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Juana mailed at least one letter on each day last week. Was the total 07 Aug 2023, 10:00
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Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.
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Juana mailed at least one letter on each day last week. Was the total 14 Nov 2023, 10:29
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Bunuel
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.

The number of letters was positive on each day.

We need to answer the question:

Is ℓ₁ + ℓ₂ + ℓ₃ + ℓ₄ + ℓ₅ + ℓ₆ + ℓ₇ > 27 ?

Statement One Alone:

=> Juana mailed fewer than 8 letters on each of the 7 days.

Let’s imagine the daily numbers of letters in chronological order.

If {7, 7, 7, 7, 7, 7, 7}, then the answer is Yes.

Whereas, if {1, 1, 1, 1, 1, 1, 1}, then the answer is No.

Statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

=> Juana mailed a different number of letters on any two of the 7 days.

If {1, 2, 3, 4, 5, 6, 7}, then the sum is 28, and the answer is Yes. Since 28 is the least possible sum for different numbers of letters, we have a definite Yes answer.

Statement two is sufficient.

Answer: B

Could you explain more regarding the meaning of "on any two of the 7 days" in Statement (2)
Show more

Ittiphatkluayy Matsu12

The phrase "different number of letters on any two of the 7 days" means that Juana did not mail the same number of letters on any 2 days OR the number of letters she mailed in 7 days were all different

If she mails \(ℓ₁, ℓ₂, ℓ₃, ℓ₄, ℓ₅, ℓ₆\) and \(ℓ₇\) letters on seven days respectively, then \(ℓ₁≠ ℓ₂≠ ℓ₃≠ ℓ₄≠ ℓ₅≠ ℓ₆ ≠ ℓ₇\)

Hope this helps :)
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Juana mailed at least one letter on each day last week. Was the total 07 Aug 2023, 10:34
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Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
Jauna can mail 1 letter each on 7 days, sum = 7
Jauna can mail 7 letters each on 7 days, sum = 49
so not sufficient.
(2) Juana mailed a different number of letters on any two of the 7 days.
Jauna has mailed at least one letter (given)
so minimum number of letters Jauna can sent = 1+2+3+4+5+6+7 =28 which is greater than 27, hence sufficient.

So B should be the Answer. IMO
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Juana mailed at least one letter on each day last week. Was the total 31 Aug 2023, 07:44
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Bunuel
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.
Show more

The number of letters was positive on each day.

We need to answer the question:

Is ℓ₁ + ℓ₂ + ℓ₃ + ℓ₄ + ℓ₅ + ℓ₆ + ℓ₇ > 27 ?

Statement One Alone:

=> Juana mailed fewer than 8 letters on each of the 7 days.

Let’s imagine the daily numbers of letters in chronological order.

If {7, 7, 7, 7, 7, 7, 7}, then the answer is Yes.

Whereas, if {1, 1, 1, 1, 1, 1, 1}, then the answer is No.

Statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

=> Juana mailed a different number of letters on any two of the 7 days.

If {1, 2, 3, 4, 5, 6, 7}, then the sum is 28, and the answer is Yes. Since 28 is the least possible sum for different numbers of letters, we have a definite Yes answer.

Statement two is sufficient.

Answer: B
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Juana mailed at least one letter on each day last week. Was the total 24 Sep 2023, 18:01
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Bunuel
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.

The number of letters was positive on each day.

We need to answer the question:

Is ℓ₁ + ℓ₂ + ℓ₃ + ℓ₄ + ℓ₅ + ℓ₆ + ℓ₇ > 27 ?

Statement One Alone:

=> Juana mailed fewer than 8 letters on each of the 7 days.

Let’s imagine the daily numbers of letters in chronological order.

If {7, 7, 7, 7, 7, 7, 7}, then the answer is Yes.

Whereas, if {1, 1, 1, 1, 1, 1, 1}, then the answer is No.

Statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

=> Juana mailed a different number of letters on any two of the 7 days.

If {1, 2, 3, 4, 5, 6, 7}, then the sum is 28, and the answer is Yes. Since 28 is the least possible sum for different numbers of letters, we have a definite Yes answer.

Statement two is sufficient.

Answer: B
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Could you explain more regarding the meaning of "on any two of the 7 days" in Statement (2)
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Juana mailed at least one letter on each day last week. Was the total 27 Jan 2024, 01:25
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What if I say
"on any three of the 7 days"
Does this has any meaning?
or just the same as
"on any two of the 7 days"?

Posted from my mobile device
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Juana mailed at least one letter on each day last week. Was the total 31 Mar 2024, 21:31
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sanebeyondone
What if I say
"on any three of the 7 days"
Does this has any meaning?
or just the same as
"on any two of the 7 days"?

Posted from my mobile device
Show more
sanebeyondone ­No, "a different number on any three of the 7 days" has no meaning, and wouldn't be used on the GMAT or normal English.

(2) Juana mailed a different number of letters on any two of the 7 days.

@Matsu12  "a different number on ANY two ____" means that none of the numbers are the same. This has the same meaning as the wording we may see in other problems, "7 different integers are selected".

Therefore, the minimum possible sum is the evenly spaced set, 1+2+3+4+5+6+7, since the stem says she mailed "at least one letter on each day".
The fastest way to calculate a sum of consecutive integers is S = AN. 
For an evenly spaced set, A = Average = Median = ((First + Last) / 2)
N = the number of numbers
S = AN = 4*7 = 28, so statement (2) is sufficient.

 ­
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Juana mailed at least one letter on each day last week. Was the total 08 Apr 2024, 22:31
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Bunuel
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.
Show more
­
Total letters mailed is more than 7 (at least 1 letter each day)
Question: Is Total > 27 ?

(1) Juana mailed fewer than 8 letters on each of the 7 days.

This means she mailed at most 7 letters each day. So max she mailed 49 letters total. We don't know whether total was less or more than 27.
Not Sufficient

(2) Juana mailed a different number of letters on any two of the 7 days.

So minimum number of letters she mailed is 1 + 2 + 3 +... + 7
Sum of n consecutive integers starting from 1 is given by \(\frac{n(n+1)}{2} = \frac{7*8}{2} = 28\)
So she mailed at least 28 letters. Sufficient to say that Yes, total is more than 27. 

Answer (B)
 
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Juana mailed at least one letter on each day last week. Was the total 13 Apr 2024, 08:40
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KarishmaB

Bunuel
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.
­
Total letters mailed is more than 7 (at least 1 letter each day)
Question: Is Total > 27 ?

(1) Juana mailed fewer than 8 letters on each of the 7 days.

This means she mailed at most 7 letters each day. So max she mailed 49 letters total. We don't know whether total was less or more than 27.
Not Sufficient

(2) Juana mailed a different number of letters on any two of the 7 days.

So minimum number of letters she mailed is 1 + 2 + 3 +... + 7
Sum of n consecutive integers starting from 1 is given by \(\frac{n(n+1)}{2} = \frac{7*8}{2} = 28\)
So she mailed at least 28 letters. Sufficient to say that Yes, total is more than 27. 

Answer (B)

 
Show more
­Statement (2) - Any two of the 7 days could mean same number of letters for 5 days and different numbers for the rest of the days. Why is it different for all the days?
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Juana mailed at least one letter on each day last week. Was the total 13 Apr 2024, 08:51
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Spoorthii

KarishmaB

Bunuel
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.
­
Total letters mailed is more than 7 (at least 1 letter each day)
Question: Is Total > 27 ?

(1) Juana mailed fewer than 8 letters on each of the 7 days.

This means she mailed at most 7 letters each day. So max she mailed 49 letters total. We don't know whether total was less or more than 27.
Not Sufficient

(2) Juana mailed a different number of letters on any two of the 7 days.

So minimum number of letters she mailed is 1 + 2 + 3 +... + 7
Sum of n consecutive integers starting from 1 is given by \(\frac{n(n+1)}{2} = \frac{7*8}{2} = 28\)
So she mailed at least 28 letters. Sufficient to say that Yes, total is more than 27. 

Answer (B)



 
­Statement (2) - Any two of the 7 days could mean same number of letters for 5 days and different numbers for the rest of the days. Why is it different for all the days?
Show more
"­mailed a diferent number of letters on any two of the 7 days" means if you pick ANY 2 days of the 7, the number of letters will be different. Hence number of letters has to be different on all days. 

"mailed a diferent number of letters on two of the days" means that on some specific 2 days the number of letters was different. ­We don't know about other days.
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Juana mailed at least one letter on each day last week. Was the total 08 Jun 2024, 11:02
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any two of seven days means ?
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Juana mailed at least one letter on each day last week. Was the total 08 Jun 2024, 11:10
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Aslamkhanshovon
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.

any two of seven days means ?
Show more
­The phrase "any two of the seven days" means that Juana mailed a different number of letters on each day, so no two days had the same count of mailed letters.
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Juana mailed at least one letter on each day last week. Was the total 03 Apr 2025, 04:15
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Bunuel
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.
Show more

My question here is why is (2) alone SUFF? Most responses focus on it being the least possible, but where in the question does it ask that?

Using (2):

(1,2,3,4,5,6,7,8) fulfills condition 2 and results in more than 27 letters sent; total is 28.
(2,3,4,5,6,7,8,9) fulfills condition 2 as well and results in more than 27 letters sent; total is 44

Here we get two different solutions which means two should not be sufficient!

Using both we can limit ourselves to the first solution and only get one answer for the total number of letters.
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Juana mailed at least one letter on each day last week. Was the total 24 Jun 2025, 02:21
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It's the "different number of letters on any TWO of the 7 days" that is highly confusing to me. This was how I saw it:
1+2+3+4+5+6+7 = 28, YES because 28>27
1+2+3+4+5+3+2 = 20, NO, because 20>27 and it satisfies the fact that she mailed at least 1 letter.
Why are we assuming that the number of letters are different on each day for the other 5 days? What if Juana repeated the number of letters on a particular day. Nowhere does it say that Juana mailed different letters on each day.
This is very confusing, because some DS questions involve considering all possibilities in order to answer the question, but in this case we are just assuming that she is mailing a different number of letters on each day. I'm open to all explanations. Thanks :)

JeffTargetTestPrep
Bunuel
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.

The number of letters was positive on each day.

We need to answer the question:

Is l1 + l2 + l3 + l4 + l5 + l6 + l7 > 27 ?

Statement One Alone:

=> Juana mailed fewer than 8 letters on each of the 7 days.

Let’s imagine the daily numbers of letters in chronological order.

If {7, 7, 7, 7, 7, 7, 7}, then the answer is Yes.

Whereas, if {1, 1, 1, 1, 1, 1, 1}, then the answer is No.

Statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

=> Juana mailed a different number of letters on any two of the 7 days.

If {1, 2, 3, 4, 5, 6, 7}, then the sum is 28, and the answer is Yes. Since 28 is the least possible sum for different numbers of letters, we have a definite Yes answer.

Statement two is sufficient.

Answer: B
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Juana mailed at least one letter on each day last week. Was the total 24 Jun 2025, 06:16
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ach12345
Bunuel
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.

My question here is why is (2) alone SUFF? Most responses focus on it being the least possible, but where in the question does it ask that?

Using (2):

(1,2,3,4,5,6,7,8) fulfills condition 2 and results in more than 27 letters sent; total is 28.
(2,3,4,5,6,7,8,9) fulfills condition 2 as well and results in more than 27 letters sent; total is 44

Here we get two different solutions which means two should not be sufficient!

Using both we can limit ourselves to the first solution and only get one answer for the total number of letters.
Show more

You're missing the point. We are not asked to find the exact number of letters. The question is whether the total is greater than 27. Statement (2) guarantees that the totals possible are 28 or more, so that's a definite Yes. That makes it sufficient.
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Juana mailed at least one letter on each day last week. Was the total 24 Jun 2025, 06:18
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sahilsehdev97
It's the "different number of letters on any TWO of the 7 days" that is highly confusing to me. This was how I saw it:
1+2+3+4+5+6+7 = 28, YES because 28>27
1+2+3+4+5+3+2 = 20, NO, because 20>27 and it satisfies the fact that she mailed at least 1 letter.
Why are we assuming that the number of letters are different on each day for the other 5 days? What if Juana repeated the number of letters on a particular day. Nowhere does it say that Juana mailed different letters on each day.
This is very confusing, because some DS questions involve considering all possibilities in order to answer the question, but in this case we are just assuming that she is mailing a different number of letters on each day. I'm open to all explanations. Thanks :)
Show more

Your doubt is addressed above, but here it is again. You're misunderstanding the wording. Statement (2) clearly says Juana mailed a different number of letters on any two of the 7 days, which means no two days had the same number. So repetition is not allowed.

That forces the 7 numbers to all be distinct, and since she mailed at least one each day, the smallest possible total is 1+2+3+4+5+6+7 = 28, which is greater than 27. So the answer is always Yes, making the statement sufficient.
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Juana mailed at least one letter on each day last week. Was the total 02 Aug 2025, 13:16
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It says different in ANY two of the days (NOT any 2 Consecutive days) . So randomly if you pick 2 days, they shouldn't have same no of letters
sahilsehdev97
It's the "different number of letters on any TWO of the 7 days" that is highly confusing to me. This was how I saw it:
1+2+3+4+5+6+7 = 28, YES because 28>27
1+2+3+4+5+3+2 = 20, NO, because 20>27 and it satisfies the fact that she mailed at least 1 letter.
Why are we assuming that the number of letters are different on each day for the other 5 days? What if Juana repeated the number of letters on a particular day. Nowhere does it say that Juana mailed different letters on each day.
This is very confusing, because some DS questions involve considering all possibilities in order to answer the question, but in this case we are just assuming that she is mailing a different number of letters on each day. I'm open to all explanations. Thanks :)

JeffTargetTestPrep
Bunuel
Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.

The number of letters was positive on each day.

We need to answer the question:

Is l1 + l2 + l3 + l4 + l5 + l6 + l7 > 27 ?

Statement One Alone:

=> Juana mailed fewer than 8 letters on each of the 7 days.

Let’s imagine the daily numbers of letters in chronological order.

If {7, 7, 7, 7, 7, 7, 7}, then the answer is Yes.

Whereas, if {1, 1, 1, 1, 1, 1, 1}, then the answer is No.

Statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

=> Juana mailed a different number of letters on any two of the 7 days.

If {1, 2, 3, 4, 5, 6, 7}, then the sum is 28, and the answer is Yes. Since 28 is the least possible sum for different numbers of letters, we have a definite Yes answer.

Statement two is sufficient.

Answer: B
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Juana mailed at least one letter on each day last week. Was the total 24 Aug 2026, 04:38
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what if she sent one letter one those any two days or what if she sent 3,4,5 we dont know it either here so does that not make it same like statement 1 which was not sufficient like she could have sent max how much min how much we dont know so how its sufficient ?
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Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.
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Juana mailed at least one letter on each day last week. Was the total 24 Aug 2026, 04:59
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what if she sent one letter one those any two days or what if she sent 3,4,5 we dont know it either here so does that not make it same like statement 1 which was not sufficient like she could have sent max how much min how much we dont know so how its sufficient ?

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Statement (2):

“Juana mailed a different number of letters on any two of the 7 days.”

means:

Pick any two days you want, and the numbers of letters mailed on those two days must be different.

So she cannot have:

Monday = 1 and Tuesday = 1

or Wednesday = 4 and Friday = 4.

All 7 daily numbers must be different. For example: 1, 2, 3, 4, 5, 6, 7 is allowed.
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Juana mailed at least one letter on each day last week. Was the total 28 Aug 2026, 02:23
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Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?

(1) Juana mailed fewer than 8 letters on each of the 7 days.
(2) Juana mailed a different number of letters on any two of the 7 days.

This is a really good DS problem that shows a lot of useful things. First, pay attention to what you write on your paper. I strongly advise making the AD / BCE grid a habit, as it helps us structure our process, eliminate answers, and avoid silly errors. Second, absolutely do not use subscript ( l1,l2, etc.) on your paper - when writing in a hurry you'll misread things and make mistakes.

The top strategy for DS problems is Testing Cases: if you can find 2 cases that agree with a statement and give different answers to the question, then that statement is insufficient. Really digest and understand this, as it goes to the heart of what 'sufficient' and 'not sufficient' mean. It's worth practicing this strategy so that you're fluent with the logic; when numbers are relatively simple (such as in this problem), you should definitely consider Testing Cases.

For statement 1, Juana could have mailed 1 letter on each day (7 letters in total) or she could have mailed 7 letters on each day (49 letters in total). This proves that statement 1 is not sufficient - eliminate answers A and D. Notice how picking extreme cases, i.e. trying to maximise or minimise the number of letters she sent, is a good habit. Picking simple cases is also important to save time. For an advanced student, it's probably not necessary to write anything on your paper for statement 1; for a less experienced student, writing out the numbers may help keep things clear.

For statement 2, pick more cases. To start, we don't need to choose specific numbers: we could think "maybe she sent really big numbers of letters, like 7 different numbers over a hundred each. That means that she definitely sent more than 27 letters". If we know the strategy of Testing Cases well, then we'll realise that now we need to try to find a scenario that agrees with statement 2 in which she sent less than 27 letters. Go for the minimum: she sent 1 letter on Monday, 2 on Tuesday, etc. So she sent 1+2+3+4+5+6+7 letters in total. If we've done our homework for evenly-spaced sets, we'll know that the sum of these numbers is 4x7 = 28, but adding the numbers up one-by-one is also fine. Check over the logic: according to statement 2, the minimum number of letters that Juana sent is more than 27, so statement 2 gives enough information to know that she sent more than 27 letters. Statement 2 alone is sufficient, answer B.
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