Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 6 Measurement
Problem
Solution
1). Dora orders a kit that will let her make a tiered strawberry garden using three concentric circles as in the picture. The borders to make the circles are cut from one long piece of aluminum border stripping measuring 48 feet long. The two rings in which to plant the strawberries are each 1 foot wide. What is the diameter of the center circle if Dora uses as much of the stripping as she can and only considers whole number of feet diameters?
Use the approximation = 3.14 		Let D1 be the diameter of the outer ring, D2 is the diameter of the second ring and D3 the diameter of the innermost ring.
1. Then D2 = D1 - 2 and D3 = D1 - 4 ft.
2. So the total circumference is
    (D1 + D1 -2 + D1 - 4).
3. Since the total circumference is 48 feet, write this equation and solve it for D1.
    (3D1 - 6) = 48
    3D1 = 6 + 48/3.14 = 21.296
    D1 = 7.09 = 7 ft.
4. Then compute the diameter of the innermost ring =
    7 - 4 = 3 ft.
2) A rectangular picture frame has outer dimensions length 30 in., width 24 in. and height 1/2 in. as shown. The frame is 2 inches across from inner edge to outer edge. If the frame is solid wood, what is the volume of wood used to create the frame? 1. Using the outer dimensions, compute the volume of the frame and the interior of the frame =
    outer volume = 30 x 24 x 1/2 = 360 cu. in.
2. Subtract 4 inches from each outer dimension and compute the volume of the interior of the frame =
    inner volume = 26 x 20 x 1/2 = 260 cu. in.
3. Subtract the interior volume from the outer volume =
    360 - 260 = 100 cu. in.

Problem
Solution
3) At Mario's pizza they serve a pizza lunch buffet. The pieces are cut from 8 inch diameter pizza and a 12 inch diameter pizza. The 8 inch pizzas are cut to have 8 congruent slices. The 12 inch pizzas are cut so that the pieces have the same area as the pieces from the 8 inch pizza and so have less crust. What is the angle at the tip of a slice from the 12 inch pizza?
1. Compute the area of 1 8-inch slice =
    (64/8) = 8
2. Compute the area of the 12-inch pizza =
    144
3. Divide this area by the area of 1 8-inch slice to get the number of 12-inch slices = 18 slices
4. Divide 360o by this number to get the angle at the tip of a 12-inch slice = 360/18 = 20o
4) A room is 25% full. If 900 more people enter the room it will be at full capacity. Full capacity is determined by one person per 1.5 square meters. The length of the room is 60 meters. What is the width of the room? 1. If 900 people enter a room that is 1/4 full and that fills it,
then those 900 people are 3/4 of the capacity of the room.
2. Compute the number of people in 1/4 of the room = 300
3. Compute the total capacity of the room = 1200 people.
4. Multiply by 1.5 to get the area in square meters = 1800 sq. m.
5. Divide this by 60 to get the width of the room = 30 meters.
5) Aleisha is exercising her business talents as the go to classmate to get things from. She trades:
  5 bags of chips = 2 candy bars
  2 bags of chips = 3 tattoo stickers
  5 tattoo stickers = 12 brand new pencils
  9 brand new pencils = 3 gel pens
If she's successful at trading, how many gel pens can she get for 4 candy bars?		 1. If 2 candy bars = 5 bags of chips, then 4 candy bars = 10 bags.
2. If 2 bags of chips = 3 tattoo stickers then 10 bags = 15 stickers.
3. If 5 tattoo stickers = 12 pencils, then 15 stickers = 36 pencils.
4. Finally, if 9 pencils = 3 gel pens, then
36 pencils = 12 gel pens.