buoyant wrote:SliceCo, a company that sells knives, is structured so that each of its x regional sales directors has y salespeople working for that director. If, for the month of May, SliceCo sold exactly 2,200 knives, how many regional directors worked for SliceCo that month?

(1) Each regional director sold 150 knives, and each salesperson sold 80 knives.

(2) Each regional director had 5 salespeople.

OA A

Number of directors = x.
Since for each of the x directors there are y salespeople, the total number of salespeople = xy.

Question rephrased: What is the value of x?

Statement 1: Each regional director sold 150 knives, and each salesperson sold 80 knives.
Number of knives sold by the x directors = 150x.
Number of knives sold by the xy salespeople = 80xy.
Since 2200 knives are sold, we get:
150x + 80xy = 2200
15x + 8xy = 220
x(15 + 8y) = 220.

The factor in red -- 15 more than a multiple of 8 -- must divide evenly into 220.
Since 220 = 110 * 2, the factor in red cannot be greater than 110.
Make a list of options for the factor in red, starting with 15+8 = 23:
23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103.
Only the value in red is a factor of 220.
Thus, 15+8y = 55.
Since x(15+8y) = 220, we get:
x * 55 = 220.
x = 220/55 = 4.
SUFFICIENT.

Statement 2: Each regional director had 5 salespeople.
No way to determine the value of x.
INSUFFICIENT.

The correct answer is A.

Hi Mitch,

Thank you for a nice explanation.

Won't testing several numbers in such questions be time consuming? Also, how to know in such questions that an equation with two variables is going to be sufficient.
Is there any other way to approach such questions?

buoyant wrote:Hi Mitch,

Thank you for a nice explanation.

Won't testing several numbers in such questions be time consuming? Also, how to know in such questions that an equation with two variables is going to be sufficient.
Is there any other way to approach such questions?

When an equation with two variables is restricted to POSITIVE INTEGERS, we must check whether it can be satisfied by more than one combination of values.
If the equation can be satisfied by only ONE combination of values, then we have SUFFICIENT information to solve for each variable.

An alternate approach to the problem above:
Let D = the number of directors and S = the number of salespeople.
Statement 1 implies the following:
150D + 80S = 2200
15D + 8S = 220.

The units digit of 15D must be 5 or 0.
To yield a sum of 220, the units digit of 8S must also be 5 or 0.
Since 8S is even, its units digit cannot be 5.
Implication:
Both 15D and 8S must have a units digit of 0.

Options:
15D = 30, 60, 90, 120, 150, 180, 210.
8S = 40, 80, 120, 160, 200.

Two combinations will yield a sum of 220:
15D + 8S = 60+160 = 220.
15D + 8S = 180+40 = 220.
The latter equation is not valid, since it implies more directors than salespeople.
Thus, we know that 15D=60 and that 8S=160, implying that D=4 and S=20.

You are correct:
Statement 1 requires a bit of a time investment.
But statement 2 compensates:
It can be evaluated without almost no work at all.

Total number of salespeople = x directors * y salespeople each = xy

so 150x + 80xy = 2200 knives

Substitute y = 5:
150x + 80x * 5 = 2200
550x = 2200
x = 4

Are statements 1 and 2 the answer options? Or is the question to find an answer that satisfies these 2 conditions? If the latter, then the answer is 4.

I keep finding questions that refer to answer choices A, B, C, ... etc but I don't see them in the question. What am I missing? It just looks staight forward to me that 4 is the answer. Thanks.