This week, we’re going to tackle a GMATPrep® question from the quant side of things. We’ll tackle a medium-level question this week in order to learn how to master weighted average questions in general, and in the next article, we’ll try a very hard one – just to see whether you learned the concept as well as you thought you did. : )
Before we begin, I want to mention that every weighted average problem I’ve seen on GMATPrep is a Data Sufficiency question. This doesn’t mean that they’ll never give us a Problem Solving weighted average problem, but it does seem to be the case that the test-writers are more concerned with whether we understand how weighted averages work than with whether we can actually do the calculations. So we’re going to work on that conceptual understanding today and then we’ll discuss a neat calculation shortcut next week (built on the same principles!), just in case we do need to solve.
Let’s start with a sample problem. Set your timer for 2 minutes…. and… GO!
* 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.
Given “a certain company,” we’re asked to determine the ratio (not a real number, just a ratio – key point!) of two subgroups that together make up all employees: males to females.
So, what do we know? We know that male employees have an average of 9.8 years of experience. We also know that female employees have an average of 9.1 years of experience. What would be useful to solve? It would be useful to know about the actual number of male and female employees. Alternatively, it would be useful to know about the relationship between the number of male employees and the number of female employees. (For example, if they told me 60% of the employees were female, then I would know the ratio of males to females was 40:60, or 2:3, even though I wouldn’t know the actual number of employees.)
Most of what we’re going to do next is just to explain how weighted averages work. Once you understand how this works, you will not actually have to do these calculations on DS questions (this will take way longer than 2 minutes!); you’ll be able to determine conceptually whether enough info was provided to solve.
In the given problem, could there be equal numbers of male and female employees? Go take a look at the problem again and see what you think.
Let’s say that there are, in fact, 50 male employees and 50 female employees. If the male employees’ average experience is 9.8 years and the female employees’ average experience is 9.1 years, then what is the average experience for the whole group? That would just be the average of 9.8 and 9.1. Is the average of those two numbers 9.3 (the “total group” average given in statement 2)? No. So now we know we’ve got a weighted average problem; in other words, the number of male employees is not equal to the number of female employees. (Bonus question: can you tell, just based on what we’ve discussed so far, whether there are more male or female employees?)
In order to understand how weighted averages work, let’s calculate a few things and let’s start by using the weighted average formula to see what happens in a case where we have equal numbers of employees (which, again, is not true for this problem – we’re just examining the concept).
We know the two sub-group averages, 9.8 and 9.1, and we’re also assuming an equal weighting of the two averages, 50:50, which simplifies to 1:1. Put that ratio, 1:1, in a form where the two parts add up to 1: ½:½. Each average gets paired with its “adds up to 1” weighting:
[(9.8)(1/2) + (9.1)(1/2)] = 9.45
Because the weightings are equal, this can be simplified to the standard average formula (below); this is why we don’t bother calculating the “adds up to 1” weighting when the weightings are equal.
(9.8 + 9.1) / 2 = 9.45
What if the weightings are not equal, though? Let’s say that there were 40 male employees and 60 female employees. Then, the ratio would be 40:60, or 2:3, and the “adds up to 1” weighting would be 2/5: 3/5. (The easiest way to determine the “adds up to 1” weighting is to first add the two parts of the given ratio, 2 and 3, to get 5. 5 becomes the denominator of both fractions and the original numbers, 2 and 3, become the numerators of each respective fraction: 2/5 and 3/5.)
The weighted average formula would become:
[(9.8)(2/5) + (9.1)(3/5)] = 9.38
Here’s the “abstract” version of this formula:
[(average #1)(a) + (average #2)(b)] = weighted average, where:
a + b = 1, and
a and b represent the relative weightings of the two sub-groups
In the given problem, we don’t know a and b, but we do know the two sub-group averages, 9.8 and 9.1, so we can write these two formulas:
9.8a + 9.1b = c
a+b = 1
We need to see whether we have enough information in the statements such that a and b could be calculated.
Statement 1 says “There are 52 male employees at the company.” That gives us an actual number for the male employees; that might be good. We want the male to female ratio, though; does this statement tell us anything about the other group, female employees? No. Not sufficient. Eliminate answers A and D.
Statement 2 says “The average number of years of experience for the company’s male and female employees combined is 9.3 years.“
That’s something we can add to one of our formulas: c, the weighted average, is 9.3:
9.8a + 9.1b = 9.3
a+b = 1
What do we have? We have two distinct, linear equations with two variables, a and b. Can we solve for a and b? Yes! Sufficient!
The correct answer is B.
We can simplify this further (for future data sufficiency questions) by saying: if we know the two sub-group averages and we know the overall weighted average, then we know we can solve for a and b, the relative weightings of the two sub-groups. (Don’t bother to write the equations, of course – it’s data sufficiency!) In this case, a:b represents the requested ratio (male:female).
Key Takeaways for Data Sufficiency Weighted Average Problems:
(1) Determine that you have a weighted average problem: this occurs when an average is discussed or could be calculated, but that average is not a standard 1:1 or equally weighted average.
(2) Carefully write down what you were asked to solve, then determine what you know, what you don’t know, and what you would need to know in order to solve (before you look at the statements). Remember that, if you have two sub-group averages and the overall average, then you can determine the relative weightings of the sub-groups.
(3) Check the given statements to see whether you can find a “match” (that is, a statement tells you what you had already decided you would need to know in order to solve).
Answer to bonus question: there are more female employees at the company because the weighted average, 9.3, is closer to 9.1 (the female employee figure) than to 9.8 (the male employee figure). Click here to read the second article in this series, where we’ll elaborate on this concept.
* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.