Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2013 Grade 5 Algebra
Problem
Solution
1) The distance around a track is 1/4 mile. Jessie plans to run 5 miles and has already done 6 laps. How many more laps must she run to complete 5 miles? 1. Compute the number of laps in 5 miles =
    5/(1/4) = 5 x 4 = 20 laps.
2. Subtract the laps she has already done. Remaining =
    20 - 6 = 14 laps.

2) At the end of February Jenny noticed that she could finally get up when it was light in the morning. She kept track of when sunrise was for a few days in the paper and saw a pattern:
Feb 24, 2013 6:59 AM
Feb 25, 2013 6:57 AM
Feb 26, 2013 6:55 AM
Feb 27, 2013 6:53 AM
If the pattern holds true, when was sunrise on February 1st?		 1. Find the amount of time the sunrise advances for 1 day = 2 min.
2. Compute the number of days from Feb 1st to the 24th = 23 days. Remember, don't count the 24th!
3. Multiply this by the number of minutes the time advances in one 1 day =
    23 x 2 = 46 minutes.
4. Compute the time of sunrise on the 1st, going forward this number of minutes =
    6:59 am + 46 minutes = 7:45 AM.

3) The figure shows a sequence of castle fronts made from cubes. The number of cubes needed for each castle front form a numerical pattern. How many cubes are needed for the castle front of Castle 8?
 1. This is actually 2 sequences added together.
2. The first sequence is the number of cubes on the top of each castle (a constant) = 2 cubes.
3. The second sequence is the number of cubes making the remainder of the castle front. Write out that sequence =
    N = 1   2   3
    # = 3   8   15
4. For each castle, N, then this second sequence is
    N x (N+2)
5. Compute the number of cubes in the 8th castle as the number of cubes in the first sequence + the number of cubes in the remainder, using N = 8 =
    2 + 8 x (8 + 2) = 2 + 8 x 10 = 82 total cubes.

Problem
Solution
4) Trina, who is 10, wants to know how old her mother is. Her mother says two times the difference between our ages is the same as the sum of our ages plus 14 more. How old is her mom? Use T for Trina and M for Mom.
1. Write the expression for "2 times the difference in our ages" =
    2 (M - T)
2. Write the expression for "the sum of our ages +14" =
    M + T + 14
3. Set these two expressions equal to each other and solve =
    2 (M - T) = M + T + 14
    2M - 2T = M + T + 14
    M = 3T + 14
    Since T = 10,
    M = 3 X 10 + 14
    M = 30 + 14
    M = 44.
5) Mrs. Severson keeps a jar with pennies and dimes in her classroom as a tool for teaching place value to her young students. Jori reaches in and pulls out a handful of coins and counts them. She has 43 coins. Tom computes and finds out the handful is worth 88 cents. How many are pennies? Method 1: Guess-and-check
# dimes#penniesTotal
53888 cents

Method 2: Use 2 equations:
Use P for the number of pennies and D for the number of dimes.

Equation 1: Write the equation for the total number of coins:
    D + P = 43
Equation 2: Write the equation for the total value of all coins:
    10D + P = 88
Solve the 2 equations for P:
  1. Turn the first equation into an equation for D: D = 43 - P
  2. Substitute this value for D in the second equation:
        10(43 - P) + P = 88
        430 - 10P + P = 88
        -9P = 88 - 430 = -342
        9P = 342
        P = 38