A consultancy firm has exactly nine employees: F, G, H, I, J, K, L, M, and N.
- K’s salary is greater than both I’s and L’s.
- L’s salary is greater than N’s.
- I’s salary is greater than F’s.
- F’s salary is greater than M’s.
- M’s salary is greater than G’s.
- G’s salary is greater than J’s.
- J’s salary is greater than H’s.
- Which one of the following employees cannot have the third highest salary?
- If M and N earn the same salary, at least how many of the employees must have lower salaries than L?
- The salary rankings of each of the nine employees could be completely determined if which one of the following statements were true?
- L’s salary is greater than F’s.
- L’s salary is greater than I’s.
- N’s salary is greater than F’s.
- N’s salary is greater than I’s.
- N’s salary is greater than M’s.
- If N’s salary is the same as that of one other employee of the firm, which one of the following must be false?
- I’s salary is less than L’s.
- J’s salary is less than L’s.
- L’s salary is less than F’s.
- L’s salary is less than H’s.
- N’s salary is less than G’s.
- What is the minimum number of different salaries earned by the nine employees of the firm?
- Assume that the employees of the firm are ranked according to their salaries, from first (highest) to ninth (lowest), and that no two salaries are the same. Which one of the following is a complete and accurate list of G’s possible ranks?
- fifth, sixth
- fifth, seventh
- fifth, sixth, seventh
- fifth, sixth, seventh, eighth
The Action: A quick look at the rules tells you immediately that this is a “free-floating” sequencing set. The rules place entities in relation to each other instead of assigning them to specific spots (as in “the mime was ranked fifth,” from Unit 4). You are asked to order nine employees in a consultancy firm—F, G, H, I, J, K, L, M, and N—in relation to each other based on their salaries.
The Key Issues will be:
1) What employee can, must, or cannot have a higher/lower salary than what other employee?
2) What employee can, must, or cannot have the same salary as what other employee?
The Initial Setup: Often, the best way to visualize this kind of “free-floating” sequencing set is vertically. On the top of the page, write “more” and at the bottom write “less.” This will serve to remind you that those employees above have a larger salary than those under (this also makes logical sense). That and listing the employees (their letters actually) off to the side are about all you can do before hitting the rules:
To give yourself an idea where to start, scan the rules and try to spot a employee that is not stated as having a lower salary than another employee. K in Rule 1 isn’t explicitly lower than anyone, so put a “K” at the top of your sketch.
1) From the “K” draw two lines, one down to an “I” and one down to an “L.”
2) From that “L” draw a line down to an “N.”
3) Now jump over to the other branch. From the “I,” draw a line down to an “F.”
4) From the “F,” you need a line down to an “M.”
5) From the “M,” draw a line down to a “G.”
6) From the “G,” draw a line down to a “J.”
7) And finally, from the “J,” draw a line down to an “H.”
The first thing to do is to check your list of entities and make sure that they’re all included in the sketch. In this case they are, so you can fully depend on the master sketch (and redrawings of it, as needed) to answer all the questions.
There are a few things that you should notice right from the start. K has the largest salary, period. Either H or N has the smallest salary. Here’s a common mistake to avoid: Just because L and N are connected to a shorter “branch,” don’t assume that they necessarily make a higher salary than any of the entities in the other “branch.” They could both make less than even H. As soon as you’ve taken some time to make sure you understand the sketch, you should be all set to rack up some easy points.
The Final Visualization: And here is our neat, accessible sketch:
There are two kinds of employees who cannot have the third highest salary: Those who must have a higher salary and those who must have a lower salary. Who must have a greater salary? Only K, who has the greatest salary, but she’s not one of the choices. So look for those entities that have three or more people over them in the sketch (they could be at best fourth highest).
The right branch is composed of L and N below K. We noted above that L and N are flexible and can fit anywhere into the left branch, so either of them could have the third highest salary. Eliminate answer choices (C) and (E). On the left branch, count down three people, and anyone after that could be the answer. M, G, J, and H all have three or more people over them, which means that any of them qualifies as a employee that cannot have the third highest salary. The testmakers happened to pick M, answer choice (D).
You can probably picture this scenario in your head, but you may have also opted for a quick re-drawing of the master sketch including the new information (just draw an “=” sign between M and N):
Now it’s simply a matter of counting the people that are definitely below L. The new sketch clearly shows that N, M, G, J, and H are all below L, a total of 5 people, answer choice (C). I and F could be below L, but they could also be above L. K, as always, is definitely above L.
To determine all the salaries, you need to connect the L—N branch of the sketch to the longer branch with all of the entities’ locations definitely set. Not much to do but check the choices.
(A) L is now above F but could be above or below I, and N could be anywhere below L. Keep on looking.
(B) L is between K and I, but N could be anywhere from third to last.
(C) N is now above F, but L and N are not set in relation to I.
(D) If N is above I, N must be third, following K and L. This leaves I through H fourth through ninth, respectively. All of the employees’ rankings are determined, so (D) is the answer.
(E) L and N could assume many positions in the ranking above M and below K, so (E) is no help.
If N has the same salary as one other employee, what CAN’T be true? At first glance, it appears that there’s not much to do here except note the new info and check the choices. However, you might have noticed that L couldn’t be the same or less than H because there would be no one left for N to share salaries with. If you recognized this, you could have just scanned the choices and found that choice (D) is impossible for this reason. If you didn’t see this, you would need to check each choice, stopping when you found the choice that must be false.
As for the wrong choices, here are orderings that show that each of them is possible:
(A) and (B) K—L—(I=N)—F—M—G—J—H
What are the minimum number of different salaries? Well, L and N can share salaries with two other employees in the left branch. The entities in the left branch, however, are all separated explicitly by the rules. So all you have to do is count them up (don’t forget to count K at the top). There are 7, which is choice (C).
Here we’re looking for G’s possible rankings, given the fact that no one shares salaries. First count the people who are explicitly placed above G. There are four, which means that G can be fifth but no higher than fifth. Does this eliminate any choices?
No, since fifth appears in each answer choice, and no answer choice includes a place higher than fifth. Who else could earn a higher salary than G? The only “wild cards” are L and N who are very flexible. One, both, or neither could be above G. That makes G’s possibilities fifth, sixth, and seventh, choice (D).
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