Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2015 Grade 5 Measurement
Problem
Solution
1) Midas is building a stone wall that will measure 3 ft long and 8 ft tall. He uses blocks measuring 1/8 ft by 1/3 ft by 1/2 ft. He wants the wall to be as thin as possible. How many blocks will he need to build the wall? 1. If Midas wants the wall to be as thin as possible the 1/8 ft. dimension will be the thickness, leaving the wall to be built by blocks of length 1/3 ft. and height 1/2 ft.
2. Compute the area of one of these blocks: 1/6 sq. ft.
3. Compute the area of the wall: 24 sq. ft.
4. Divide the area of the wall by the area of one block to get the number of blocks:
    24/(1/6) = 24 x 6 = 144 blocks.

2) You have an oven that can cook 5 pizzas in 12 minutes. At this rate, what is the maximum number of pizzas that you can cook in 14 hours?		 1. Compute the number of pizzas that can be cooked in an hour = 5 pizzas/12 min = 25 pizzas/hour.
2. Multiply this by the number of hours =
    25 x 14 = 350 pizzas.

3) Zona lives in a bubble where time moves 1/3 slower than normal. If a clock in Zona's bubble matches a normal clock at 1:23 pm, what time will a normal clock display when Zona's clock reads 3:51 pm?		 1. If Zona's clock is 1/3 slower, then the normal clock will display a later time.
2. For every normal hour, Zona's clock advances 40 minutes.
3. The equation for this is:
    Z = (1 - 1/3)N where Z = Zona's time and N = normal time.
4. Compute the amount of Zona's time between
    1:23 pm and 3:51 pm =
    2 hours 28 minutes = 148 minutes.
5. Turn the above equation around to be an equation for N:
    N = 3Z / 2
6. Using your turned-around equation, compute the amount of normal time that transpired = (3 X 148) / 2 = 222 minutes = 3 hours 42 minutes.
7. Add this to the start time = 1:23 + 3 hours 42 minutes = 5:05 pm.

Problem
Solution
4) Geoff buys boxes of lemonade mix for $3.20. Each box can make 25 cups of lemonade, which he sells for 50 cents per cup. How much profit will he make if he buys and sells 3 boxes of lemonade? 1. Compute the number of cups that 3 boxes will make =
    25 x 3 = 75 cups.
2. Compute how much those cups will sell for =
    75 x $.50 = $37.50
3. Compute how much Geoff paid for the 3 boxes of mix =
    3 x $3.20 = $9.60.
4. Subtract this from the amount his cups will sell for to get his profit =
    $37.50 - $9.60 = $27.90

5) Find the area of this figure.
(Figure is not to exact scale)
 Method 1: Cut it up into a triangle and rectangles.
This figure can be cut up in a number of ways to make a triangle and several rectangles. I guess the best way to cut this figure up is with vertical cuts.
1. Make 2 vertical cuts along the triangle long side and along the long side of the cutout on the right side. This makes 1 triangle and 4 rectangles.
2. Compute the area of the triangle =
    (9 x 4)/2 = 18 sq. ft.
3. Compute the areas of the 4 rectangles and add it to the triangle to get the total figure area: (left to right):
    rectangle #1. 5 x 4 = 20 sq. ft
    rectangle #2. (5 - 2) x (9 + 5) = 3 x 14 = 42 sq. ft.
    rectangle #3. 6 x 2 = 12 sq. ft.
    rectangle #4. 3 x 2 = 6 sq. ft.
    total area: 18 + 20 + 42 + 12 + 6 = 98 sq. ft.

Method 2: Use area subtraction:
1. Complete the outer rectangle and compute the total area of that rectangle:
    9 x 14 = 126 sq.ft.
2. Fill in the missing dimensions.
3. Subtract (1) triangle and (1) rectangle.
    126 - 18 - (5 x 2) = 126 - 28 = 98 sq. ft.