G. Grouping
and Counting
Overlapping Groups
When a question relates to
overlapping groups, try diagramming the problem with overlapping
circles. This will make it easy to account for the overlap.
Example 25
If, in a certain school, 20 students play soccer, 10 play basketball,
and 7 play both, how many students play basketball, soccer or
both?
(A) 20
(B) 22
(C) 23
(D) 25
(E) 29
Solution
Using the diagram above we have deduced some new facts:
3 play only basketball
13 only play soccer
7 play both
total of 23 players
Possible Range Questions
When questions ask for a possible
range, be sure to examine the lowest and highest possibilities.
Example 26
A cabinet contains 3 to 5 bottles, each of which contains 30
to 40 mushrooms. If 10 percent of the mushrooms are flawed, what
is the greatest possible number of flawed mushrooms in the cabinet?
(A) 51
(B) 40
(C) 30
(D) 20
(E) 12
Solution
There are, at most, 5 bottles, each of which contains at most
40 mushrooms; so, there are, at most, 5 × 40 = 200 mushrooms
in the drum. Since 10 percent of the mushrooms are flawed, there
are at most 20 (20 = 10% × 200) flawed mushrooms. The answer
is (D).
Count Inclusively
When doing counting problems, always be sure to
count the first and last of the range of items.
Example 27
Fence posts are being placed at 20 foot intervals along a road
1000 feet long. If the first fence post is placed at one end
of the road, how many fence posts are needed?
(A) 49
(B) 50
(C) 51
(D) 52
(E) 53
Solution
The average test taker would simply take 1000 and divide it by
20 to get 50. However, to get the right answer, you must include
the first and last choice. Since the road is 1000 feet long and
the fence posts are 20 feet apart, there are 1000/20 = 50, or
50 full sections in the road. If we ignore the first fence post
and associate the fence post at the end of each section with
that section, then there are 50 fence posts (one for each of
the fifty full sections). Counting the first fence post gives
a total of 51 fence posts. The answer is (C).
