MathCounts Preparation: How to Excel at Middle School Math Competitions by Huasong Yin
Solutions to Exercises on Page 161
#1. A cube has six congruent square faces and its surface area is 6a 2 , where a is
the side length. 6a 2  54  a 2  9  a  3 . The volume is a 3  27 . Answer: 27
#2. If the original rectangular prism has integer dimensions a, b, and c units, then
the interior dimensions are a-2, b-2, and c-2 units.
There should be (a-2)(b-2)(c-2) interior unit squares. We know that there are
only 5 interior unit squares and 5 is a prime number. The interior dimensions
are 1, 1, and 5 units.
The original rectangular prism has dimensions 3, 3, and 7 units. A unit cube with
only one painted face should have the painted face as an interior unit square of
one face of the prism. The number of such unit cubes is the same as the surface
area of the interior rectangular prism.
21  1  1  5  1  5  22 .
Answer: 22
#3. There are only 7 interior unit cubes. Because 7 is a prime number we know
that the interior dimensions are 1, 1, and 7 units. The original rectangular prism
has dimensions 3, 3, and 9 units. A unit cube has exactly two faces painted if it is
one of the edge cubes but not at the end of the edge (i.e. not a corner cube). The
number of such unit cubes is the same as the total side lengths of the interior
rectangular prism. 41  1  7  36 .
Answer: 36
#4. The surface area is 6a 2  6  32  54 .
Answer: 54
#5. The ratio between the edge lengths of the two cubes is
3
1 1
the volumes should be    .
2 8
Answer:
1
. The ratio between
2
1
8
#6. If the edge length of the cube is a, then a 3  5x and 6a 2  x .
a 6 5x
a

  5  a  30 .
2
6a
x
6
© 2014,Huasong Yin
x  6 52  150 .
Answer: 150
MathCounts Preparation: How to Excel at Middle School Math Competitions by Huasong Yin
#7. If the right circular cylinder has radius r and height h, then the volume is
r 2 h and the lateral surface area is 2r  h . r 2h  3.5  2rh  r  2 .
Answer: 2
#8. If the dimensions are all tripled, the volume should be multiplied by the cube
of 3.
3  33  81 .
Answer: 81
#9. The volume of the cube is 8  2  32  8  8  8 and its edge length is 8. Its
surface area is 6a 2  6  64  384 .
Answer: 384
#10. An octagonal prism has 3  8  24
edges.
A rectangular prism has 3  4  12 edges.
24-12=12.
Answer: 12
#11. R=2r and H=3h. The new volume is R 2 H   2r  3h  22  3r 2 h  .
2
So the ratio of the new volume to the original volume is 22  3  12 .
Answer: 12
#12. ab  48 , ac  49 and bc  50 .
Multiply the three equations: ab  ac  bc  48  49  50 .
abc2  49  24  2  50  72 102  24 .
abc  72  102  24  70 24  70 4  6  70  2 6  140 6
Answer: 140 6
© 2014,Huasong Yin
MathCounts Preparation: How to Excel at Middle School Math Competitions by Huasong Yin
2
6
6 6
#13. The price should be 0.60      0.60  22   0.60  6  3.60 dollars.
4
 3 4
Answer: $3.60
 6   5  10 
#14.          2  1  3  6 cubes can be put in the rectangular box. The
 3   3  3 
6  33
27 54
ratio of the volumes is


 54% .
6  5  10 50 100
© 2014,Huasong Yin
Answer: 54%