Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 8 Number Sense
Problem
Solution
1) Grandpa tells Nelli he has an investment in a fund that has a history of increasing in values by 6.2% every year. He has $32,000 invested in that fund. He asks her to compute how much the fund will earn him over five years if he makes no changes to his investment. Let V be the value of Grandpa's intestment after 5 years.
V = P (1 + r)t where:
  P is the principal you start with
  r is the interest rate (specified as a decimal)
  t is the number of time periods
  • V = $32,000(1 + 0.062)5
  • V = $32,000 x 1.0625 = 32000 x 1.351 = $43,228.74
  • He will have earned $43,228.74 - $32,000 =
    $11,228.74

    • 2) A dime is 1.25x10-3 meter thick and is worth 1x 10-1 dollars. How many meters high is a stack of dimes worth 1x106 dollars? Express your answer in meters without using scientific notation.
    1. 1x106 dollars / 10-1 dollars =
      1x106+1 = 1x107 dimes
    2. 1x107 x 1.25x10-3 meters =
      1.25x107-3 meters = 1.25x104 meters =
          12,500 meters

    Problem
    Solution
    3) The numbers 1 - 32 can be represented with a set of five lights using the binary system. An "on" light is recorded with a 1, an "off" light is recorded with a 0. With the pattern given in the table, how would the number 23 be represented in 1's and 0's?
    • Each binary digit, or "bit", starting on the right side, is a power of 2, so the first bit is 20 = 1, the second is 21 = 2, and so on up to the 5th bit which is 24 = 16
    • Here's how you turn a decimal number into binary:
      1. Write down the decimal number.
      2. Divide the number by 2.
      3. Write the result underneath.
      4. Write the remainder on the right hand side.
        This will be 0 or 1.
      5. Divide the result of the division by 2 and again write down the remainder.
      6. Continue dividing and writing down remainders until the result of the division is 0.
    Here's how that looks for 23:
    23/2=11 Remainder:1
    11/2=5  Remainder: 1
    5/2=2     Remainder 1
    2/2=1    Remainder 0
    1/2=0    Remainder: 1
    The number 23 (decimal) = "1 0 1 1 1"
    4) The Golden Ratio is the ratio of a Golden rectangle's length to width that is most pleasing to the eye. This ratio is (1 + √ 5) / 2. If a Golden Rectangle has width 2√ 5 , then what is its length? Give an exact answer leaving radicals in your answer if needed.
    		Let L be the rectangle's length.
    Set up the ratio of the rectangles length (L) to its width and set that equal to the golden ratio and solve for L:
        L     =   (1 + √ 5)
    2√ 5               2

    L = (2√ 5) (1 + √ 5)
                  2

    L = √ 5 (1 + √ 5)

    L = 5 + √ 5
    5) In the equation (2x)3(4x2)-2= 25 4-3 what is x?
    1. (2x)3(4x2)-2 = 32/64 = 1⁄2
    2. 8x3 / (16x4) = 1⁄2
    3. 8/16x = 1⁄2
    4. x = 1