Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2008 Grade 6 Algebra
Problem
Solution
1) The children are harvesting apples. They harvest 72 dozen apples. They lost 1/8 of the apples to bruises while packing. 1/7 of the remaining apples were thrown out because of scabs. Twenty dozen apples were donated to the local food bank. How many apples did they have left to sell? 1. Compute what 1/8th of the apples is and subtract it:
    72/8 = 9. 72 - 9 = 63 apples
2. Take 1/7th of this and subtract it:
    63/7 = 9
    63 - 9 = 54 apples
3. Subtract 20 dozen apples, 54 - 20 =
    34 dozen apples.
2) The expression 1.25C + 1.85R computes the cost of purchasing C bottles of Cola and R bottles of Root Beer. Max has $10.00 to spend and wants to buy at least 2 bottles of each and then as many bottles of soda as he can with the money left. How many bottles of Cola and how many bottles of Root Beer should he purchase? 1. Compute the cost of 2 cans of cola and
    2 cans of root beer and subtract that from the $10:
    2 x 1.25 + 2 x 1.85 = $6.20.
    $10 - $6.20 = $3.80.
2. Divide this remaining amount by the cost of a
    bottle of cola = $3.80/$1.25 =
    3 bottles with $.05 left over.
3. Since root beer costs more than cola, there can be no more root beer, so Max bought
    2 cans of root beer and 5 cans of cola.
3) What number is in the one's digit place for the expanded form of 864?
This kind of problem relies on the fact that some numbers have repeating one's place digits when taken to increasing powers. Keep taking powers of 8 (2,3,4, ...) until you notice a pattern. Divide the power you want by the number of repeating digits. The remainder will be the number of the digit in the repeating sequence.
  1. The sequence is 8,4,2,6.
    4 digits long
  2. 64/4 = 17 with 0 left over.
  3. This means the 1's digit of 864 is 6
N   8N  
18
264
3512
44096
532768
6262144
72097152
816777216
Interesting note: Any number whose 1's digit squared results in the same 1's digit (for example 52 = 25) will have that 1's digit in any power! So a problem like "what is the one's digit of 6 to the power of 4545" will be easy! It's a 6 because 6x6 = 36 x 6 = 216 x 6 = 1296 ...
The 1's digit that behave this way are 1,5 and 6.

Problem
Solution
4) Sosna is moving. She can carry 16 books at once in her arms to the car. She can also carry 26 DVDs in one load. She doesn't mix books and DVDs to keep things organized. Sosna made 7 trips to the car with arms fully loaded and carried a total of 142 objects. How many DVDs did she carry to the car? Method 1: Use Guess-and-Check.
Guess # #trips
to the car		#book
trips		# DVD
trips		Total number of objects
27434 x 16 +
3 x 26 = 142

Sosna carried 3 x 26 = 78 DVDs to the car.

Method 2: Use 2 equations.
Use D for the number of DVD trips and
    B for the number of book trips.
1. The equation for the number of trips is
    D + B = 7
2. The equation for the number of objects is
    26D + 16B = 142
3. Make the first equation an equation for D.
    D = 7 - B
4. Substitute this for D in the second equation and solve for B
    26(7 - B) + 16B = 142
    182 - 26B + 16B = 142
    -10B = -40
    B = -40 / -10 = +4 (minus divided by minus is +)
    B = 4 book trips.
5. Subtract this from 7 to get the number of DVD trips =
    7 - 4 = 3 DVD trips x 26 DVDs per trip = 78 DVDs.
5) There is a population of 832 rabbits in the city of Lakewood. The population of rabbits doubled every six months. How many months ago was the population at 52 rabbits? 1. Divide 832 successively by 2 until you get to 52 rabbits =
    4     6-month cycles.
2. Multiply this by 6 to get the number of months ago there were
    52 rabbits = 4 x 6 = 24 months.