Logic Problem Solution:
Christmas Tree, O Christmas Tree
Neither the blue spruce (clue 5), Douglas fir (6), Fraser fir (7), nor
Scotch pine (8) was the fifth and last tree bought; the white pine was.
By clue 3, the Starrs bought their tree immediately before the family
who chose the Douglas fir did; while by clue 7, the family who bought
the Fraser fir picked their tree immediately ahead of the Lights. The
only possible commonality between the two clues is if the Starrs bought
the Fraser fir and the Lights chose the Douglas fir. However, by clue
3, then, the Lights' Douglas fir would have cost $25 more than the
Starrs' Fraser fir, but by clue 7, the Starrs' Fraser fir would have
cost $50 more than the Lights' Douglas fir. So, four of the five
treebuying families are named between clues 3 and 7. Since the Ball
family didn't pick the Fraser fir (2) and bought their tree later in
the evening than the family choosing the Douglas fir (6), the Balls
are the fifth family to the four in clues 3 and 7. There are three
possible orderings for the five families given clues 3, 6, and 7: 1)
the Starrs, the Douglas fir, the Balls, the Fraser fir, and the Lights;
2) the Starrs, the Douglas fir, the Fraser fir, the Lights, and the
Ball family; or 3) the Fraser fir, the Lights, the Starrs, the Douglas
fir, and the Ball family. In the first arrangement, the Lights would
have bought the white pine, contradicting clue 4. Trying the second
arrangement, the Ball family would have gotten the white pine. The
Starrs would have picked the Scotch pine (8) and the Lights then the
blue spruce. Since the Scotch pine cost $30 less than the Garlands'
tree (8), the Garlands wouldn't have bought the Douglas fir, which
cost $25 more than the Starrs' Christmas tree (3). The Garlands would
have bought the Fraser fir and the Hollys the Douglas fir. Letting the
blue spruce equal X in price, the Fraser fir would have cost
X + 50 (7), the Hollys' Douglas fir 2X (5), and the
Starrs' Scotch pine 2X  25 (3). The Starrs' Scotch pine also
would have cost X + 20 (8), so that 2X  25 = X + 20.
Solving, X would equal $45. So the Lights would have spent $45,
the Garlands $95, the Hollys $90, the Starrs $65, and the Balls $60 for
their white pine (4). However, there would be two differences between
tree prices of $5impossible by clue 1. Therefore, arrangement 2)
also fails, and arrangement 3) remains: the family selecting the Fraser
fir bought first, followed in order by the Lights, the Starrs, the
family who chose the Douglas fir, and the Ball family, who paid $60
for the white pine they picked (4). The Garlands didn't buy the first
tree (8); they picked the Douglas fir, and the Hollys chose the Fraser
fir. Since the Garlands' Douglas fir cost $25 more than the tree the
Starrs bought (3), the Starrs didn't take home the Scotch pine, which
cost $30 less than the Garlands' choice (8). The Starrs' Christmas
tree is the blue spruce and the Lights' the Scotch pine. Letting the
Starrs' blue spruce cost X, the Garlands' Douglas fir cost
X + 25 (3). The Hollys' Fraser fir cost 2X (5), so the
Lights' Scotch pine was 2X  50 (7). The Lights' Scotch pine
also cost X  5 (8). So, 2X  50 = X  5.
Solving, X equals $45. So, the Starrs spent $45, the Garlands
$70, the Hollys $90, and the Lights $40. The five families bought
Christmas trees last evening as follows:
 1st  Holly family, Fraser fir, $90
 2nd  Light family, Scotch pine, $40
 3rd  Starr family, blue spruce $45
 4th  Garland family, Douglas fir, $70
 5th  Ball family, white pine, $60

