Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 5 Algebra
Problem
Solution
1) How many times would a piece of paper have to be folded in half to create 128 layers? This is an exponent problem.
Each fold doubles the previous number of layers, so how many folds (powers of 2) does it take to get to 128? =
    2 ... 4 ... 8 ... 16 ... 32 ... 64 ... 128.
This is 7 folds.

2) If three pieces of chalk are the same length as 2 Popsicle sticks, how many Popsicle sticks would be the same length as 12 pieces of chalk? 		1. Compute the number of popsicle sticks/pieces of chalk = 2/3
2. Multiply by the number of pieces of chalk =
    2/3 x 12 = 8 popsicle sticks

3) There were 36 heads and 104 legs in a group of horses and riders. How many riders were in the group? 		There are 16 horses and 20 riders

Problem
Solution
4) Mr. Jackson is taking inventory of the scarves in his store. He has 1/8 as many gray scarves as brown ones. He has four times as many brown ones as yellow ones. There are 24 that are either rust or yellow. There are 10 rust-colored scarves. How many scarves does he have in his inventory? Work backwards.
1. If there are 24 scarves that are either rust or yellow and there are 10 rust-colored scarves, then there are 14 yellow scarves.
2. If he has 4 times as many brown ones as yellow ones, then he has 14 x 4 = 56 brown scarves.
3. If he has 1/8 as many gray scarves as brown ones then he has 7 gray scarves.
4. Total up the number of gray, brown, yellow and rust colored scarves
    = 14 yellow + 10 rust-colored + 56 brown + 7 gray = 87 scarves.
5) The temperature has been steadily rising since 6 am. Emily records temperatures in a table as shown below. What is the temperature 10 hours after 6 am? This is an arithmetic sequence problem.

Important: if you are looking for the temperature at 10 hours after 6 am you are looking for the 11th term in the sequence!
There are 2 ways to solve this problem:
Method 1:Write out the sequence elements:

1. Find the difference between consecutive hours = ____ degrees.
2. Write out the hours up to 10 on a line and the temperatures at those hours using the above difference.

  -3 1 5 9 13 17 21 25 29 33 37

3. Temperature at 10 hours =
37 degrees. 		Method 2:Use the arithmetic sequence formula:
An = A1 + d x (n - 1) where:
An is the term you are looking for.
A1 is the first term
d is the difference between successive terms
n = number of terms in the sequence
A11 = -3 + 4(11-1)
A11 = -3 + 4 x 10 =
37 degrees.