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LRDI Practice Set 22

A soft drink manufacturer surveyed consumer preferences for exactly seven proposed names for its new soda: Jazz, Kola, Luck,Mist, Nipi, Oboy, and Ping. The manufacturer ranked the seven names according to the number of votes they received. The name that received the most votes was ranked first. Every name received a different number of votes. Some of the survey results are as follows:

  1. Jazz received more votes than Oboy.
  2. Oboy received more votes than Kola.
  3. Kola received more votes than Mist.
  4. Nipi did not receive the fewest votes.
  5. Ping received fewer votes than Luck but more votes than Nipi and more votes than Oboy.
  1. Which one of the following could be an accurate list of the seven names in rank order from first through seventh?
  1. Jazz, Luck, Ping, Nipi, Kola, Oboy,Mist
  2. Jazz, Luck, Ping, Oboy, Kola,Mist, Nipi
  3. Luck, Ping, Jazz, Nipi, Oboy, Kola,Mist
  4. Luck, Ping, Nipi, Oboy, Jazz, Kola,Mist
  5. Ping, Luck, Jazz, Oboy, Nipi, Kola,Mist
  1. Which one of the following statements must be true?
  1. Jazz received more votes than Nipi.
  2. Kola received more votes than Nipi.
  3. Luck received more votes than Jazz.
  4. Nipi received more votes than Oboy.
  5. Ping received more votes than Kola.
  1. If the ranks of Ping, Oboy, and Kola were consecutive, then which one of the following statements would have to be false?
  1. Jazz received more votes than Luck.
  2. Jazz received more votes than Ping.
  3. Nipi received more votes than Oboy.
  4. Nipi received more votes than Mist.
  5. Oboy received more votes than Nipi.
  1. What is the total number of the soft drink names whose exact ranks can be deduced from the partial survey results?
  1. one
  2. two
  3. three
  4. four
  5. five
  1. What is the maximum possible number of the soft drink names any one of which could be among the three most popular?
  1. three
  2. four
  3. five
  4. six
  5. seven
  1. If Ping received more votes than Jazz, then what is the maximum possible number of names whose ranks can be determined?
  1. two
  2. three
  3. four
  4. five
  5. six

The Action: The action here is revealed in the second sentence: The seven proposed soda names are ranked in terms of their popularity, from first to seventh. This is straightforward sequencing. The Key Issues will be:

1) Which names are more or less popular than which other names?

2) Which names must, can and cannot be ranked in each slot from first to last?

The Initial Setup: Make a roster of the entities, but then look at the rules: Each one relates one or more entities to others. There are no rules such as “Nipi is ranked third or sixth”—in fact, only one rule directly relates to any entity’s position in the ranking, and that one negatively (“Nipi is not 7th”). What we have here is a loose sequence, also known as a “freefloating” sequence. So we don’t want to create a vertical or horizontal list of slots just yet; rather, we want to put the rules together en route. Let’s see how this happens, working with the most concrete rules first.

The Rules:

1) On the roster, Jazz is higher than Oboy. How much higher? No way to tell. They could be consecutive, or widely separated. To take note of this on the page (and to avoid throwing yourself mixed messages), jot down the J/O relationship kind of freely, something like this:

2) Ordinarily we’d seek out another rule that ties in with what we already know. Here, the testmakers give us a break and throw it in right away: The same Oboy whom we just learned is outranked by Jazz, ranks higher than Kola. Therefore:

3) And now Rule 3 is obliging too. Since Kola ranks higher than Mist, we can be sure of the following:

—and voila, four of our seven are solidly ranked!

4) deals with Nipi, who hasn’t entered the relative ordering yet. Let’s postpone this one for now.

5) requires care. It’s complex, and if you blow it you can write the game off. Let’s take it one step at a time—and the best place to start is with any mention of the names we’ve already heard. For that reason, let’s start with the information that Ping got more votes than Oboy, since Oboy’s already in the picture. Jazz got more votes than Oboy as well. Who’s ranked higher, Jazz or Ping? Can’t tell yet, but we do know this:

Next: Now that Ping’s in the picture, we can incorporate the fact that Ping received fewer votes than Luck, but more than Nipi:

Back to Rule 4. Nipi doesn’t come in 7th. So who does? Everyone else, except for Mist, is ranked above one or more names. Therefore, it has to be true that Mist was the least popular name, #7 on our hit parade.

Key Deductions:

Well, we’ve already begun those, haven’t we, as we moved seamlessly into Step 4. (That’s how it happens with some games, notably loose sequences). In freefloating sequence games, we usually form a few deductions right off the bat simply by establishing the relationships on the page. For example, we weren’t explicitly told, but can deduce from our sketch, that Ping received more votes than Kola, and Luck more than Nipi, Oboy and Kola. There are others, of course, and they should all be clear from the relationships etched in the sketch. It’s also valuable to consider the game’s overall parameters, for instance noticing that there are two and ONLY two contenders for the #1 slot—Jazz and Luck—so I guess we’ll all be drinking Jazz or Luck sometime soon. Beyond that, all of the other names’ rankings are limited only by what we have learned so far, which means there’s still a lot of flexibility.

The Final Visualization:

We should be ready to redraw our sketch as needed, when new information arrives. But we should also be ready to get some quick points from what we’ve done so far, which is:

The Questions:

1. (C)

Ordinarily, in an Acceptability question, you should check each rule against the choices to seek the violators. Here it’s even easier, since we know we must see J--O--K--M in that rough order somewhere among the right answer, and (A) and (D) both violate that sequence, so they can go in one swoop. (B) violates Rule 4 (not to mention the fact that we deduced that Mist must be ranked 7th). And (E) reverses Rule 5’s requirement that Ping get fewer votes than Luck. As the only one left standing, (C) must be right.

2. (E)

(E)’s relationship is easily seen in the sketch or deducible like so: P gets more votes than O (Rule 5) and O gets more than K (Rule 2), hence . . . choice (E).

(A), (B), (D) We cannot be sure of Nipi’s precise relationship to Jazz, Oboy, or Kola. These three statements are possibly true, but not necessarily so.

(C) would be true if Luck is ranked #1. But if Jazz is ranked #1, then (C) is false.

3. (C)

You can sketch out the question stem like so:

Next you can add the things you already know—and remember, the stem mandates that “P O K” is an unbroken sequence:

All that’s left is to add N, which ordinarily will rank somewhere below P (Rule 5) but not 7th (Rule 4). But here, since “P O K” is an unbroken sequence, we can be sure of:

(A) is possibly true (if Jazz is ranked #1), while (B), (D), and (E) are all definitely true in this sequence. But (C) is false: Oboy must rank above Nipi here.

4. (A)

We saw this up front: All we can be sure of is that Mist got the fewest votes. Perhaps you ran with this and chose (A) and immediately moved on to save time. Kudos if you did! If not, you probably double-checked to confirm that every other entity has at least two possible rankings. N’s ranking is unfixed relative to that of J, O, and K, so the exact position of these four names can’t be determined without more info. L can be first or second, while P could be second or third. Only M’s dead last ranking can be precisely determined from the partial results in the rules—choice (A).

5. (B)

We’ve already seen, in Qs. 1 and 3, that Luck, Ping, and Jazz can all rank among the top three, and we know from the setup that Mist must rank 7th. So all we need do here is figure out which of the remaining names—Kola, Nipi, and Oboy—could sneak into the top three. Kola cannot, because at least four other names (J, O, P, and L) rank higher. Oboy cannot, because at least three others rank higher (J, L, and P). But Nipi can, since it is only for-sure outranked by Luck and Ping; Nipi could rank 3rd. Final box score: Three names (M, K, O) cannot be among the top three; four can—choice (B).

6. (B)

Create a new sketch here, putting P atop J. That means that L, always atop P, is now at the top of the list. It reads: L--P--J--O--K--M. But what about Nipi? As long as its rank is below Ping’s (Rule 5) and not last (Rule 4), it can take any position. So all we know for sure is that L is 1st, P 2nd, and M (as always) 7th. That’s three, choice (B). Positions 3 through 6 remain uncertain, because N can be inserted anywhere in this middle part of the ordering.

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