8/2/2019 Each Employee of Company Z is an Employee of Either
Division
1/2
Quote: Each employee of Company Z is an employee of either
Division X or Division Y, but not both. If each division has some
part-time employees, is the ratio of the
number of full-time employees to the number of part-time
employees greater for Division X than for Company Z?
(1) The ratio of the number of full-time employees to the number
of part-time employees is less for Division Y than for Company
Z.
(2) More than half of the full-time employees of Company Z are
employees of Division X, and more than half of the part-time
employees of Company Z are employees
of Division Y.
Method-1This is a mixture problem.
X is being combined with Y to form Z.
Unless all the ratios are equal, the ratio for Z must be between
the ratios for X and Y.
Statement 1: The ratio of the number of full-time employees to
the number of part-time employees is less for Division Y than for
CompanyZ. Ratio for Z > Ratio for Y.
Since the ratio for Z is between the ratios for X and Y, Ratio
for X > Ratio for Z > Ratio for Y.
Thus, Ratio for X > Ratio for Z.
Sufficient.
Statement 2: More than half of the f ull-time employees of
Company Z are employees of Division X, and more than half of the
part-timeemployees of Company Z are employees of Division Y. Let F
= full-time employees and P = part-time employees.
Ratio for X = (more than half of F)/(less than half of P)
Ratio for Y = (less than half of F)/(more than half of P)
To compare ratios, we cross-multiply.
The numerator used in the greater product belongs to the greater
ratio.
Cross-multiplying, we get:
(more than half of F)(more than half of P) vs. (less than half
of F)(less than half of P).
The product on the left clearly is greater.
Since (more than half of F) is the numerator of X, Ratio for X
> Ratio for Y.
Since the Ratio for Z is between X and Y, Ratio for X > Ratio
for Z > Ratio for Y.
Thus, Ratio for X > Ratio for Z.
Sufficient.
The correct answer is D.
Method 2
here's a fact that you should know. i can furnish a proof if you
reallyreallyreally want me to, but it should be clear:
if a data set can be split into two groups, both of which have
at least the ratio a:b for some 2 characteristics, then the entire
data set hasat least the ratio a:b for those 2 characteristics.in
other words, if the ratio of FT to PT employees is at least, say,
3:1 in both divisions, then the overallratio of FT to PT employees
must also be 3:1.
here's a corollary:
i f a data set can be split into two groups, and one of the
groups has a ratio HIGHER than the overall ratio for some 2
characteristics, thenthe o the r group has a ratio LOWER than the
overall ratio for those 2 characteristics - and vice versa.this
follows logically from the above statement, because it violates the
first result (and common sense) if both divisions' ratios are
somehow higher (or both lower)
than the overall ratio.
statement (1)
this statement must be true, because if div. y has a lower
ratio, then div. x must have a higher ratio to balance things out
(see the corollary above).so, sufficient.
if you want actual inequalities to prove this, i would be glad
to provide them, but you should be able to conceptualize this
resultso that you have a fighting chance of
completing the problem within the allotted time.
statement (2)
because FT and PT are mutually exclusive, this statement implies
that div. x has more FT employees, but fewer PT employees, than
does div. y.
therefore, the ratios are (higher / lower) for div. x and (lower
/ higher) for div. y, so the overall ratio must be higher for div.
x.
sufficient
