Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2006 Grade 8 Number Sense
Problem
Solution
1) The Monday after Thanksgiving is now called "Cyber Monday" and is the busiest on-line shopping day of the year. On Cyber Monday 2004, credit card users for one company spent 4/5 the amount of money they spent on Cyber Monday in 2005. What percent more money was spent in 2005? Let X = the amount spent in 2004 and Y = the amount spent in 2005.
4⁄5 Y = X
Y / X = 1 / (4⁄5) = 5⁄4 = 1.25 = 25% more in 2005
2) Solve for N in the following equation.
 Working from the bottom up:
  • 1 + 1/1 = 2
  • 1 + 2/2 = 2
  • 1 + 3/2 = 5/2
  • 6 = N / (5⁄2) = N(2⁄5)
  • 6 = 2N / 5
  • 2N = 30
  • N = 15
3) Joji Kato set a world record in the 500 meter speed skating race at Salt Lake City. He finished in 34.30 seconds. If he could keep the same speed for the 10,000 meter speed skating race, how many minutes and seconds would it take him to finish the race? Divide the 10,000 meters by the rate, making sure that the units cancel correctly to yield seconds:

34.30 sec  x   10,000 meters = 686 seconds =
500 meters

11 minutes 26 seconds

Problem
Solution
4) What is the one's digit of 826?

 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 8 (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.

    81 = 8
    82 = 64
    83 = 512
    84 = 4096
    85 = 32768
    86 = 262144
The one's digits are 8, 4, 2, 6 and then they repeat. It's a cycle of 4 digits.
26/4 = 6, remainder 2, so the one's digit for the 26th power is 4 (the 2nd digit of the sequence)

Interesting note: Any number whose 1's digit squared results in the same 1's digit (for example 52 = 25) will have that 1's digit in any power! So a problem like "what is the one's digit of 6 to the power of 4545" will be easy! It's a 6 because 6x6 = 36 x 6 = 216 x 6 = 1296 ...
The 1's digit that behave this way are 1,5 and 6.
5) An alien species has 3 "fingers" on each "hand". They visit a school while on a friendship mission. The children discover the aliens also use the digits 0, 1, 2, 3, 4, and 5 and that they mean the same thing but they don't use the digits 6 - 9. When they multiply 4x5 they get 32. The teacher says they are right. What does the aliens' number 53 mean to us? This problem is talking about number systems.
  1. The number system the aliens use is the base-6 system.
  2. In this system, the units 'digit' is 60 = the number of ones. For our case this is 3.
  3. The second digit from the right is the 61 digit which is the number of 6s. For our case, this is 5 6's = 30.
  4. The number, in the decimal system, is 3 + 30 = 33