Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 7 Number Sense
Problem
Hint
1) Jessica is camping with her family and needs to get water for the camp. She needs 11⁄2 gallons. She has a pot that holds 5/8 gallon. The spring where she gets water is 1/5 mile from camp. How many miles will she walk while getting the water using only the 5/8 gallon pot? Express your answer as a mixed number.
  1. Each round trip is _____ of a mile.

  2. She needs _________ gallons
  3. Divide the number of gallons by the round trip time to get the number of whole pot trips. If any is left over, add a trip. Total trips = ___. Total distance = _____ miles

2) The prime factors of one number are 2, 3, and 7. A second number has the prime factors 2, 7, and 11. If their least common multiple is 9702 what is the smallest the first number could be? 		For the first number to be a minimum, you must add multiple prime factors, starting with the smallest ones (2 and 3).

At some point, the product of your factors of the first number will divide evenly into 9702. That product will be the smallest the first number can be =



_______
3) At the summer carnival there is a game of chance. You pay $5.00 to play a single game. A game consists of one toss at the target. If you hit the target you win $11.00. If you miss you get $0.00. What is the fewest number of games you can play and break even? (Meaning the carnival didn't gain or lose any money to you). This is a least common multiple (lcm) problem.
  1. What is the lcm of 5 and 11? ______
  2. However many games you play, let's call it N, you pay $5xN to play
  3. In order to break even you must win W of those games to equal $5 x N
  4. $11 x W must equal $5 x N, therefore both W and N must be factors of the lcm of 5 and 11.
  5. The only value for N that this is true is _____________.

Problem
Hint
4) Jorrie is an ambitious girl. She has decided that by the time she is 35 she will retire and will live off of the interest of her considerable fortune. The first year of her retirement she wants the interest to be $3.3x105. She will spend 2/3 of the interest, and put the other 1/3 back into her investment. If she earns 5% per year on her investment, how much should her investment be worth at the beginning of the second year of retirement? Express your answer to the nearest dollar without using scientific notation.
  1. At the beginning of the first year of her retirement, she earns 5% on the principal (P) which is _______
  2. Compute P from this information = _____________

  3. The amount she has (after spending 2/3 of the interest) at the beginning of the second year is
    P + 1/3 of the first year interest = ____________

5) In the expression below place minus signs and parentheses on the left hand side of the equal sign to create a true equation.
3       4       5       6 = 2
 This is an exercise in negative number arithmetic.

The first 2 and the second 2 numbers differ by 1, so if you can place parentheses around the first 2 and parentheses around the second 2 with appropriate signs to make them both 1s then you have it!
The answer is ____________________________