Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 6 Number Sense

Problem
Solution
1) A company creates parts for robots. 24% of the parts created are broken. If there are 42 broken parts, how many parts has the company made? Use T for the total number of parts.
The equation is:
    24% x T = 42.
Solve for T = 42/.24 = 175

2) In the 6th grade class at Prime Middle School, the number of boys is 2/3 the number of girls. 1/4 of the girls wear glasses, and 1/8 of the girls who wear glasses have braces. If 3 girls have glasses and braces how many students are in the 6th grade class? 		1. The fraction of the girls who wear glasses and braces is 1/8th of 1/4th of the total number of girls = 1/32 of the girls.
2. That percentage is 3 girls, so compute the number of girls in the class from this information = (1/32)G = 3; G = 3 X 32 = 96 girls.
3. The number of boys is 2/3 of this = 2/3 X 96 = 64
4. Add them to get the total number of students = 96 + 64 = 160 students.

3) Complete the Magic Square. A Magic Square is a square array of numbers where the sum of the entries in each row, each column, and each diagonal are all the same. Express all four answers as fractions.
1. Ah, the key is that the row, column and diagonal totals are given to you by that one completed diagonal.
Compute it's total = 2/3 + 5/12 + 1/6 =
    8/12 + 5/12 + 2/12 = 15/12
2. For the two rows that have 2 numbers, compute the 3rd using the above total. Add the 2 together and subtract from the total.
    Row 1 Column 2: 15/12 - 2/3 - 1/2 = 15/12 - 8/12 - 6/12 = 1/12
    Row 2 Column 1: 15/12 - 5/12 - 7/12 = 3/12 =     1/4
3. Now you have enough information to complete the columns, so do that in the same way.
    Row 3 Column 1: 15/12 - 3/12 - 8/12 = 1/3
    Row 3 Column 2: 15/12 - 1/12 - 5/12 = 9/12 = 3/4

Problem
Solution
4) There are 25 children on a swim team. 10 swimmers are at most ten years old. 8 are older than twelve and 4 are older than thirteen. How many eleven and twelve year olds are on the team? 1. From the 25 total kids subtract the number
    that are at most 10 (10), the (8) that are older than 12.
2. You can ignore those that are older than 13
    because they are included in the 8 that are older than 12.
3. What's left is the 11 and 12 year olds =
    25 - 10 - 8 = 7
5) Liz was playing with a pile of pennies. She noticed when she stacked them in piles of three there was 1 left over. When she stacked them in piles of five there were 2 left over. There was more than one pile of 5. What is the smallest number of pennies she could have? Make a table of stacks of 5, like this:
# stacks
of 5
(N)		Total (T)
pennies
5N + 2		Remainder
after dividing
by 3 (R)
21212/3 = 4
R = 0
31717/3 = 5
R = 2
42222/3 = 7
R = 1



<=== Divide the totals (T)
by 3 until the remainder is 1.
T = 22 (divided by 3 = 7, remainder 1)