Right away, the word "ratio" tips us off that we're dealing with
ratios in this problem, and the word "greater" indicates that we're dealing with
inequalities. However, as we read through the rest of the problem, things start to get a little more confusing: one company, two divisions, full-time and part-time employees ... this is a lot to process.
We do see the words "either" and "both" though, which should get some overlapping sets wheels turning in our minds. We see that, like the problem above, we have two criteria: employees can belong to
Division X or Division Y and can be
full-time or part-time. Overlapping sets problems have a lot of information, so it's really easy to get lost in them. A good trick is to
use a visual representation to keep track of what you know:
- For two overlapping criteria, use a table, where each axis represents one criterion.
- For three overlapping criteria, use a venn diagram, where each circle represents a criterion.
Since this problem doesn't have any concrete numbers, it isn't strictly necessary to make a visual representation. However, it can still be helpful to define the relationships between our sets and build equations:
We know that the two numbers in each row should add up to the total at the end of the row and the two numbers in each columns should add up to the total at the bottom of the column. So we can now build 6 different equations:
- Full-Time @ X + Part-Time @ X = Employees @ X
- Full-Time @ Y + Part-Time @ Y = Employees @ Y
- Full-Time @ Z + Part-Time @ Z = Employees @ Z
- Full-Time @ X + Full-Time @ Y = Full-Time @ Z
- Part-Time @ X + Part-Time @ Y = Part-Time @ Z
- Employees @ X + Employees @ Y = Employees @ Z
Now that we have this set up, let's figure out what the question is asking for. Like with all word problems, we want to translate words into math. Whenever we're dealing with ratios, we should remember that
ratios can (and should) be expressed as fractions:
- Is full-time @ X/part-time @ X > full-time @ Z/part-time @ Z?
or in other words, are there more full-time employees for every part-time employee at Division X than at the entire company?
Statement 1
This Statement gives us information about the ratio of full-time employees to part-time employees at
Division Y compared to Company Z:
full-time @ Y/part-time @ Y < full-time @ Z/part-time @ Z
Now, before we rule this statement out because it doesn't tell us anything about Company X, let's see how we can use our equations to substitute X back into the inequality. Looking at equations 4 and 5, we see that we can rearrange the equations to give:
- Full-Time @ Y = Full-Time @ Z - Full-Time @ X
- Part-Time @ Y = Part-Time @ Z - Part-Time @ X
Subbing those into our inequality gives us:
full-time @ Z - full-time @ X/part-time @ Z - part-time @ X < full-time @ Z/part-time @ Z
Let's think about what we know about fractions. To make a fraction smaller, we need to either
- decrease the numerator relative to the denominator
- increase the denominator relative to the numerator
We know that we are decreasing both the numerator and denominator, so we must be decreasing the numerator by a greater percentage than we are decreasing the denominator. This means that the number of full-time employees at Division X is larger relative to the number of part-time employees at Division X than the number of full-time employees at Company Z to the number of part-time employees at Company Z. In other words, the ratio of the number of full-time employees to the number of part-time employees is greater for Division X than for Company Z.
Statement 1 is sufficient.
Statement 2
Like with Statement 1, let's translate this into math:
full-time @ X > 1/2full-time @ Z
part-time @ Y > 1/2part-time @ Z
Given equation 5, the second half of our statement also tells us that
part-time @ X < 1/2part-time @ Z
This means we can write the ratio of full-time employees at Division X as
>1/2full-time @ Z / <1/2part-time @ Z
or, cancelling the 1/2 in both the numerator and denominator,
>full-time @ Z / <part-time @ Z
To make a fraction larger, we need to either:
- increase the numerator relative to the denominator
- decrease the denominator relative to the numerator
Here, we're doing both: full-time employees at Division X is
greater than full-time employees at Company Z
and part-time employees at Division X is
less than part-time employees at Company Z. This means that
full-time @ X/part-time @ X > full-time @ Z/part-time @ Z
which is exactly what we're trying to solve for.
Statement 2 is sufficient.
Since both statements are sufficient to solve the problem individually, the correct answer is D.
We actually featured this problem recently on the
PrepScholar GMAT blog as one of the
5 Hardest Data Sufficiency Questions. I recommend checking out the article for more strategies and trends we can take away from this and other 700+ level problems!
