Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 8 Number Sense
Problem
Solution
1) At Bob's school the price of school t-shirts goes up by 10% rounded to the nearest cent each year. This year the price of the t-shirt is $10. What will the price be 4 years later?

 This is a compound interest problem:
Value after 4 years = $10(1.10)4 = 10 x 1.4642 = $14.64
2)In Kyle's English class 4 students have green eyes. 5/8 of the students have brown eyes and 1/4 of the students have blue eyes. There are no other eye colors. How many students are in his class?

  1. The fraction of the students that have either blue or brown eyes is:
    5⁄8 + 1⁄4 = 7⁄8 of the students
  2. Therefore, the 4 students with green eyes are 1⁄8 of the students, so there are 8x4 = 32 students in his class.
3) On a number line, what fraction is halfway between

    1⁄3 - -3⁄4 and -3⁄8 - 5⁄6?

Add the two expressions together and divide by 2. Be careful! Lots of common denominator arithmetic!
    [ (1⁄3 - -3⁄4) + (-3⁄8 - 5⁄6)] x 1⁄2 =
    [ (13⁄12) + (-58⁄48 )] x 1⁄2 =
    [ -6⁄48] x 1⁄2 = [ -3⁄48] = -1⁄16

Problem
Solution
4) Liz was playing with a pile of pennies. She noticed when she stacked them in piles of three there was 1 left over. When she stacked them in piles of five there were 2 left over. When she stacked them in piles of seven there were 3 left over. What is the smallest number of pennies she could have? Start with stacks of 7 and compute the remainders of stacking in 5s and 3s:
# stacks
of 7
(N)		 Total
pennies

(7N + 3)		 # stacks of 5 + remainder # stacks of 3 + remainder
2173+25+2
3244+48+0
4316+17+3
5387+312+2
6459+015+0
75210+217+1
Liz has 52 pennies.
5) How many perfect square numbers will there be in the first century of the third millennium, so between 2000 and 2100? List them.
  1. The square root of 2000 is 44.72
  2. The square root of 2100 is 45.82
  3. There is 1 perfect square = 452 = 2025.