Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2007 Grade 8 Number Sense
Problem
Solution
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 = 1.5L x 1.25W = 1.875 LW
  3. The area of the picture is enlarged by 87.5%
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: √60006 = 244.96 and the square root of 69996 = √69996 = 264.5
  2. So the square root must be greater than 244 and less than 265.
  3. Now, let's get clever! The square of the number ends in 6 so we must only consider the numbers between 245 and 264 whose squares end in 6. Those are numbers that end in either 4 or 6 (4x4=16, 6x6=36). No other single digit product ends in 6. Those squares between 245 and 265 are:
    246, 254,256, and 264.
  4. Those numbers squares are:
    2462 = 60516 : not a palindrome
    2542 = 64516 : not a palindrome
    2562 = 25536 : not a palindrome
    2642 = 69696 (*) <=== the one!
  5. The missing numbers are 969

Problem
Solution
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. 2⁄5 of 2⁄3 = 4⁄15 of his candy left.
  2. When he gave 6 pieces to his sister he must have had at least 7 to have 1 left over.
  3. Therefore he must have started with 7x15/4 = 26.25 pieces of candy.
  4. This doesn't work because he can't give out part of a piece of candy so assume he had 8 pieces:
    8x15/4 = 30 pieces. This works!
  5. He started with 30 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 6 then the inequality is satisfied.
  2. By the first expression (n+4) if n is greater than -5 then the inequality is satisfied.
  3. So the range of n is -4 ≤ n ≤ 6, so the number of integers that satisfy the the inequality is 11
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.
  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 = 2 + 2 x 99 = 200
  4. The sum of the first 100 even numbers is:
    Sum = (2 + 200) x 50 = 202 x 50 = 10100