Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 6 Algebra
Problem
Solution

1) Here are two representations of the same multiplication problem.
y and z both stand for single digit numbers. What is y?
 Note: Be careful with Representation 2. It is misleading. It looks like algebraic relationships. For example:
      y x (yz) = 402
but this is not correct! yz means 10y + z not y times z. In any case, you can solve the problem without Representation 2, so ignore it!

1. Notice in the multiplication of yz times z
    (yz x z = 469) that the last digit is a 9.
    This means that z times z must end in a 9.
    Therefore, z must be either 3 or 7 (3x3=9, 7x7=49)
2. Look at the first multiply of yz times y: (yz x y = 402)
    Therefore y x z must end in a 2
3. Examine the 2 possibilities for z: z = 3 or 7:
    if z = 3, then y must be 4 for z x y to end in a 2
    if z = 7, then y must be 6 for z x y to end in a 2
    Try both of these and see which one works!
    Putting these values back in the original multiply results in the fact that only z = 7 and y = 6 result in 5092
    so y = 6
Here's the full calculation:
        67
   x   76
      402
+  4690
    5092

Problem
Solution
2) Each week Roxy mows Mrs. Smith's lawn and does some extra yard work. Mrs. Smith pays Roxy a fixed amount for mowing her lawn and an additional hourly amount for extra work. Based on the table below, how much is the fixed amount for mowing the lawn?
 1. Subtract the amount Roxy earned in the second week from the amount she earned in the first week and divide by the difference in the hours she worked. =
    45 - 27 = 18/3 = 6.
This is the amount she gets paid for extra work per hour.
2. Apply this to the amount she earned in week #1
    to find the fixed amount = 27 - 2 x 6 = $15

3) Vicky and Tammy are shopping for new clothes. Vicky says to Tammy, "If I had $5 more, we would have an equal amount of money." Tammy responds, "That may be true,but if I had $5 more, I would have twice as much money as you." How much money does Tammy have?		 Let V = the amount of money Vicky has and
    T the amount Tammy has.
1. Write the first equation: V + 5 = T.
2. Write the second equation: T + 5 = 2V.
3. Put the value for T from the first equation into the second equation and solve for V =
    (V + 5) + 5 = 2V
    V + 10 = 2V
    V = 5
4. Put this value in the first equation to find T =
    T = 10 + 5 = 15

Problem
Solution
4) Emilio wants to drown his sorrows by eating the full pan of brownies his mom left out on the counter to cool. In five minutes he eats 1/3 of the pan. In the next five minutes he eats 1/3 of the remaining pan. If this pattern continues how much of the pan of brownies will be left after 30 minutes? Express your answer as a fraction. The table provided may help you think it out.
 Let L1 = what's left from the last period,
L2 = what's left in this period
E = what he eats in this period.
1. At each 5 minute point do the following calculation:
    E = L1 x 1/3 (he eats a third of what was left)
    L2 = L1 - E (what's left after he eats a third)
    (at each time period the amount left at the previous time period (computed as L2) becomes L1 for this new time step.
2. Repeat this 5 times to get the amount left after 30 minutes =
    10 min: E = 2/3 x 1/3 = 2/9; L2 = 2/3 - 2/9 = 4/9
    15 min: E = 4/9 x 1/3 = 4/27; L2 = 4/9 - 4/27 = 8/27
    20 min: E = 8/27 x 1/3 = 8/81; L2 = 8/27 - 8/81 = 16/81
    25 min: E = 16/81 x 1/3 = 16/243; L2 = 16/81 - 16/243 = 32/243
    30 min: E = 32/243 x 1/3 = 64/729

5) Sam gives his mother a bouquet of flowers; some of the flowers have 3 petals and the rest have 5 petals. He counts 12 flowers in all. His little sister counts 44 petals. How many 3 petal flowers are in the bouquet?		 Method 1: Use 2 equations:
Let T = number of 3-petal flowers and
      F = number of 5-petal flowers.
1. Then T + F = 12, and
    3T + 5F = 44
2. Turn the first equation into an equation for F:
    F = 12 - T
3. Substitute this expression for F in the second equation and solve.
    3T + 5(12 - T) = 44
    -2T = -16
    therefore T = 8

Method 2: Use guess-and-check.
Guess ## 3-petal#5 petalTotal
#petals
1843 x 8 + 4 x 5 = 44