Data Sufficiency

A science lab is conducting an experiment monitoring the appearance of genetic traits in laboratory rats. Of the three traits being monitored all of the animals bear at least one of the three traits being monitored, one third of the rats were obese, three fourths of the rats had white fur, and one half of the rats had an elongated mandible. How many had all three of these traits?

(1) Of the 120 rats in the experiment, 40 had white fur and an elongated mandible.
(2) Of the 120 rats in the experiment, 40 had exactly two of the traits.

Mo2men wrote:A science lab is conducting an experiment monitoring the appearance of genetic traits in laboratory rats. Of the three traits being monitored all of the animals bear at least one of the three traits being monitored, one third of the rats were obese, three fourths of the rats had white fur, and one half of the rats had an elongated mandible. How many had all three of these traits?

(1) Of the 120 rats in the experiment, 40 had white fur and an elongated mandible.
(2) Of the 120 rats in the experiment, 40 had exactly two of the traits.

A useful formula for 3 overlapping groups:

T = A + B + C - (AB + AC + BC) - 2(ABC)

The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in A, everyone in B, and everyone in C:
Those in exactly 2 of the groups (AB+AC+BC) are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups (ABC) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.

In the problem above:
Let T = 120.
Obese = (1/3)(120) = 40.
White fur = (3/4)(120) = 90.
Elongated mandible = (1/2)(120) = 60.
Exactly 2 of the groups = OW + OE + WE.
All 3 groups = OWE.

Plugging these values into the formula, we get:
120 = 40 + 90 + 60 - (OW + OE + WE) - 2(OWE)
-70 = - (OW + OE + WE) - 2(OWE)
(OW + OE + WE) + 2(OWE) = 70.

Since the least possible value for (OW + OE + WE) is 0, the greatest possible value for OWE is 35.

Statement 1: Of the 120 rats in the experiment, 40 had white fur and an elongated mandible.
Case 1: Of these 40 rats, 5 have all 3 traits (OWE=5), implying that 35 have only white fur and an elongated mandible (WE=35).
Case 2: Of these 40 rats, 10 have all 3 traits (OWE=10), implying that 30 have only white fur and an elongated mandible (WE=30).
Since OWE can be different values, INSUFFICIENT.

Statement 2: Of the 120 rats in the experiment, 40 had exactly two of the traits.
Substituting (OW + OE + WE) = 40 into (OW + OE + WE) + 2(OWE) = 70, we get:
40 + 2(OWE) = 70.
2(OWE) = 30
OWE = 15.
SUFFICIENT.

The correct answer is B.