Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 6 Number Sense
Problem
Solution
1) A rectangular strip of paper has an area of 3/4 sq. ft. The rectangle has a width of 3/5 foot. Find the length of the rectangle in feet
(express your answer as a reduced fraction).		 1. The equation for area is
    A = L x W
2. Put the above values in and solve for L
    3/4 = L x 3/5
    L = (3/4) / (3/5)
    (to divide one fraction by another, invert the second and multiply)
    L = (3/4) x (5/3) = 5/4 or 1 1/4
2) The sum of two numbers is 38. Their product is 360. What are the two numbers? Let the 2 numbers be N1 and N2
Method 1: Use factoring:
1. Find all the factors of 360 =
    180   90   45   9   5   2   2   2
2. Find a combination of these factors that result
    in 2 factors that sum to 38 = 9 x 2 = 18 and 5 x 2 x 2 = 20

Method 2: Use the quadratic formula
1. N1 x N2 = 360
2. N1 + N2 = 38
3. Rewrite the second equation to solve for N1 and substitute it in the first equation and solve for N2 =
    N1 = 38 - N2
    (38 - N2) N2 = 360
    38 N2 - N22 = 360
We now have a quadratic equation of the form:
    N22 + 38N - 360 = 0     (a = -1, b = 38, c = -360)
Using the quadratic formula:
    N2 = - 38 ± √ 38 2 - (4) (-1) (-360)      So:
                  -2
    N2 = - 38 ± √ 1444 - 1440)      So:
                  -2
    N2 = - 38 ± 2     = +19 ± -1 = 18   N1 = 38 - 18 = 20
                  -2

Problem
Solution
3) What is the sum of the first 25 odd numbers? Method 1: Construct a sequence:
1. Make a sequence of the sums of the odd number and all the odd numbers that precede it:
element #Odd number
n		Sum
n + n-1 + . . .
1 11
2 34
3 59
4 716
5   25
Do you see the pattern here? Yes!
The sum is the square of the last element number!
2. The sum of the first 25 odd numbers = 252 = 625

Method 2: Gauss' method:
1. Cut the numbers in half, making groups of the first 12 and the last 13 odd numbers (1 - 23) and (25 - 49)
2. Turn the second set around (49 + 47 + 45 + ...)
3. Now add them to the first set:
          1     3       5   ...
    + 49 + 47 + 45 + ...
4. Now you can see that each column adds to the same sum, but only 12 of them add this way, the last one is extra (because there are 25).
Multiply this out and add the last number.
    The 13th term to be added to the sums is 1 + 2(12) = 25
    Sum = 50 x 12 + 25 = 625

Method 3: Use the formula:
The formula for the sum of the first N terms of an arithmetic sequence is:
    Sum = (A1 + AN) N / 2, where:
    A1 is the first term
    AN is the last term.
Using this formula, the sum is
    (1 + 49) x 25/2 = 50 x 25 / 2 = 625

Problem
Solution
4) Add sets of parentheses to this expression to make the statement true:

12 - 6 - 2 x 8 + 6 / 2 = 56
1. The factors of 56 are 28, 14, 7, and 2.
2. Find a grouping of numbers on the left that result in one of these factors.
3. Find another grouping that multiplies this to get 56 or 112. (You might use that divide by 2 on the end).
4. Rewrite with parentheses:
    (12 - (6 - 2)) x (8 + 6) / 2 = 56
    (8 x 14)/2 = 56

5) A square with area 1 square meter is repeatedly cut in half with one half of the remainder shaded each time as shown in the figure.What is the area of the shaded region in figure 5?
 This is a geometric sequence.
1. Write a sequence of the ratios of the white area to the shaded for each figure (I'll give you the first 2):
    1/2   1/4   1/8 ...
2. Find the multiplication factor between each sequence element = 1/2
3. Apply this out to the 5th region =
    A5 = (1/2)5 = 1/32
4. Subtract this from 1 = 1 - 1/32 = 31/32 sq. m.