WASHINGTON STATE MATHEMATICS COUNCIL 2000 MIDDLE SCHOOL MATH OLYMPIAD   Session II ANSWERS: SEVENTH GRADE   NOTE:  Take off half-credit if units were asked for in the question but not provided by the team. 7

Number Sense

1.    The mean is 20.  (The 4 #'s are 17, 19, 21, 23.  Give 2 points if 4 #'s are given instead of mean.)

2.    143 (Formula for the nth  term is 3n - 7.)

3.    3   (Note: The cycle is 1-3-9-7.)

4.    24  (Note:  16 multiples of 6 and 12 multiples of 8 less the 4 repeated common multiples of 6 & 8)

5.    \$30

Measurement

1.    34  (Note:  17 cuts at 2 minutes each.)

2.    Wednesday

3.    8 cups  or  8 cups

4.    2,464 lbs  or 1 ton 464 lbs

5.    Thursday, October 12, at 4:20 a.m.    or    October 12 at 4:20 a.m.

(Note:  150,000 seconds is 1 day, 17 hours, 40 minutes)

Geometric Sense

1.    123.6 (if used π button) or 123.8 (if used 3.14 for π)  (Answers may include sq. in. label.)

2.    22.5  or 22

3.    Any right triangle with an area of 88 (whose legs multiply to equal 176).  Examples are:

4.    304    (Give 2 points for 16 x 19.)  (Note:    The theater contains 16 rows of 19 seats.)

5.    9 (people)

Probability & Statistics

1.    30    (5 one-digit numerals and 25 two-digit numerals)

2.    Several possibilities:  e.g.,  36, 38, 40, 42, 44, 46  or  &nbP 20, 30, 40, 42, 52, 62.

The two middle numbers must add to 82; the 4 other numbers must add to 164.

3.    1/1,296  or  6/7,776

4.    1/5  or  4/20  or  .2  or  20%

5.    96  (60 within-conference games and 36 across-conference games)

Algebraic Sense

1.    12

2.    20    (20th place)

3.    6  (ways)  (8 cents, 48 cents, 88 cents, 128 cents, 168 cents, and 208cents in 8 cent stamps, with the rest being filled in with 5 cent stamps)

4.    \$75

5.    4950 (is the 4-digit number)  (Note:  49.50 is the mean of 49 and 50)

 WASHINGTON STATE MATHEMATICS COUNCIL 2000 MIDDLE SCHOOL MATH OLYMPIAD   Session II ANSWERS: EIGHTH GRADE   NOTE:  Take off half-credit if units were asked for in the question but not provided by the team. 8

Number Sense

1.    (a)  1  (Note:  day 42)

(b)  6  (Note:  days 12, 24, 36, 48, 60, 72)  (All 3 girls work out on day 84.)

2.    15 (shots)

3.    1892    (Notes:  442 = 1936;  1936-44=1892)

4.    base 3 and base 8  (Notes:  base 3 = 64, base 4 = 145, base 5 = 276, base 6 = 469,

base 7 = 736, base 8 = 1089, base 9 = 1,540, base 10 = 2,101)

5.    1  5  15  30  45  51  45  30  15  5  1

1    6    21    50    90    126  141  126    90  50  21  6  1

(Method:  Each number in a row is determined by numbers above it in the previous row.  It is the sum of the number directly above it and the two numbers directly to the left of that number.  If there are no numbers to the left of that number (or if there is only 1 number to the left of that number), then only those 2 (or 1) numbers are summed to get the number in the row below.)

Measurement

1.    56 yards    (Note:  7 yards of length needed 8 times, filling the 18' or 216" length by  2'3" width pieces)

2.    400  (hare-leaps)

3.    Wednesday

4.    8 cups  or  8 cups

5.    14 (groups of rafters)  (Note:  Groups leaving at 4 a.m. through 5 p.m.)

Geometric Sense

1.    Any right triangle with an area of 88 (whose legs multiply to equal 176).  Examples are:

2.    8,000  (cubes)

3.    There are (at least?) 3 possible answers -- the 7 pointed star at right and the two figures at left:

4.    60 + 6  or 78  or 79  (depending on their calcaultion method)

5.    28 or 28 units

Probability & Statistics

1.    5/14 or  or  10/28  or  20/56  (Give 2 points for 5/28 or 10/56.)

There are 8 X 7=56 possible ways to choose 2 coins without replacement.  Of these 56 ways, there are 20 ways to get 15 cents:

(2 dime choices for 1st coin) (5 nickel choices for 2nd coin) = 10 choices with dime 1st,

(5 nickel choices for 1st coin) (2 dimes choices for 2nd coin) = 10 choices with nickel 1st.

[The 56 possible 2-coin chocies are as follows.   Note that d1 means dime #1, n2 means nickel #2, etc.  The 20 choices that are worth 15 cents are in boldface.]

14 ways with dime first       15 nickel-dime or nickel-quarter ways with nickel first

d1,n1  d2,n1    d1,q        n1,d1  n3,d2    n1,q

d1,n2   d2,n2    d2,q            n1,d2  n4,d1      n2,q

d1,n3  d2,n3    d1,d2          n2,d1   n4,d2      n3,q

d1,n4  d2,n4    d2,d1           n2,d2    n5,d1     n4,q

d1,n5  d2,n5           n3,d1    n5,d2      n5,q

20 ways with 2 nickels        7 ways with quarter first

n1,n2    n2,n1   n3,n1    n4,n1    n5,n1      q,n1  q,n5

n1,n3    n2,n3    n3,n2    n4,n2  n5,n2        q,n2   q,d1

n1,n4    n2,n4    n3,n4  n4,n3    n5,n3    q,n3    q,d2

n1,n5    n2,n5    n3,n5    n4,n5    n5,n4     q,n4

2.    13

3.    (a)  2,073  (items total)

(b)  136  (families had 2 items)

4.    Several possibilities:  e.g.,  36, 38, 40, 42, 44, 46  or  20, 30, 40, 42, 52, 62.

The two middle numbers must add to 82; the 4 other numbers must add to 164.

5.    0  (zero)  (If 3 men get their hats, the fourth man does too!)

Algebraic Sense

1.    39

2.    \$75

3.    61  (day 61)

4.    4950 (is the 4-digit number)  (Note:  49.50 is the mean of 49 and 50)

5.    8 (days)  (Give 2 points for 8.57 or   days.)