GMAT Club Olympics 2024 (Day 1): In a company, the average (arithmetic

Question

In a company, the average (arithmetic mean) salary of all employees equals the average of the average (arithmetic mean) salary of the men and the average (arithmetic mean) salary of the women. If the sum of the average salaries of the men and the women is $88,000, what is the average salary of the men in the company?

(1) The number of men in the company is greater than the number of women.
(2) The average salary of the men is equal to the average salary of the women in the company.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Bunuel · Accepted Answer

GMAT Club Official Explanation:

In a company, the average (arithmetic mean) salary of all employees equals the average of the average (arithmetic mean) salary of the men and the average (arithmetic mean) salary of the women. If the sum of the average salaries of the men and the women is $88,000, what is the average salary of the men in the company?

Generally, in a weighted average problem like this one, the average salary of the combined group will always be closer to the average of the larger group. For instance, if the average salary of men is $50,000 and that of women is $60,000, and there are more men than women, the overall average salary will be closer to $50,000 than $60,000. This means it will be less than $55,000, ranging from $50,000 to just below $55,000. However, the question states that the overall average salary equals the midpoint between the average salaries of men and women. This scenario can only occur if either the numbers of men and women are equal, or the average salaries of men and women are the same.

Expressed algebraically, assuming there are m men and w women, with average salaries of x and y respectively, the overall average salary can be calculated as (mx + wy)/(m + w). We are told that this equals (x + y)/2. For, (mx + wy)/(m + w) = (x + y)/2 to hold, either m = w or x = y must be true:

\frac{mx + wy}{m + w} = \frac{x + y}{2}

2mx + 2wy= mx + my + wx + wy

mx + wy= my + wx

mx -wx = my -wy

x(m -w) = y(m -w)

x(m -w) - y(m -w)=0

(m -w)(x - y)=0

Either m = w or x = y.

(1) The number of men in the company is greater than the number of women.

This statement implies that the number of men and women are different, hence, it must be true that their average salaries are the same. Thus, the average salary of the men equals $88,000/2 = $44,000. Sufficient.

(2) The average salary of the men is equal to the average salary of the women in the company.

Given that the average salaries are equal, and the sum of the average salaries is $88,000, then the average salary of the men equals $88,000/2 = $44,000. Sufficient.

Answer: D.

Kinshook · Answer

Given: In a company, the average (arithmetic mean) salary of all employees equals the average of the average (arithmetic mean) salary of the men and the average (arithmetic mean) salary of the women.
Asked: If the sum of the average salaries of the men and the women is $88,000, what is the average salary of the men in the company?

Let the number of men and number of women in the company be m & w respectively.
Let the average salaries of men and average salaries of women be x & y respectively.

The sum of the average salaries of the men and the women is $88,000
x + y = 88000
y = 88000 - x

Average salary of all employees = (mx + ny)/(m+n) = (x+y)/2 
2(mx+ny) = (m+n)(x+y) = 88000(m+n)
mx + n(88000-x) = 44000(m+n)
(m - n)x = 44000(m+n) - 88000n = 44000(m-n)
x = $44000;(1) The number of men in the company is greater than the number of women.
m > n;
The average salary of the men in the company = x = $44000
SUFFICIENT

(2) The average salary of the men is equal to the average salary of the women in the company.
x = y ; y = 88000 - x = x ; x = $44000
The average salary of the men in the company = x = $44000
SUFFICIENT

IMO D

karanbadalia · Answer

From the info available, we can conclude that either the avg of men = avg of women, or number of men = number of women.

Therefore from statement 1 and 2 individually, we get that avg is equal.
So answer is D.

gaubeo308 · Answer

Solution:  Consider Statement (2). Since the sum of the average salaries of the men and the women is $88,000, the average salary of the men in the company is $88,000 divided by 2, equal $44,000. Statement (2) alone is sufficient to answer the question.

Consider Statement (1). Let the number of men be n and the number of women be m, where n>m. Then, we have   \frac{(x_1+...+x_n+y_1+...+y_m)}{(m+n) }= \frac{(x_1+...+x_n)}{n} + \frac{(y_1+...+y_m)}{m} . Simplifying this equation, we eventually get to the answer that the average salary of men is $44,000. Therefore, Statement (1) is sufficient to answer the question. Answer choice D is correct.

Fido10 · Answer

let's say M1, M2, ...Mm are the salaries of Men
and W1, W2, ...Ww are the salaries of women
and M is the number of Men 
and W is the number of Women
and AM is average of the men 
and AW the average salary of the women
We know that : the average (arithmetic mean) salary of all employees equals the average of the average (arithmetic mean) salary of the men and the average (arithmetic mean) salary of the women then
 /2 = (M1+ M2+ ...Mm+W1+ W2+ ...Ww)/(M+W)

(1) The number of men in the company is greater than the number of women.

(AM+AW)/2=(AM*M + AW*W)/(M+W)
=>2*AM*M + 2*AW*W = M*AM+M*AW+W*AW+W*AM
=>M*AM+AW*W =M*AW+W*AW
=>M(AM-AW) = W(AM-AW) (*)
in this case we have that M>W then for (*) to be true AM-AW should be 0, then AM=AW
Then, from AM +AW = 88.000 we can conclue that  AM=44.000, sufficient 
 (2) The average salary of the men is equal to the average salary of the women in the company. 
AM +AW = 88.000 
in this case, AM=AW then AM+AW=44.000,  sufficient

ANSWER is D

wwcd · Answer

can someone please tell me why this step is wrong :

mx - my = wx - wy

m ( x - y ) = w ( x - y )

m = w

@ Kinshook , @ karanbadalia  , @ gaubeo308

Bunuel · Answer

Note that we cannot divide  m(x - y) = w(x - y)  by (x - y) because (x - y) can be 0 and division by zero is not allowed. By dividing by (x - y), you would be incorrectly assuming that (x - y) does not equal zero, potentially excluding a valid case (observe that x = y satisfies the equation). As a rule, never reduce an equation by a variable (or by an expression containing a variable) if you are not certain that the variable (or expression with the variable) does not equal zero. Remember, we cannot divide by zero.

Hope this helps.

mvgkannan · Answer

Let A = average salary of all employees, X= Avg salary of Men, Y = Avg salary of Women, m= Total men and w = Total women.

By definition, A = (mX + wY)/(m+w), 
but here extra detail is A = (X+Y)/2. Equating both shows that mX + wY = mY + wX. Now we have two scenarios,

Scenario I, m=w, then it is directly visible that A=X=Y X = 44000. 
Scenario II, m is not equal to w. Here, from mX+ wY = mY + wX, we get (m-w)X = (m-w)Y, which again shows that X=Y and from above We get again X = 44000.

IMO no need of the statements at all. Directly we can say Option D. Whatever statement is given, ans is already clear.

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