Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 7 Number Sense
Problem
Hint
1) What is the smallest exponent n for which the following expression is true?
    (0.5)3 > (0.6)n 		Use your calculator that does exponents for this problem. Compute the value of the first expression = _____


Compute the value of the second expression for values of n starting with 2 and find the first one that produces a value less than the first value.



  n = _____
2) In Mr. Hemple's orchestra there are 4/7 as many boys as girls. 1/4 of the girls wear glasses, and 2/7 of the girls who wear glasses also have braces. If 2 girls have glasses and braces, how many students are in Mr. Hemple's orchestra? This is a classic 'work backwards' problem.
  1. Start with the fraction of the girls who wear glasses also have braces and compute the number of girls who wear glasses =
        _____ girls wear glasses
  2. These girls who wear glasses are _____ of the girls, therefore there are ______ girls total.

  3. There are _____ as many boys, so
    there are _____ boys.

  4. The total number of students is ______
3) On a number line, what fraction is halfway between
    1 - 3 and   -3 + 5 ?
    3   4           8     6 		Simplify both terms, convert to a common denominator, add them and divide by 2.

Problem
Hint
4) 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. When she stacked them in piles of seven there were 3 left over. What is the smallest number of pennies she could have? Start with stacks of 7 and compute the remainders of stacking in 5s and 3s (I'll do the first stack for you):
# stacks
of 7
(N)		 Total
pennies

(7N + 3)		 # stacks of 5 + remainder # stacks of 3 + remainder
2173+25+2
3    
4     
5    
6    
7    
8    
9    
Liz has _____ pennies.
5) It is the end of market day and Grandma Wilcox wants to sell all her remaining eggs. One at a time to the last two customers come to her stand. To the first customer she sells half of all of the eggs she has right then plus half an egg without breaking any eggs. To the last customer she sells half of all her remaining eggs plus half an egg without breaking any eggs. She's now out of eggs. What is the smallest number of eggs she could have had left at the time the last two customers came? Very confusing explanation.
What the problem means is that when you take half the remaining eggs (which can involve a fraction of an egg) and add half an egg you end up with a whole number of eggs with each customer transaction. Therefore the number of eggs she started with must be odd to have half an egg remaining after dividing by 2.
So, the number of eggs must be one of: 3,5, 7 ...
Do the computations (carefully!) for 3,5, 7 eggs and see which one works.





The least number of eggs she could have had before these 2 customers arrived was ______