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29 Jul 2024, 05:31
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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.
(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.
29 Jul 2024, 05:31
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Official Solution:
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 \(\frac{mx + wy}{m + w}\). We are told that this equals \(\frac{x + y}{2}\). For, \(\frac{mx + wy}{m + w} = \frac{x + y}{2}\) 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 \(\frac{$88,000}{2} = $44,000\). 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 \(\frac{$88,000}{2} = $44,000\). Sufficient.
Answer: D
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 \(\frac{mx + wy}{m + w}\). We are told that this equals \(\frac{x + y}{2}\). For, \(\frac{mx + wy}{m + w} = \frac{x + y}{2}\) 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 \(\frac{$88,000}{2} = $44,000\). 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 \(\frac{$88,000}{2} = $44,000\). Sufficient.
Answer: D
25 Nov 2024, 11:35
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I am not able to understand statement 1's solution, could you please explain why "This statement implies that the number of men and women are different, hence, it must be true that their average salaries are the same"
