
  2. The only numbers that have exactly 1 more factor are squares. For example the factors of 25 are 1, 5 and 25. But only numbers whose square roots are primes, otherwise there will be additional factors.
  3. Find the numbers that have prime square roots:




  4. ____ numbers between 100 and 1000 that have 3 factors.
4) How many dots would there be in the 20th step of the sequence below?
This problem involves an arithmetic sequence
  1. Starting with 2 dots at step 1.
  2. 2 dots are added to the first 2 steps ( = 4 and 6)
  3. 3 dots are added to steps 3 and 4 (= 9 and 12)
  4. The number of dots added is incremented every other step
  5. Using this information, compute the number of dots in the 20th sequence





    = _____ dots.
5) What is the one's digit of 3 to the power of 99?

 This kind of problem relies on the fact that some numbers have repeating one's place digits when taken to increasing powers. Keep taking powers of 3 (2,3,4, ...) until you notice a pattern. Divide the power you want by the number of repeating digits. The remainder will be the number of the digit in the repeating sequence.




The one's digit of 399 = _________