Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2009 Grade 7 Number Sense
Problem
Solution
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 2⁄5 of a mile.
  2. She needs 11⁄2 = 12⁄8 gallons
  3. 12⁄8 ÷ 5⁄8 = 12/5 = 2, remainder 2, so it takes 3 trips =
    3 x 2⁄5 = 11⁄5 mile

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?
  1. 2x3x7 = 42 and 2x7x11 = 154 and 42x154= 6468, which is less than 9702 so clearly one of the prime factors (or more) are repeated in the first number.
  2. For the first number to be a minimum, we must investigate multiple 2s and 3s first.
  3. Try 2 2s:
    2x2x7x3 = 84;   9702/84 = 115.5, so this doesn't work
    Try 3 2s:
    2x2x2x7x3 = 168;   9702/168 = 57.75, so this doesn't work either.
  4. Try 2 3s:
    2x7x3x3 = 126; 9702/126 = 77, so this works!
  5. The first number is 126 and the second 77:
        126 x 77 = 9702
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? 5x11 = 55
  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 55.
  5. The only value for N that this is true is 11
    because $5 x 11 = $11 x 5.

Problem
Solution
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 $3.3x105.
  2. P = 3.3x105 ÷ 5x10-2 = $0.66x107
  3. The amount she has (after spending 2/3 of the interest) at the beginning of the second year is
    P + ($3.3x105/3) =
    P + $1.1x105 =
    0.66x107 + $1.1x105 =
    66x105 + 1.1x105 = 67.1x105 = $6,710,000.

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 -(3-4) -(5-6) = 2, so the result is:
+1 + 1 = 2