Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2014 Grade 5 Algebra
Problem
Solution
1) You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on. What is the total distance the penny will fall in 6 seconds? It is important to note that the indicated number of feet given in this problem statement is the increase from the previous second, not the absolute distance fallen in that time. At each second it travels 32 feet farther than the previous second. With that in mind, construct the following table:
Timedistance increase from
previous second		 Total distance
1 sec1616
2 sec16+32=4816+48=64
3 sec48+32=8064+80=144
4 80+32=112144+112=256
5 112+32=144256+144=400
6 144+32=176400+176=576 feet
2) The distance from Spokane to Pasco by train is 150 miles. The train ride is direct with no stops along the way except that the train stops every 30 miles to let freight trains pass. The trip takes 2 hours and 40 minutes. Suppose its average speed is 100 mph. What is the average wait time for letting freight trains pass? Express your answer to the nearest second. 1. Compute the number of stops by dividing the total distance by the distance between stops = 150 / 30 = 5 stops.
2. Compute the time for the trip assuming no stops by dividing the distance by the speed of the train = 150/100 = 1.5 hours.
3. Subtract this from the total time of the trip, including stops = 2 hours 40 minutes - 1 hour 30 minutes = 70 minutes. = total time of the stops.
4. Divide this time by the number of stops = 70/5 =
    14 minutes/stop.

Problem
Solution

3) In May of 2000, I was eleven times as old as my son, Taylor. Nine years later I was six more than three times his age. How old was Taylor in May 2000?		 Use M for Mom's age in 2000.
Use T for Taylor's age in 2000.
1. Write the equation for Mom's age in 2000: M = 11T
2. Write the equation for Mom's age 9 years later:
    (M+9) = 3(T+9) + 6
so, putting 11T in for M:
    (11T + 9) = 3T + 27 + 6
    8T = 33 - 9
    8T = 24, so T = 3 years old
4) Rosie has a prime location for a lemonade stand in front of her house and she rents it out when she isn't using it for $5 an hour. Chadd decides to try it. He borrows $10 worth of supplies from his mom that he'll have to pay back, and decides to charge $1 per cup. What should his average selling rate of cups per hour be if he wants to make a profit of $30 in 4 hours? 1. Compute Chadd's profit/hour = 30/4 = $7.50/hour.
2. Compute Chadd's total cost for 4 hours =
(fixed cost + rent) = $10 + 4 X $5 = $30.
3. Compute Chadd's total expected income if he is to make this profit (cost + profit) =
    30 + 30 = $60.
4. Divide this by 4 to get the hourly cup rate = 60/4 =
    15 cups/hour.

5) In the pattern of circles shown, each new encircling layer adds more circles. The layers alternate in color beginning with white, then grey, and then white. How many circles will be in the next white layer?
The figure has 6 sides and with each layer still has 6 sides.
1. Each layer adds 1 to the length of a side, so the next gray layer has sides that are 4 circles long and the next white layer has sides that are 5 circles long.
2. But the circles on the corners are shared by the adjacent sides, so 6 of them must be subtracted.
3. Doing this, the next gray layer has 6 X 4 = 24 circles on each side - 6 shared circles, leaving 18 circles in the gray layer.
4. The next white layer has 5 circles on each side
    - 6 shared circles, leaving 6 X 5 - 6 =
    24 circles in the next white layer.