A street cleaning crew works only Monday to Friday, and only during the day. It takes the crew an entire morning or an entire afternoon to clean a street. During one week the crew cleaned exactly eight streets—First, Second, Third, Fourth, Fifth, Sixth, Seventh, and Eighth streets. The following is known about the crew’s schedule for the week:
- The crew cleaned no street on Friday morning.
- The crew cleaned no street on Wednesday afternoon.
- It cleaned Fourth Street on Tuesday morning.
- It cleaned Seventh Street on Thursday morning.
- It cleaned Fourth Street before Sixth Street and after Eighth Street.
- It cleaned Second, Fifth, and Eighth streets on afternoons.
- If the crew cleaned Second Street earlier in the week than Seventh Street, then it must have cleaned which one of the following streets on Tuesday afternoon?
- First Street
- Second Street
- Third Street
- Fifth Street
- Eighth Street
- If the crew cleaned Sixth Street on a morning and cleaned Second Street before Seventh Street, then what is the maximum number of streets whose cleaning times cannot be determined?
- What is the maximum possible number of streets any one of which could be the one the crew cleaned on Friday afternoon?
- If the crew cleaned First Street earlier in the week than Third Street, then which one of the following statements must be false?
- The crew cleaned First Street on Tuesday afternoon.
- The crew cleaned Second Street on Thursday afternoon.
- The crew cleaned Third Street on Wednesday morning.
- The crew cleaned Fifth Street on Thursday afternoon.
- The crew cleaned Sixth Street on Friday afternoon.
- If the crew cleaned Fifth, Sixth, and Seventh streets in numerical order, then what is the maximum number of different schedules any one of which the crew could have had for the entire week?
- Suppose the crew had cleaned Fourth Street on Tuesday afternoon instead of on Tuesday morning, but all other conditions remained the same. Which one of the following statements could be false?
- The crew cleaned First Street before Second Street.
- The crew cleaned Second Street before Fifth Street.
- The crew cleaned Third Street before Second Street.
- The crew cleaned Sixth Street before Fifth Street.
- The crew cleaned Seventh Street before Second Street.
This set requires us to order seven streets—1, 2, 3, 4, 5, 6, and 7—based on when they’re cleaned—either the morning or afternoon of one Monday through Friday work week. This is a sequencing set with a slight twist. The morning and afternoon aspect means that you’ve got ten spaces in which to plug the streets. The Key Issues will be:
1) When is each street cleaned?
2) What streets can, must, or cannot be cleaned before or after what other streets?
3) What streets can, must, or cannot be cleaned in the morning?
4) What streets can, must, or cannot be cleaned in the afternoon?
The Initial Setup:
If you were keeping track of these days, you’d probably make a little calendar. A calendar is just a type of gird, so put the days—M, T, W, Th, and F—across the top of a 5 X 2 grid, and along the side write AM and PM, for morning and afternoon. Since the testmakers were kind enough to number the streets, your job will simply be to write the numbers into the square depending on when each street is cleaned. If you know that no streets can be cleaned on a certain day and time, just put an “X” in that square. Remember to list the streets (in this case, the numbers) off to the side:
1) No streets are cleaned on Friday morning, so put a big X in Friday’s AM square.
2) Wednesday afternoon is out, so place another X in Wednesday’s PM square.
3) Easy enough; write a 4 in Tuesday’s AM square, and cross it off the list of streets.
4) And now 7 is definitely set as well. Cross 7 off your list of streets and put it in Thursday’s AM square.
5) Make sure you interpret this correctly. For now write, “8 . . . 4 . . . 6,” and we’ll come back to this rule under Key Deductions.
6) 2, 5, and 8 are cleaned in the afternoon, so at the far end of the PM row, write “2, 5, and 8.”
Let’s now look closer at Rule 5. Street 4 is set on Tuesday AM and 6 is cleaned after 4. Rule 6 says that 2, 5, and 8 are cleaned in the afternoon. Now, combine Rules 5 and 6, and the only afternoon slot that is available before 4 is Monday PM, so 8 must be cleaned Monday PM; write an 8 in Monday’s PM square. And cross off 8 from the list of afternoon streets.
It’s also a good idea to identify the “floaters,” those entities that aren’t bound by any rules. Here the floaters are 1 and 3. They can go just about anywhere. However, there is one thing about these two floaters that you may have noticed. 4, 7, and 8 are all set. 6 must be cleaned after 4 on Tuesday PM, Wednesday, Thursday, or Friday. 2 and 5 must be cleaned in the afternoon. So what about Monday AM?
The only streets that can be cleaned on Monday AM are 1 and 3. They are the only entities free to be cleaned on Monday (6 can’t) AM (2 and 5 can’t). Write “1/3” in Monday’s AM square. This is a tricky bit of deductive thinking, but if you saw it, you’re that much ahead of the set.
The Final Visualization: So here’s the very helpful sketch:
2nd street must be in the afternoon (Rule 6), and the only available afternoon slot before 7th street, which is cleaned on Thursday AM (Rule 4), is Tuesday afternoon. What street must be cleaned on Tuesday afternoon?—2nd street, choice (B).
The only morning that 6 can be cleaned on is Wednesday (that’s the only morning after 4—Rule 5), and 2 is under the same conditions as in Q. 12: before 7, which means Tuesday PM. The only streets that we don’t know are 1, 3 (our two floaters), and 5. That makes three undetermined streets, answer choice (C).
Streets 4, 7, and 8 are all set, so none of them can be cleaned on Friday afternoon. All the other streets (1, 2, 3, 5, and 6) could be cleaned on Friday PM without violating any of the rules. Try it. That’s a total of five streets, choice (E).
In the Key Deductions section above, we saw that 1 or 3 must be cleaned on Monday AM (all of the other streets are explicitly prohibited from being cleaned then). This question stem lets you know that 3 can’t be the one cleaned on Monday AM, so for this question, it has to be 1. Let’s stop right here and see if that’s enough by itself to answer this question.
Quickly scan the choices. Answer choice (A) must be false. If 1 is cleaned before 3, then 1 must be cleaned on Monday AM, which means 1 can’t be cleaned on Tuesday PM.
If you hadn’t made the “1/3 on Monday AM” deduction before, you could have still checked each of the choices. (B) and (C) 1 cleaned on Monday AM, 2 cleaned on Thursday PM, and 3 cleaned on Wednesday AM shows that these two could be true; eliminate both of them.(D) and (E) 1 on Monday AM, 3 on Wednesday AM, 5 on Thursday PM, and 6 on Friday PM eliminates (D) and (E).
7 is set on Thursday AM, so 6 must be cleaned immediately before that: Wednesday AM (no streets are cleaned on Wednesday PM—Rule 2). 5 must be cleaned right before 6: Tuesday PM. That’s 5, 6, and 7 cleaned in numerical order. We’re left with 1, 3 (one of which is on Monday AM), and 2 (which must be cleaned on one of the remaining afternoons).
If Street 1 is cleaned Monday AM, then 2 and 3 are left to float between Thursday and Friday afternoons. That’s two possibilities so far. But if 3 is cleaned Monday AM, then 1 and 2 are left to float between Thursday and Friday afternoons, giving us another two options. That’s four total. Written out, the possibilities look like this:
1) Monday AM—1, Thursday PM—2, Friday PM—3,
2) Monday AM—1, Thursday PM—3, Friday PM—2,
3) Monday AM—3, Thursday PM—2, Friday PM—1, and
4) Monday AM—3, Thursday PM—1, Friday PM—2.
Four possibilities, choice (D).
A new rule, so it’s not a bad idea to quickly alter your master sketch, moving street 4 to Tuesday PM. Go back and see how that affects everything else. 8 is still on Monday PM (still the only afternoon before 4), and 7 is still on Thursday AM (Rule 4 hasn’t changed). Rule 6 said that 2 and 5 are in the afternoon. Since 4 is now in the afternoon that leaves two afternoons for 2 and 5 to split—Thursday PM and Friday PM. The only place left for 6 (must be after 4—Rule 5) is Wednesday AM.
The only streets left are 1 and 3, and they’ll split Monday AM and Tuesday AM. Now to the choices. Street 5 can be cleaned on Thursday PM while 2 is cleaned Friday PM, showing that choice (B) could indeed be false and is therefore the credited answer. (A), (C), (D), and (E) all must be true, as evidenced by the new master sketch.
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