In a network of car dealerships, a group of \(d\) sales

[BTGmoderatorLU]

Source: Veritas Prep

In a network of car dealerships, a group of \(d\) sales directors each has a team of \(a\) associates. In a given month, if the directors each sold 10 cars and each sales associate sold 20 cars, and all cars were sold by either a sales director or a sales associate, how many people total sold cars?

1) The total number of cars sold was 270.
2) \(a > d > 2\)

The OA is C

[Ian Stewart]

If we have d directors, and each director has a associates, we have ad associates. If the directors each sell 10 cars, they all sell 10d cars in total, and if the associates each sell 20 cars, they sell 20ad cars in total. So 10d + 20ad = 10d(1 + 2a) is the total number of cars sold.

Statement 1 tells us 10d(1 + 2a) = 270, so d(1 + 2a) = 27. So d and 1+2a are positive integers that multiply to 27, and so must be, in some order, 1 and 27 or 3 and 9. It's possible d=3 and 1+2a = 9, so a = 4. If d= 9, then 1+2a = 3, and a = 1. But then each dealer has only 1 associate, and the question tells us each dealer has a "team of associates". One person is not a "team", so this answer does not make sense. If we have 27 directors, then 1+2a would equal 1, and a=0, and from the wording of the question, we cannot have zero associates. And if we have 1 director (so a=13), we do not have, as the question describes, a "group" of directors. So there is only one possible value for d and one possible value for a, and Statement 1 is sufficient.

Statement 2 is clearly insufficient alone, so the answer should be A. If they want the answer to be C here (which is the OA), they need to change the wording of the question (which they need to do anyway, because "each" nonsensically refers back to the singular "a group" in the first sentence, when it presumably should be referring back to each of the sales directors).

[deloitte247]

Statement 1
Total number of cars sold was 27
$$d\left(1+2a\right)=27$$
possible cases are d=1 and a=13, d=3 and a=4, d=9 and a =1,d =27 and a=0
information given is not enough to arrive at a specific value, Statement 1 is INSUFFICIENT.

Statement 2
value of a and d is unknown, statement 2 is INSUFFICIENT.
combining statement 1 and 2 together:-
possible values are
d=1 and a=13
d=3 and a=4
d=9 and a=1
d= 27 and a = 0
from statement 2 $$a>d>2$$
the only possible case that returns true for this inequality is the case of d = 3 and a = 4
Total = 10 d +20 ad
270 = 10 d + 20 ad
directors = 10 d = 10*3 =30
associates = 20 ad
270 = 10d + 20ad = 20*4*3 = 240
Total people = 240 + 30 = 270; statement 1 and 2 together are SUFFICIENT. $$answer\ is\ Option\ C$$