Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 8 Algebra
Problem
Solution
1) A batch of brownies uses 3 eggs and weighs 25 oz. A cake takes 4 eggs and weighs 35 oz. If Wanda has 22 eggs and enough of all the other ingredients how many batches of brownies and how many cakes should she make so that the total weight of baked goods is as large as possible?
  1. Brownies result in 25/3 oz. per egg = 81⁄3 oz/egg
  2. Cakes result in 35/4 oz. per egg = 83⁄4 oz/egg
  3. She should maximize the number of cakes.
  4. 22/4 = 5 cakes with a remainder of 2 eggs, but you can't make a batch of brownies with 2 eggs, so you must back off 1 cake to
        4 cakes (using 16 eggs) +
        2 batches of brownies (using 6 eggs)
    for a total of 22 eggs.
2) Sally wants to change her allowance so that she gets 1 penny on the first day and on each following day double the amount of the previous day. Find an expression that allows you to compute how many total pennies Sally has after n days and then show how to use it to compute how many pennies she gets on day n = 16. This is a geometric series. The sum of the terms of a geometric series is
Sum = A1 [ (1 - m n) / (1 - m)]
For our problem this is (1-2n) / (-1)
which is 2n - 1
On the 16th day she gets 216 - 1 = 65535 pennies.
Note: This is the cumulative total, not the amount she gets on the last day

Problem
Solution
3) Juan and Tai want to build a skate board ramp. They get a 10 foot long piece of plywood under which they will need to build supports. Tai's dad asks them to determine the height of their ramp. Following some advice they are going to give their ramp a slope of 1:3 so for every 3 feet in length across the flat ground they will raise the plywood by 1 foot - see drawing. What is the height of the ramp? Express your answer to the nearest quarter foot. Using the pythagorean theorem:
  1. the width is 3h and the height is h
  2. 10 = √ (3h) 2 + h 2)  = h√10
  3. h = 10 / 3.16 = 3.16 = 3.25 ft
4)A boy lives on the same road as his school and the road is straight. Every morning he bikes to school. This morning when he got halfway to school he notices that he dropped his math book and bikes back a quarter of the distance he just biked to pick it up. From this position he goes half the remaining distance to the school and notices that he has dropped his notebook. He bikes back 500 yards to pick it up. He's now 1100 yards from school. How far from school is his house? Let D = the distance to the school.
Leg Current positionRemaining distance
Leg 1:
Go half way 		1⁄2D 1⁄2D
leg 2:
Go back 1⁄4		 1⁄2D - 1⁄8D = 3⁄8D 5⁄8D
Leg 3: Go half way to school 3⁄8D + 5⁄16D = 11⁄16D 5⁄16D
Leg 4: Go back 500 yards 11⁄16D -500 5⁄16D +500
Leg 5: Go remaining 1100 yards 11⁄16D +600 = D 0
11⁄16D +600 = D
D = 1920 yards
5) The table shows some data on sales of tickets for a charity event. Paul assumes the data is linear. If that is true, what is the price of 9 tickets?

Number
of tickets 		4 6     9     13
Price $36.75 $51.25 ? $102.00
  1. Assume the price change between the 6 ticket price and the 13 ticket price is linear.
  2. Set up the ratio of the 6-to-9 ticket price to the 6-to-13 ticket price:
        ($102.00 - $51.25) / 8 = (x - $51.25) / 3 =
        $50.75/7 = (x - $51.25) / 3
        (3/7) $50.75 = x - $51.25
        $21.75 = x - $51.25
        x = $73