Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 8 Number Sense
Problem
Solution
1) 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 = 1⁄3
  2. Second term = 1 - 1/1⁄3 = 1 - 3 = -2
  3. Third term = 1 - 1/(-2) = 1 1⁄2
  4. Fourth term = 1 - 1/3⁄2 = 1 - 2⁄3 = 1⁄3
  5. The terms repeat every 3rd term.
  6. 10/3 = 3, remainder 1, so the tenth term repeats the first:  1⁄3

2) What is the missing term in this sequence?
    1, 3, 3, 9, ___________, 243.
Each number is the product of the 2 preceding numbers, so the next number is 9 x 3 = 27
As a check, 27 x 9 = 243, the next term.

Problem
Solution
3) What is the digit in the one's place of 6734?

 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.

    671 = 67
    672 = 4489
    673 = 300763
    674 = 20151121
    675 = 1350125107
    676 = 90458382169
The one's digits are 7, 9, 3, 1 and then they repeat. It's a cycle of 4 digits.
34/4 = 8, remainder 2, so the one's digit for the 34th power is 9 (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.
4) Counted cross stitch is a way of painting by stitching with silk thread. Each cross stitch fills a (3/16) inch x (3/16) inch square. How many stitches go in to a 12 x 15 inch cross stitch painting?
  1. Area of the cross stitch = 3⁄16 x 3⁄16 = 9⁄256 sq. in.
  2. Area of the painting = 12x15 = 180 sq. in.
  3. 180 ÷ 9⁄256 = 180 x 256⁄9 = 256 x 20 = 5120 stitches
5) Ruby selects earrings from a rack that are all priced the same. The store has to charge 10% tax. If there was no sales tax, then Ruby could buy one more pair of earrings for the same amount of money she is spending. How many pairs of earrings is she buying? Let N = the number of pairs of earrings Ruby is buying.
Then N(1.1) = N + 1
1.1 N - N = 1
0.1N = 1, so N = 10