How many of the days, Monday through Friday, are such that at most two batches of cookies could be made on that day?

kens on September 26, 2020

I have the same question. Don't quite understand the wording of the question. Please help.

lsatstudier1 on September 29, 2020

Bumping this

Victoria on October 16, 2020

Hi @Abigail-Lee, @kenken, and @lsatstudier1,

Happy to help! This question is definitely confusing and very open-ended. Hopefully I can help clarify it for you.

We know that a bakery makes exactly three kinds of cookie: O, P, and S.

Exactly three batches of each kind of cookie are made each week (Monday through Friday).

Therefore, we are dealing with nine variables: O1, O2, O3, P1, P2, P3, S1, S2, and S3.

Each batch is made, from start to finish, on a single day.

So, we are trying to schedule nine batches of cookies over five days. No batches overlap any days.

Now let's go through our conditions.

Rule 1 - No two batches of the same kind of cookie are made on the same day.

This means that no day will feature the same kind of cookie being made twice. However, it is entirely possible that one batch of each type of cookie is made each day for three days, leaving two days of the week completely open.

What we can diagram from this is:

O1 > O2 > O3
P1 > P2 > P3
S1 > S2 > S3

This means that they don't overlap.

Rule 2 - At least one batch of cookies is made on M

So, because M is the first day of the week, at least O1, P1, or S1 must be made on M.

Rule 3 - O2 is made on the same day as P1

This means that P1 cannot be made on M because O1 must precede O2

|O2P1|

We can also conclude that at least either O1 or S1 must be made on M

Rule 4 - S2 is made on R

This means that S3 must be made on F

M: O1/S1
T:
W:
R: S2
F: S3

What else can we deduce to make things a little easier on ourselves?

We know that O2 and P1 must be made on the same day. This means that P2 and P3 must be made after this day. Therefore, the latest that O2 and P1 can be made is Wednesday. They could also be made on Tuesday if O1 is made on Monday.

Now we have two options to diagram

Option 1 - |O2P1| on Tuesday

M: O1
T: O2P1
W:
R: S2
F: S3

S1 could be made M, T, or W
P2 and P3 could be made on WR, WF, or RF
O3 could be made on W, R, or F

Option 2 - |O2P1| on Wednesday

M:
T:
W: O2 P1
R: S2 P2
F: S3 P3

S1 could be made M, T, or W.
O1 could be made M or T.
O3 could be made R or F.

Now that we have a couple options diagrammed, let's address the question stem. We are looking for the number of days which can have a maximum of two batches of cookies made on that day.

If we go back to the deductions we made above, we can see that M can only have O1 and/or S1 made on that day because P1 must be made no earlier than T.

Therefore, our answer choice be (A), but let's double-check the other ones based on the diagrams above.

This is a bit confusing to explain, but we can work our way through the remaining days and eliminate them by proving that they could have three batches of cookies made on them.

We already know that M can have at most two batches of cookies made on that day because P1 cannot be made on M and we cannot have two batches of the same kind of cookie made on the same day.

Can three batches of cookies be made on T?

Yes, we can see this based on Option 1 that we've diagrammed above. If S1 is made on T, then three batches of cookies can be made on T.

M: O1
T: O2 P1 S1
W: P2
R: S2 P3
F: S3

This means that the answer choice cannot be (E) because it is possible that three patches can be made on 1/5 days.

Can three batches of cookies be made on W?

Yes, we can see this based on Option 2 that we've outlined above. If S1 is made on W, then three batches of cookies can be made on W.

M: O1
T:
W: O2 P1 S1
R: S2 P2 O3
F: S3 P3

Therefore, we can eliminate answer choice (D) because it is possible that three batches of cookies can be made on either T or W.

We can also eliminate answer choice (C) because we can see from the above diagram that R could also have three batches of cookies.

Finally, we can eliminate answer choice (B) because there is nothing to stop us from moving O3 to F.

Therefore, it is possible that three batches of cookies are made on T, W, R, or F. This means that only one day (M) is such that at most two batches of cookies could be made on that day.

Hope this helps! Please let us know if you have any further questions.