Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2007 Grade 8 Number Sense
Problem
Hint
1) The length of a picture is enlarged by 50% and the width is enlarged by 25% producing an odd looking picture. By what percent is the area enlarged? Express your answer to the nearest tenth of a percent.

 Use L for length and W for width.
  1. Original area = LW

  2. New area = __________

  3. The area of the picture is enlarged by ________%
2) The number 6  __ ,  __  __  6 is a perfect square that is a palindrome, so it reads the same forwards as backwards. What are the missing digits in order?
  1. Take the square root of 60006 (the smallest possible value):
    √60006 = ______ and the square root of 69996 (the largest possible value):
    √69996 = ________

  2. So the square root must be greater than _____ and less than ______
  3. Now, let's get clever! The square of the number ends in 6 so we must only consider the numbers between _____ and _____ whose squares end in 6. Those are numbers that end in either 4 or 6 (4x4=16, 6x6=36)



  4. The missing numbers are ___ ___ ___

Problem
Hint
3) Jona takes 1/3 of his Halloween candy to school to sell. It went so well that the next day he takes 2/5 of the Halloween candy left over to school to sell. When he gets home he gives 6 pieces to his baby sister because he's in such a good mood. What's the smallest amount of candy he could have had to start with if he still has candy left over?
  1. If he takes 1⁄3 of his candy to school, then he is left with ____________

  2. 2⁄5 of this = _____________ of his candy left.
  3. When he gave 6 pieces to his sister he must have had at least ____ to have 1 left over.
  4. Therefore he must have started with ______ pieces of candy.
  5. This doesn't work because he can't give out part of a piece of candy so keep adding pieces until this fraction is a whole number:


  6. Once you have the number he had before he gave 6 pieces to his sister, you can compute what he started with.

  7. He started with ______ pieces of candy.
4) How many integers n satisfy
(n + 4)(n - 6) ≤ 0 ?
  1. By the second expression (n - 6) If n is less than or equal to _____ then the inequality is satisfied.
  2. By the first expression (n+4) if n is greater than _____ then the inequality is satisfied.
  3. So the range of n is _____ ≤ n ≤ ______, so the number of integers that satisfy the the inequality is ______.
5) The sum of the first 100 positive odd numbers is 1002. What is the sum of the first 100 positive even numbers?
  1. The sum of an arithmetic sequence is
    Sum = (A1 + An) x n/2 where
    n is the number of terms and
    A1 and An are the first and last terms.

  2. The nth term of an arithmetic sequence is:
    An = A1 + d x (n-1) where d is the number added to each term.
  3. For our sequence, the last term is:
    A100 = __________

  4. The sum of the first 100 even numbers is:

    Sum = ______________