2011-2012 Mathcounts warm-up 16

#221: Let the two-digit number be 10x + y, the reversed two-digit number is thus 10y + x.
100% + 75% = 175% = 7/4 so 7/4 (10x + y) = 10 y + x
Multiply both sides by 4 and you have 70x + 7y = 40y + 4x, 66x = 33y so 2x = y.
There are 12, 24, 36, and 48, 4 two-digit numbers that fit.

#222: The play parallel to the base of the pyramid and also bisects it altitude so the height of the smaller
square pyramid is 1/2 of the larger square pyramid, and their volume is 1³
to 2³ or 1 to 8 ratio.
The volume of the smaller pyramid is 1/8 of 64 so 8 cm³.

#223: x + 4/x = y + 4/y (given) Times both sides by xy and you have
x² y + 4y = xy² + 4x
x² y - xy² =  4x - 4y, xy ( x - y ) = 4 ( x - y ), Divide both sides by ( x - y ) and
you have xy = 4.

#224: This one is more of a geometry question and you can get the area of the shaded region
pretty easily by just visually working on it.
The area of the shaded region is 6/16 or 3/8 = 37.5%
Use the smallest white unit square as a reference. The largest square is 4 x 4 = 16 and each of the
shaded region is 3/4 of a 2 by 2 square.
Also, 1/8 is 12.5% ( 1/4 is 25%, 1/8 is half of 1/4 so it's half of 25% or 12.5%)
3/8 is 3 times that.
You should try doing this question in your head.

#225: To form a 4-person dance team, the order doesn't matter so you use choose method.
8C2 times 11C2 = ( 8 x 7 ) / 2 times ( 11 x 10 ) / 2 = 1540.

#226: Let x be Matt's new test average. From given you can set up the following equation:
(4x [the sum of the scores of the other 4 tests not including 60] + 60) / 5 = x - 5
4x + 60 = 5x - 25, x = 85

#227: If the dimension of the larger blocks is evenly divided by the smaller blocks, then it's a pretty
straightforward question. For example, how many 1 x 1 x 1 unit cubes could fit into a 4 x 4 x 4 larger cube.
(4/1) times (4/1) times (4/1) = 64 unit cubes.

This question is tricky because the dimensions don't line up nicely so you need to change the positions
of the smaller rectangular prisms in the later stages.
It's too difficult to draw out the solutions so here is the explanation.
Break the large rectangular prism into two parts, 9 x 5 x 5 and 1 x 5 x 5.
For the first part, you'll have (9/3) x ( 5/1) x (5/1) or 75 smaller rectangular prisms nicely fit in the large
rectangular prism.
For the 1 x 5 x 5 rectangular prism, you break it into 1 x 3 x 5 and 1 x 2 x 5.
For the first one, you can have (5/3) x (5/1) x (1/1) -- just multiply the integer quotient, or 1 x 5 x 1 =
5extra 3 x 1 x 1 rectangular prisms.
For the second one, you have have (1/1) x (2/1) x (5/3) or 2 extra rectangular prisms.
total comes up to 75 +5 + 2 = 82

#228: The middle three index cards are of the same length so the triangle is an equilateral triangle.
each angle is 60 degrees. The structure rests on a flat, level surface so if you make the two angles next to
the bottom angle of the triangle a and b, you will have the two angles a + b = 180 - 60 = 120.
Angle x and y each belongs to an isosceles triangle so x = 180 - 2a and y = 180 - 2b.
x + y = 180 - 2a + 180 - 2b = 360 - 2 (a + b) = 360 -240 = 120.

#229: (22 x 10- total distance- minus 4 x 16) / 6 = 26 mi/h

#230: 2/3 of the boys = 3/5 of the girls  Times both sides by 15 and you have 10 b = 9 g
b = 9 and g = 10,  2/3 of 9 = 6 and 3/5 of 10 = 6  (19 - 6 - 6) / 19 = 7/ 19