Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 8 Number Sense
Problem
Solution

1) What is k if (k2 x 6)2 = (2732) / k ?

  1. (k2 x 6)2 = (2732) / k
  2. 36k4 = 1152/k
  3. k5 = 1152 / 36 = 32
  4. k = 2
2) A Googol is 10100. Toni has a Googol of pennies which she wants to share fairly with as many folks as possible. She splits the pile in half. Then splits both piles in half again. Then splits those piles in half. How many times can she do this splitting in half before she reaches a pile that she can't split in half evenly? (Hint: try some smaller piles first like 102)
  1. For 100, dividing by 2 successively, you get: 50,25 12.5 =
    2 divisions before one you can't split
  2. For 1000, = 500, 250, 125, 62.5 = 3 divisions.
  3. For 10000 = 5000, 2500, 1250, 625, 312.5 = 4 divisions
  4. For 100000 = 50000, 25000, 12500, 6250, 3125, 1562.5 = 5 divisions
  5. Aha! We have a pattern!
  6. The number of divisions by 2 that can be done before it can't be split evenly is equal to the number of powers of 10
    = 100
Note: This makes 2100 divisions = 1.267x1030 friends. Since the earth's total population is 7.674 billion people = 7.674x109, Joni is going to have to leave earth to find more friends!

Problem
Solution
3) Seymore has just gotten a new job. On his first day, the boss gave him 2 choices for his salary. First Choice: Start at $30,000 and get a 5% raise at the end of every year to take effect the following year. Second Choice: Start at $23,000 and get a $5000 raise at the end of each year to take effect the following year. Which choice gives the greatest salary at the beginning of the 5th year? How much more? Round your answer to the nearest cent.
  1. Choice 1: Salary after 4 years (and a raise at the start of the 5th) = $30,000(1.05)4 = $36,465.19
  2. Choice 2: $23,000 + $5000 x 4 = $43,000
Choice 2 is the best by $43000 - $36465.19 = $6534.81
4) A natural number, N, is prime if it is not evenly divisible by any prime number less than √N. How many prime numbers must we test to determine whether 2011 is prime? Well, the √2011 = 44.844..., so we must test all the primes less than 44, which are:
2,3,5,7,11,13,17,19,23,29,31,37,41,43 =
14 primes.

5) What is the first positive integer n for which
(-4⁄3)n < -3?
  1. n must be an odd integer because all even n's result in a positive value (minus times minus = plus).
  2. (-4⁄3)3 = -2.3702
  3. (-4⁄3)5 = -4.2087
  4. n = 5