Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2015 Grade 7 Algebra
Problem
Hint

1) Lucy is riding an elevator up a very tall building. She has just reached 1/3 of the way to the top. She goes 6 more floors up and is now 2/5 of the way to the top. How many floors does the building have?
  1. Compute what fraction of the building's height Lucy goes from 1⁄3 of its height to 2⁄5 of its height: ___________
  2. This fraction of its height = 6 floors, so the height of the building is _____ floors

2) In the following sequence, how many squares are needed to construct the nth figure?
(Express your answer in terms of n)
  1. Each sequence has ___ squares on top
  2. Bottom row: The expression for the number of squares in the bottom row in terms of n is _______
  3. The number of times this row is repeated is (in terms of n) ________
  4. So, the nth figure contains ________ squares.

Problem
Hint
3) The sum of three integers is 107. The first integer is 1 more than 1/4 of the third. The second integer is 1 less than twice the first. What is the mean of the three integers? Round your answer to the nearest tenth. The mean (average) of a set of 3 integers is merely their sum divided by 3 = _____________
(You don't need all that info about the relationship between elements of the set)

4) Sam takes two steps forward and one back, three steps forward and two back, and four steps forward and three back. If Sam repeats this pattern 4 more times (forward and back), how many total steps will Sam have taken?
  1. Counting the number of steps (forward and back) for each time, we get an arithmetic sequence:
        ___ ___ ___
  2. Extending this 4 more steps:
        ___ ___ ___ ___ ___ ___ ___
  3. So the last element of this sequence is ___.
    The equation for the sum of a sequence is:
        Sum = (A1 + An) n / 2
  4. Substituting our values:

        Sum = ____________

5) How many integers satisfy the following equation?
    (x + 2) (x - 5) ≤ 0
  1. Note in the second expression that when x gets above ___ the second expression goes positive and the total equation is not true.
  2. Note in the first equation that when x goes below ___ that the first expression goes negative, and along with the second negative expression, the whole expression becomes positive.
  3. So the range of x for which the expression is valid is from ___ to ___ which is ___ values.