This post was written by Rich Zwelling, one of Knewton's awesome GMAT prep instructors. For more GMAT help, check out the Knewton GMAT blog.
If you've done some GMAT preparation already, you've likely come across the concept of "weighted averages." But what does that term really mean? In short, the term "weighted" is simply meant to indicate that separate groups of numbers have different numbers of elements and thus should be weighed differently.
Let's say one group of numbers has an average of 4, and a second group of numbers has an average of 6. We cannot just average 4 and 6 and conclude that the overall average of all the numbers in the two groups is 5. There could be more numbers in one group than in another, and thus the two groups would have different "weights."
For example, if the first group has 1,000,000 numbers while the second has only 1 number, the first group is weighted much more heavily, and thus the overall average will be much closer to 4 than to 6. (When illustrating general principles, outlandish examples always do the trick )
On the GMAT, this manifests itself in several ways, but as with so many GMAT topics, the conceptual understanding is often much more important than the ability to do raw calculations. As such, you can bet that weighted average questions will show up in Data Sufficiency!
Hey, what do you know?! I just happen to have a sample GMATPrep problem right here! Feel free to give it a shot before reading my explanation:
Each employee on a certain task force is either a manager or a director. What percentage of the employees are directors?
(1) The average salary for a manager is $5,000 less than the average salary of all employees.
(2) The average salary for a director is $15,000 greater than the average salary of all employees.
A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked.
And now, the breakdown: It's easy enough to narrow the answer choices down to C and E; each statement gives information on one of the two groups, but not the other. When looking at the statements together, it's very tempting to think that the answer is E, since you are given only averages and not absolute numbers of directors or managers. However, the question asks about the percentage of employees who are directors, and percentages can be gotten without absolute numbers.
Believe it or not, the specific numbers involved give away the exact percentages. How do we know? Weighted averages! The difference between the average salary of a manager and the overall employee average ($5,000) is one-third that between the average salary of a director and the overall employee average ($15,000), and yet they end up "balancing each other" to get that overall average. That means that the $5,000 group must carry more weight. And since the two dollar amounts are in a 3:1 ratio, that means that there are 3 times as many managers as directors!
As such, we can conclude that 75% of the employees are managers, and 25% are directors. Final answer: C.
If you recall my previous post on averages, this makes sense. If the average director's salary adds an amount $15,000 greater than the average, it will take three amounts of $5,000 below the average to balance things back to the overall average. So for every one director, there must be three managers.
Again, notice that we didn't have to do a single calculation. Our conceptual understanding of weighted averages is what bailed us out.
Next week, I'll show you how a conceptual understanding of weighted averages can even help you on Problem Solving questions!
In the meantime, I have another DS problem for you. But this time, try it on your own! Post your step-by-step solutions in the comments, and be sure to apply the concepts behind weighted averages!
At a certain company, the average (arithmetic mean) number of years of experience is 9.8 years for the male employees and 9.1 years for the female employees. What is the ratio of the number of the company’s male employees to the number of the company’s female employees?
(1) There are 52 male employees at the company.
(2) The average number of years of experience for the company’s male and female employees combined is 9.3 years.
A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked.
The answer is B)
A0 does not give any information of the female employees except average work experience.
As per B) Sum of male employees’ work experience+ Sum of female employees’ work experience/ No. of male+ no. of female employees+ 9.3
=> 9.8 n(male)+ 9.1 n(female) = 9.3 n(male) + 9.3 n(female)
=> .5 n(male) =.2 n (female)