2011-2012 Mathcounts warm-up 12

#161: Let width be x and the length is 1.5x (given). 1.5x² = 216
x² = 216 divided by 1.5 = 216 times 2/3 = 144. x = 12 -width and the length is 12 x 1.5 = 18
2 x (12 + 18 + 5 x 4) = 100 feet

#162: The answer is 3: 45 pm.












#163: Tom digs twice s fast as Dick, which means that he digs 2 holes and Dick
digs 1 hole in 6 hours. (12 holes/2) x 6 = 36 hours.

#164: From 1 to 299, there are 20 x 3 = 60 times the number "3" appears.
03, 13, 23,...93, 10 times for the unit digit and 30, 31, 32...39, another 10 times
for the tenth digit for ever hundred numbers.

From 300 to 363, there are 63 - 0 + 1 = 64 times the number "3" appears for the
hundredth digit.

303, 313,...363, or 6 - 0 + 1 = 7 times the number "3" appears for the unit digit and 330, 331, 332...339, or 9 - 0 + 1 = 10 times the number "3" appears on the tenth digit

60 + 64 + 7 + 10 = 141 times

#165:The answer is 20 squares.












#166: Median is the middle number. 36 x 1.5 = 36 x 1 + 36 x 0.5 = 54, 54 x 1.5 = 54 + 27 = 81
36 divided by 1.5 = 36 x 2/3 = 24, 24 divided by 1.5 = 24 x 2/3 = 16
(16 + 24 + 36 + 54 + 81) / 5 = 42.2

#167:  According to the given information, M + 3H = 47 --first equation
In 2 years, (M + 2) = 2 ( H + 2) = 2H + 4, M = 2H + 2--second equation
Substitute M with 2H + 2 to the first equation and you have 2H + 2 + 3H = 47
5H = 45, H = 9 

#168: 3O + 2C = 78---equation 1
          2O + 3C = 82---equation 2
equation 2 - equation 1 and you have C - O = 4, which is the answer.

#169: If it's just one digit number, you have "5". 1 way
Two-digit numbers, you have "14, 41, 23, 32". 4 ways
Three-digit numbers, you have 113, and 122, both have three arrangements. (113, 131, 311, etc... or 3!/2!
--the denominator is how many numbers repeating.) so altogether 6 ways.
Four-digit numbers, you have 1112-- 4!/3! or 4 arrangements.
Five-digit numbers, you have 11111.  1 way
1 + 4 + 6 + 4 + 1 = 16 ways.

#170: The answer is square root of 73.