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GMAT Club Olympics 2025 (Day 9): Five mathematics textbooks, T1, T2

1 2 3 4 5

muuss

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11 Jul 2025, 02:55

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1.T2 = 8/60
T4 = 18/60
total = 26/60 < 30/60
books left= T1(11/60), T3 (14/60), T5 (9/60)
T1=26+11=37>30
T2=26+14=40>30
T3=26+9=35>30
So any of the book works.
Insufficient
2.T1 = 11/60
T5 = 9/60
Total=20/60
lower-shelf=40/60 which is greaten than 30/60
Since T1 is on the upper shelf we are left with T2,T3,T4
T2=8/60,T4=18/60
T3 would be 40-26=14
Sufficient
IMO:B

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T1=11/60
T2=2/15=8/60
T3=7/30=14/60
T4=3/10=18/60
T5=3/20=9/60

lower shelf > 30/60

T3 in lower shelf?

(1)
T2+T4=24/60

If lower shelf = T2, T4, T3 (38/60) the answer is yes
If lower shelf = T2, T4, T1 (35/60) the answer is no

INSUFFICIENT

(2)
T1+T5=20/60

T3 must be in the lower shelf because the other two T2, T4 are 26/60 < 30/60
lower shelf can be T3, T4 (32/60) or T2, T3, T4 (40/60)

SUFFICIENT

IMO B

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Given,
Five mathematics textbooks, T1, T2, T3, T4, and T5, have been placed on two shelves, neither of which houses any other books.
The table displays the number of pages in each book as a fraction of the total number of pages in the five books together.

Textbook	T1	T2	T3	T4	T5
Fraction of the total pages in the five textbooks	11/60	2/15	7/30	3/10	3/20
The same fraction with common denominator	11/60	8/60	14/60	18/60	9/60

To find:
If more than 1/2 of the total pages that comprise the five books are found on the lower shelf, is T3 found on the lower shelf?
Statement 1:
T2 and T4 have been placed on the lower shelf.
T 2 + T 4 = 8/60 + 18/60
= 26/60 < 1/2
We have no information about any other book where it can be kept.
Not Sufficient
Statement 2:
T1 and T5 have been placed on the upper shelf.
T 1 + T 5
= 11/60 + 9/60
= 20 /60
Lower shelf = 40/60 >1/2
Book on lower shelf = T2, T3, T4
Sufficient

Ans: B

Bunuel

Five mathematics textbooks, T1, T2, T3, T4, and T5, have been placed on two shelves, neither of which houses any other books. The table displays the number of pages in each book as a fraction of the total number of pages in the five books together. If more than 1/2 of the total pages that comprise the five books are found on the lower shelf, is T3 found on the lower shelf?

(1) T2 and T4 have been placed on the lower shelf.
(2) T1 and T5 have been placed on the upper shelf.

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11 Jul 2025, 04:41

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Five mathematics textbooks, T1, T2, T3, T4, and T5
2 shelves with only these 5 books
The number of pages in each book is a fraction of the total number of pages in the five books together
More than 1/2 of the total pages that comprise the five books are found on the lower shelf

We know that each book is a fraction of the total pages, so lets bring them in order
T1 = 11/60 = 11/60
T2 = 2/15 = 8/60
T3 = 7/30 = 14/60
T4 = 3/10 = 18/60
T5 = 3/20 = 9/60

So, we need to find out if T3 is on the lower shelf, given that more than 1/2 of the total page count is on the lower shelf

Given the statements,
(1) T2 and T4 have been placed on the lower shelf.

=> Lower shelf = T2 + T4 = 8/60 + 18/60 = 26/60
We know that lower shelf has more than half the page count
So lower shelf > 30/60

To satisfy this,
T1, T3 and T5 any of them can be in lower shelf to satisfy the condition i.e,
T1 + Lower shelf = 11/60 + 26/60 = 37/60 > 30/60
T3 + Lower shelf = 14/60 + 26/60 = 40/60 > 30/60
T5 + Lower shelf = 9/60 + 26/60 = 35/60 > 30/60

We cant know if T3 is on the lower shelf,
Statement (1) is insufficient

(2) T1 and T5 have been placed on the upper shelf.

=> Upper shelf = T1 + T5 = 11/60 + 9/60 = 20/60

We know that lower shelf has more than half the pages (>30/60) and we have T2, T4 and T3 as the possibilities to be in the lower shelf
T2 = 8/60 , T3 = 14/60 , T4= 18/60

Only 2 combinations for the below shelf is possible to have more than half the pages (>30/60)

T2+T3+T4 = (8+14+18)/60 = 40/60 > 30/60
T3+T4 = (14+18)/60 = 32/60 > 30/60

In both these combinations T3 is in the lower shelf
=> Statement (2) is sufficient

(B) Statement (2) alone is sufficient, but Statement (1) is not

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11 Jul 2025, 04:46

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(1)
T2+T4=2/15+3/10=24/60 < 1/2
We need to add at least one more. This can include T3 or not.

Statement is insufficient

(2)
T1+T5=11/60+3/20=20/60
T3 cannot be in the upper shelf because the other books T2+T4=2/15+3/10=8/60+18/60=26/60 < 1/2

T3 is in the lower shelf

Statement is sufficient

The right answer is B

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11 Jul 2025, 04:53

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First lets make the computations easier by making the fractions have the same denominator.
T1 = 11/60
T2 = 8/60
T3 = 14/60
T4 = 18/60
T5 = 9/60

It is given that more than half the pages are in the lower shelf. Which means > 30/60 in the lower shelf.
We need to determine if T3 is in the lower shelf.

Statement 1:
T2 and T4 are in the lower shelf, which means 26/60 are in lower shelf. Given that more than 30 parts must be there in lower shelf, the third book can be any of the books and not necessarily T3. NOT SUFFICIENT.

Statement 2:
T1 and T5 are in the upper shelf. 20/60 are in the upper shelf. IF T3 is in the upper shelf, lower shelf will have less than half the total pages. Hence T3 must be in the lower shelf instead. SUFFICIENT

Answer is B.

Bunuel

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Bunuel

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Now,
Take a LCM for all fractions
=>
T1 =11/60
T2= 8/60
T3= 14/60
T4= 18/60
T5= 9/60

Lower shelf has more than 1/2= 30/60
Is T3 on lower shelf?

Statement 1:
T2 and T4 are on the lower shelf.
T2 and T4 give 26/60, but even if T5 is on the lower shelf, it satisfies the criteria.
So, T3 may be there or may not be there
Not Sufficient.

Statement 2:
T1 and T5 are on the upper shelf.
T1+T5= 29/60
=> T3 has to be on the lower shelf to fulfil the criteria.
Sufficient.

ANSWER: B

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11 Jul 2025, 05:20

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Bunuel

Shelves = 2
Books = T1, T2, T3, T4, and T5

The table displays the number of pages in each book as a fraction of the total number of pages in the five books together.
T1 = 11/60, T2 = 2/15, T3 = 7/30, T4 = 3/10, and T5=3/20

=> T1 = 11/60, T2 = 8/60, T3 = 14/60, T4 = 18/60, and T5=9/60

Shelf2 > 1/2 of all pages
Shelf2 > 30/60

is T3 found on the lower shelf?

(1) T2 and T4 have been placed on the lower shelf.
T2 + T4 = 8/60 + 18/60 = 26/60

T3 in lower shelf
T2 + T4 + T3 = 8/60 + 18/60 + 14/60 = 40/60 --> Satisfies

T3 not in lower shelf
T2 + T4 + T1 = 8/60 + 18/60 + 11/60 = 37/60 --> Satisfies

Insufficient

(2) T1 and T5 have been placed on the upper shelf.

Shelf 1
T1 + T5 = 11/60 + 9/60 = 20/60
Shelf 2
T2 + T4 + T3 = 8/60 + 18/60 + 14/60 = 40/60 --> Satisfies

Shelf 1
T1 + T5 + T2 = 11/60 + 9/60 + 8/60 = 28/60
Shelf 2
T4 + T3 = 18/60 + 14/60 = 32/60 --> Satisfies

Let's try
Shelf 1
T1 + T5 + T3 = 11/60 + 9/60 + 14/60 = 34/60
Shelf 2
T2 + T4 = 8/60 + 18/60 = 26/60 --> does not satisfy

Hence T3 has to be in shelf 2

Sufficient

Option B

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11 Jul 2025, 05:30

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We have to convert all the fractions in to same denominator for a better understanding.

T1,T2,T3,T4,T5 will be 11/60,8/60,14/60,18/60,9/60

statement 1: T2 and T4 on lower shelf. If T2 and T4 on lower shelf total no of pages=26. But it should be greater than 30. any of the remaining books can be on the lower shelf. So we cannot say if T3 is in lower shelf.

Statement 2:T1 and T5 are on upper shelf. Total number of pages=20. If T3 is kept in upper shelf then the remaining pages in the lower shelf will be less than 30. Therefore T3 should be on the lower shelf. Hence we can come to a conclusion.

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11 Jul 2025, 05:55

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Option B is the correct answer.

First lets understand the information mentioned in it and then lets solve for the answer.

So first the question starts by telling us that "Five mathematics textbooks, T1, T2, T3, T4, and T5, have been placed on two shelves, neither of which houses any other books. The table displays the number of pages in each book as a fraction of the total number of pages in the five books together". Then it tells us the Fraction of Total Pages in the five books, the question then tells us that if more than 1/2 of the total pages that comprise the five books are found on the lower shelf and concludes by asking us that "is T3 found on the lower shelf".

T1 = 11/60
T2 = 2/15 => (2/15)*(4/4) => 8/60
T3 = 7/30 => (7/30)*(2/2) => 14/60
T4 = 3/10 => (3/10)*(6/6) => 18/60
T5 = 3/20 => (3/20)*(3/3) => 9/60
Now all the Fractions have same denominator which will make the calculations easier for us while solving the question.

Lets check the statements now and see whether we can get our answer from them or not.

Statement 1: "T2 and T4 have been placed on the lower shelf". So as per this statement 26/60 fraction of pages are on the lower shelf and from the question we know that more than 1/2 of the total pages that comprise the five books are found on the lower shelf, as per these information we can say that there must be atleast one more book in the lower shelf. Now see after adding which of the remaining three books i.e. T1, T3 and T5 will meet the condition mentioned in the question. So if we assume T1, T2 and T4 are in the lower shelf then it would mean that (11+8+18)/60 i.e. 37/60 which is more that 1/2 of the total number of pages so this combination satisfies all the conditions. Now lets assume that T2, T3 and T4 are on the lower shelf, which will give us (8+14+18)/60 i.e. 40/60 which is also more that 1/2 of the total number of pages so this combination also satisfies all the conditions. So after seeing these to combinations we can conclude that we are getting multiple combinations that satisfies the question that's why this statement is Not Sufficient to answer the question.

Statement 2: "T1 and T5 have been placed on the upper shelf". So this statement tells us that 20/60 fraction of pages on the top shelf. Now lets see if there is any other book available with them on the top shelf or not which will satisfy the conditions mentioned in the question. So if we assume T1, T2 and T5 to be on the top shelf then it would mean that 28/60 fraction of pages are on the top shelf which will satisfy the condition and will also tell us that T3 & T4 are in the lower shelf. Now lets assume T1, T3 and T5 to be on the top shelf which will give us that 34/60 fraction of pages on the top shelf, which does not satisfy the condition mention in the question i.e. "more than 1/2 of the total pages that comprise the five books are found on the lower shelf" so this cannot be the case and the same will happen when we will take T1, T4 and T5 to be in the top shelf i.e. 38/60 in top shelf. So from this statement alone we can conclude that T3 is not on the top shelf which would mean that it is on the lower shelf. So this statement is Sufficient to answer the question.

After checking both the statements we can now Conclude that Only Statement 2 gives us the confirmed answer that's why Option B is our answer.

Bunuel

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11 Jul 2025, 06:10

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Simplify the fractions by finding the LCM and using it a base. We end up with
T1= 11/60
T2= 8/60
T3=14/60
T4=18/60
T5=9/60
S1 T2+T4= 26/60 which means T3 and T5 or T1 can be in lower shelf and meet the criteria hence insufficient
S2= T1+T5=20/60 which means T3 or T4 can be used to meet the criteria hence insufficient
S1+S2= Based on S1 Since T1 and T5 are already in the upper shelf only T3 remains to fill the required number of pages in the lower shelf
Both are sufficient C

Bunuel

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11 Jul 2025, 06:25

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Let the total pages of the 5 books be P
Given, lower shelf comprise of more than 1⁄2 P, which means upper shelf has < 1⁄2 P

Q: is T3 on lower shelf?

St1: T2 and T4 are on lower shelf
Which means, 2/15 + 3/10 = 4+9/30 = 13/30
But, 13/30 < 1⁄2 P, so there has to be one more book placed on the lower shelf
Case 1: If T1 is on lower shelf
13/30 + 11/60 = 26+11/60 = 37/60 P
37/60 P > 1⁄2 P
So, may or may not be placed on lower shelf
Case 2: If T3 is on lower shelf
13/30 + 7/30 = 20/30 P
20/30 P > 1⁄2 P
So, T3 is placed on lower shelf
Since, both the cases are possible, we cannot confirm the placement of T3 on lower shelf. Insufficient.

St2: T1 and T5 have been placed on the upper shelf.
Which means, 11/60 + 3/20 = 11+9/60 = 20/60
20/60 < 1⁄2 P This is true.
But what if we add T3 on upper shelf?
20/60 + 7/30 = 20+14/60 = 34/60
34/60 > 1⁄2 P which is not possible as the question explicitly mentions that 1⁄2 P is placed on lower shelf and upper shelf has < 1⁄2 P.
So, T3 cannot be placed on upper shelf.
St2 is sufficient.

Option B

Bunuel

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11 Jul 2025, 06:51

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Correct Answer: Yes T3 will found on lower shelf and we can answer this using statement II alone.

Given we have,
T1= 11/60
T2= 2/15
T3= 7/30
T4= 3/10
T5= 3/20

We take LCM of T1-T5,
= 11/60+2/15+7/30+3/10+3/20
= (11+8+14+18+9)/60
= 60/60= 1, hence we take total pages= 60

So lower shelf must have total of more than 30 pages.

Now let's take each statement:

(1) T2 and T4 have been placed on the lower shelf.
If we take this statement alone, we don't know whether T1, T3 and T5 are there in lower shelf or not.
T2= 8
T4= 18
T2 + T4= 26

Here there are chances we can include T1 or T3 or T5. So we cannot answer using this statement alone.

(2) T1 and T5 have been placed on the upper shelf.

If we take this statement alone, we can answer this let’s check,
T1= 11
T5= 9
T1 + T5= 20

Now we know that
T2= 8
T4= 18
T3= 14

As per given information lower shelf consist for more than half of total pages.
So lower shelf must consist of T3. Either we include T2 + T3 + T4 or T3 + T4

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11 Jul 2025, 07:09

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(1)
The only constraint is that more than 1/2 of the total pages are found on the lower shelf.
T2+T4<1/2
It is possible to add T3 to lower shelf or T1 or T5.

Insufficient

(2)
T1, T5 -> upper shelf
T2, T3, T4 -> lower or upper shelf

If T3 is in upper shelf, the only books in lower shelf T2+T4=26/60 < 1/2
It's impossible that T3 is in upper shelf.

Sufficient

Correct answer is B

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11 Jul 2025, 07:23

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Given:

Or we can also say:

	T1	T2	T3	T4	T5
Fraction of total pages in 5 textbooks	11/60	8/60	14/60	18/60	9/20

More than half of the total pages in lower shelf

We can also say lower shelf will have more than 30 pages.
Or upper shelf contains less than 30 pages

Question:
Is T3 found on the lower shelf?

Statement 1: T2 and T4 have been placed on the lower shelf.

T2=8 pages
T4=18 Pages
26 pages.

Lower shelf has more than half pages, so it can have T1 or T3 or T5 in lower shelf we dont know the exact number all we know is it is more than half.

This is not sufficient.

Statement 2: T1 and T5 have been placed on upper shelf

T1=11 Pages
T5=9 pages
Total=20 pages

Max pages it can have is 29
29-20=9 any book or combination of books have 9 or less pages can be placed here. or maybe none.

we are certain T3 and T4 can be placed here T2 can be

Yes, T3 will be placed in lower row.

This is sufficient

Answer B

Bunuel

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11 Jul 2025, 07:34

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(1)
lower shelf can be T1, T2, T4 or T2, T3, T4.

Statement (1) alone is insufficient.

(2)
T3 must in the lower shelf, otherwise T2 and T4 would be the only ones in lower shelf and their sum is 2/15+3/10=13/30 which is less than 1/2.

Statement (2) alone is sufficient.

Answer is B

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11 Jul 2025, 07:49

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To find: Is T3 on the lower shelf, given that more than 1⁄2 of total pages are there?
Total = 60/60 → More than 30/60 must be on lower shelf.

Statement (1): T2 (8/60) and T4 (18/60) are on the lower shelf
Total = 26/60 → Not enough
To exceed 30/60, need T3 (14/60) ⇒ So T3 must be on the lower shelf
Sufficient

Statement (2): T1 (11/60) and T5 (9/60) are on the upper shelf
Upper = 20/60 ⇒ Lower = 40/60
T2 + T4 = 26/60 → Need 14/60 more ⇒ T3 must be on lower shelf
Sufficient

So, correct answer option is D.

bebu24

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[1]

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[1]

11 Jul 2025, 07:58

Kudos

Bookmarks

Statement I:
T2 + T4 = 2/15 + 3/10 = 13/30

More than half of the pages must be on lower shelf
T2 + T4 + T3 = 13/30 + 7/30 = 2/3 > 1/2. T3 may be on lower shelf

T2 + T3 + T1 = 13/30 + 11/60 = 37/60 > 1/2 . T3 may not be on lower shelf

Insufficient.

Statement II:
T2, T4, T3 may be present on lower shelf

T2+ T4 = 13/30 < 1/2, so T3 must be present on Lower shelf

T2 + T4 + T3 = 13/30 + 7/30 = 2/3 > 1/2.

Sufficient

Therefore correct answer is B.

Bunuel

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Suyash1331

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Schools: IIMA '26 IIM IIM Calcutta '26 Judge '26 ISB '27 Said '27

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GMAT 1: 250 Q20 V34 Verified score

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11 Jul 2025, 08:00

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We can find the solution by placing the books given in first and second statement and find the answer by the pages remaining

iCheetaah

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11 Jul 2025, 09:36

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Hi Bunuel, I just wanted to check why I missed out on the Kudos on this explanation. Is it too wordy? I think I might have used larger fractions. Please let me know, and I will incorporate your suggestions into my other explanations

iCheetaah

Let's first convert all the fractions to a common denominator for easier calculations. We can use 60, since that is the LCM for all the denominators given:

T1 = 11/60
T2 = 8/60
T3 = 14/60
T4 = 18/60
T5 = 9/60

There are 2 shelves, upper and lower; and lower shelf contains >50% of all the pages of the 5 books. So, the lower shelf contains >1/2 or >30/60 pages

Statement 1:

This tells us that T2 and T4 are on the lower shelf.

Adding T2 and T4, we get, 26/60. If we were to put any other book doesn't matter which one (T1,T3, or T5) in the lower shelf, we will get >1/2 of all pages in that shelf.

So, insufficient.

Statement 2:

This tells us that T1 and T5 are on the upper shelf.

Let's see if we can put T3 on the upper shelf

The upper shelf with T1 and T5, already has 20/60 pages, if we put T3 there with them then we will have 34/60 pages in the upper shelf, which breaches our given statement. Since, lower shelf has to have >1/2 of all pages, we can say that T3 is on the lower shelf.
Sufficient.

Answer B.