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

1 2 3 4 5

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If more than 50 pages are on Lower Shelf, then is T3 on lower shelf?

Multiply all fraction by 60
T1= 11
T2=48
T3=14
T4=18
T5 =9
Total=100

(1) T2 and T4 have been placed on the lower shelf.
T2+T4=48+18=66
More than half is already on Lower Shelf
Sufficient

(2) T1 and T5 have been placed on the upper shelf.
T1+T5= 11+9=20
Any from T2, T3, T4 could be on upper or lower shelfs
Insufficient

A

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Statement 1 alone is insufficient because we could have multiple combinations of books to receive a fraction of over 50% of total pages on the lower shelf, this is due to the fact that T2 and T4 together are 26/60, so we could add any off the other books which would be enough to fulfill the criteria.
Statement 2 alone is sufficient because we only have two possible combinations for the lower shelf to comprise more than 50% of the pages which is either T3 and T4 or T2, T3 and T4. This is due to the fact that T2 and T4 alone do not contain enough pages to fulfill the shelf requirement, thus we can answer the question sufficiently with yes T3 will be found on the lower shelf. Answer B.

Regards,
Lucas

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First convert all fractions to have the same denominator ie. 60.

[11][/60] [8][/60] [14][/60] [18][/60] [9][/60]

Statement (1)
Insufficient as we cannot find which other book accompanies T2 and T4 on the lower shelf. A and D are out.

Statement (2)
If T1 and T5 are on the upper shelf all the other books MUST be on the lower shelf. Sufficient. B is right.

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We are given:

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

We know more than 1/2 of the total pages(>30/60) are on the lower shelf. Is T3 on the lower shelf?

Statement (1): T2 and T4 are on the lower shelf

T2=8/60 and T4= 18/60
Total= 26/60
So to exceed 30/60 on the lower shelf, we need more than 4/60 extra. which T1, T3 and T5 any can compensate. Therefore, we can't tell with surety for T3.
Insufficient.

Statement (2): T1 and T5 are on the upper shelf

T1=11/60 and T5=9/60
Total=9/60

Then lower shelf = remaining books = T2 + T3 + T4
= 8/60 + 14/60 + 18/60 = 40/60
This is more than 30/60, so the condition is satisfied. T3 is on the lower shelf.
Sufficient.

B) Statement 2 alone is sufficient.

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Simplified data: 11/60, 8/60, 14/60, 18/60, 9/60
Upper shelf pages: <30
Lower shelf pages: >30

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

Upper shelf	-	-
Lower shelf	T2 + T4	26

The lower shelf can accommodate T1, T3, or T5 and exceed 30. Not sufficient

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

Upper shelf	T1 + T5	20
Lower shelf	-	-

We know that the upper shelf page total is < 30. If T3 is added to the upper shelf, the total becomes 34. So, sufficient

Correct answer is (B)

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Lets make denominator same for all first.
LCM is 60.

Then we have : 11/60 , 8/60 , 14/60 , 18/60 , 9/60

It basically means the no of pages are :
T1=11, T2=8 ..... and so on.

Total = 60 pgs.

Lower Shelf > 30 pages
Q asks if T3 is on lower shelf ?

Statement 1 :
T2 + T4 = 26 pgs
So T3 can be a part of lower shelf but that’s not necessary.
Thus Insufficient.

Statement 2 :
T1 + T5 = 20 on Upper.
But we know Upper < 30
So only T2 can now be on Upper.
This means T3 needs to be on Lower.
Thus Sufficient.

Answer is B

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Textbook	T1	T2	T3	T4	T5	Total
Fraction of the total pages	11/60	8/60	14/60	18/60	9/60	60/60

(1) T2 and T4 have been placed on the lower shelf.
T2 + T4 = 8/60 + 18/60 = 26/60
T1 + T2 + T4 = 37/60 > 1/2
T3 + T2 + T4 = 40/60 > 1/2
T5 + T2 + T4 = 46/60 > 1/2
T1 or T3 or T5 can be on the lower shelf
NOT SUFFICIENT

(2) T1 and T5 have been placed on the upper shelf.
T1 + T5 = 11/60 + 9/60 = 20/60
Since more than 1/2 of the total pages that comprise the five books are found on the lower shelf, only T2=8/60 can also be found on upper shelf
T3 is found on the lower shelf.
SUFFICIENT

IMO B

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Making all the values in the table comparable:

T1	T2	T3	T4	T5
$\frac{11}{60}$	$\frac{8}{60}$	$\frac{14}{60}$	$\frac{18}{60}$	$\frac{9}{60}$

Rule: more than 1/2 of the total pages that comprise the five books are found on the lower shelf

(1) Lower Shelf: T2+T4 = $\frac{8}{60} + \frac{18}{60} = \frac{26}{60}$
which is less than half, any textbook from the table or more than one textbook can be added to lower shelf to attain the rule.
There is no certainity for position of T3.
Insufficient.

(2)Upper Shelf: T1+T5 = $\frac{11}{60} + \frac{9}{60} = \frac{20}{60}$
To abide by the rule, total value of pages on Upper shelf should remain <$\frac{30}{60}$
Therefore, T3 cannot be on to upper shelf and is on the lower shelf.
Sufficient.

Answer: Option B

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From T1 to T5, 11/60, 2/15, 7/30, 3/10, 3/20
after base same, 11/60, 8/60, 14/60, 18/60, 9/60
(1) T2 and T4 have been placed on the lower shelf.
T2+T4 = 8/60 + 18/60 = 26/60< 1/2, we need one more book for lower shelf as it should be greater than 1/2
in this case we have 3 books, and all can satisfy the equation

Not sufficient

(2) T1 and T5 have been placed on the upper shelf.
T1+T5 = 11/60 + 9/60 = 20/60

we don't have any information how many should this shelf should have or what fraction of the total pages. Not sufficient

Combine, 2 in the lower shelf and 2 in upper shel,f and 1 pending, it will be in lower shelf and fraction should be >1/2

Sufficient

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Let’s take the total pages of all the text books combined be 60 pages.

The five textbooks are T1, T2, T3, T4, T5 and the number of pages are 11, 8, 14, 18 and 9 respectively.

The books are arranged in two shelf’s - Upper and Lower shelf. And the Lower shelf’s contain more than 30 pages.

Is T3 present in the Lower shelf ?

Statement 1:

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

We place T2 and T4 are placed on the lower shelf, the total pages are 8+18 = 26 pages.

Constraint of Lower shelf is pages > 30. So, there can be multiple combinations such as:

26 + T1 = 26+11 = 37

26 + T3 = 26+14 = 40

26 + T5 = 26 + 9 = 35

So, any of T1, T3, and T5 can belong to Lower shelf. Hence, Insufficient.

Statement 2:

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

T1 and T5 are placed in the upper shelf. T1 + T5 = 11+9 = 20 pages in the upper shelf.

The lower shelf, should have more than 30 pages. Out of T2, T3, and T4 - the combinations of T3 and T4 are 14+18 = 32 pages.

So, with and without T2, the number of pages of Lower shelf is more than 30 pages. And, T3 remains in the lower shelf.

Hence, Sufficient.

Option B

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10 Jul 2025, 11:34

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The fractions have been simplified
T1 -> 11/60
T2 -> 8/60
T3 -> 14/60
T4 -> 18/60
T5 -> 9/60

Let's not consider an statement
If T3 is in upper shelf
Let all be in lower shelf
Upper = 14
Lower = 46 (greater than half of total i.e. >30)
satisfies
If T3 is in lower, and let only T1 be in upper
Upper = 11
Lower = 49 (greater than half of total i.e. >30)
Satisfies

Let's only consider Statement 1
Lower -> T2+T4 = 26
if T3 is in Lower and T1,T5 in upper
Lower = 40 (greater than half of total i.e. >30)
Upper = 20
Satisfies
if T3 is in Upper and T1 is in lower
Lower = 26+11 = 37 (greater than half of total i.e. >30)
Upper = 14+9 = 23
satisfies

Let's consider statement 2
Upper -> T1+T5 = 20
if T3 goes to Upper it will become 20+14 = 34>half hence wont satisfy
Hence T3 must be in Lower
so Lower -> T3,T2,T4 or T3,T4
to be greater than half of total i.e. >30
Hence enough to satisfy

We can also observe that in case 2, T4 also must be in Lower

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More than 1/2 of total pages are on lower shelf
Question: is T3 on lower shelf?

Statement1:
(1) T2 and T4 have been placed on the lower shelf.
2/15 + 3/10 = 13/30 =26/60 < 1/2
So the third book is needed in lower shelf to fulfil criteria which can be any book among T1, T3 or T5
Insufficient info

Statement 2:
(2) T1 and T5 have been placed on the upper shelf.
11/60 + 3/20 = 20/60 < 1/2
=> on lower shelf we have 40/60 books which is greater than 1/2
So if T3 is on lower shelf, then criteria fulfils
if T3 is on upper shelf, then also criteria fulfils
So insufficient info

Combining both 1&2
T1 and T5 on upper shelf
T2 and T4 on lower shelf
T3 can now either go to upper or lower shelf
if T3 is on upper shelf,
total on upper shelf = 11/60 + 9/60 + 14/60 = 34/60 > 1/2
but on lower shelf < 1/2
so this violates the criteria and hence not possible.
T3 be on lower shelf, then
total on lower shelf = 8/60 + 18/60 + 14/60 = 40/60 > 1/2
So yes T3 is in lower shelf

So C is correct answer

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First, convert each fraction into a common denominator of 60 leaving us with: 11 +8+14+18+9/ 60

now we know that "more than half" = 30 pages or more and can move onto the statements

statement 1: tells us that 8+18 = 26 pages are on the lower shelf; however, any other book could be added to the lower shelf to sum to more than 30 pages, so this statement is insufficient.

statement 2: tells us that 11+9 = 20 pages are on the top shelf --> we can therefore deduce that T3 MUST be on the lower shelf because any of the remaining books would preclude the possibility of more than 1/2 the pages being on the lower shelf. --> sufficient Answer: B

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given that more than 1/2 of the pages are found on the lower shelf.
is T3 on the lower shelf=?
1: T2+T4-->lower shelf
2/15+3/10=13/30 which is less than half the fraction of all the pages
now adding any of the books to the lower shelf will make the fraction of pages greater than 1/2
so not sure which book is on the lower shelf NON SUFF
2: T1+T5-->upper shelf
11/60+3/20=20/60=1/3 which is also less than half the fraction of all the pages
however if we add T3 in the upper shelf the fraction changes to 17/30 > 1/2
that violates the very basic condition above in bold
hence SUFF to conclude that T3 cant be on the upper shelf SUFF
hence B

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First, let's multiply all fractions by 60, so we have easier calculations.

Textbook	T1	T2	T3	T4	T5
Ratio of pages	11	8	14	18	9

From the question, we know that on the lower shelf, the sum of the ratios will be more than 30.

Statement 1:
This statement tells us T2 and T4 are on the lower shelf. So, there are already 8+18=22 on the lower shelf. We know that the lower shelf should have more than 30, so we need to add a book with a rate of 9 or more pages. Since all the rest of the books (T1, T3, and T5) could be the answer, we can't say which one, or maybe which ones are also on the lower shelf.
Statement 1 is not enough.

Statement 2:
This statement tells us that T1 and T5 are on the upper shelf. So there is already 20 on the upper shelf. The only book we can add to the upper shelf and still keep the lower shelf more than 30 is T2. So, T3 should be on the lower shelf, whether T2 is up or down.
Statement 2 is enough.

So the answer is B.

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"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?"

First of all, lets equalize all denominators. LCM(60, 15, 30, 10, 20) = 60

Then, if we assume that 60 is also the total amount of pages, each book will have:

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

Rephrasing the question: if lower shelf contains more than 30 pages, is T3 there?

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

If T2 + T4 in Lower then Lower starts with 8 + 18 = 26

In this situation we cannot infer if T3 is there or not. It could achieve 40 with a set of { T2, T3, T4 } but could also achieve a page amount greater than 30 with any other book replacing the T3.
Is it sufficient to answer? No, it's not. Eliminate answer choices A and D.

(2) T1 and T5 have been placed on the upper shelf.
If T1 + T5 in Upper then Upper starts with 11 + 9 = 20

If T3 was placed in Upper, then Upper would have 20 + 14 pages -> it implies in lower shelf with less than 30 pages. So, to have lower shelf with more than 30 pages, we must have T3 there.
Is it sufficient to answer? Yes, it is. Eliminate answer choices C and E.

Answer = B Statement (2) alone is sufficient but Statement (1) alone is not.

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LCM (60,15,30,10,20) = 60

Number of pages in

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

Total pages = 60

Lower shelf has atleast 30 pages

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

Lower shelf = 8 + 18 = 26

T3 can be in the lower shelf or in the upper shelf.

The statement alone is not sufficient.

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

Upper shelf = 9 + 11 = 20

Now if T3 is in upper shelf the number of pages in the upper shelf will exceed 30. This is not possible.

Hence, T3 is in the lower shelf

Option B

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10 Jul 2025, 13:41

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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.

fraction of total pages per book :

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

Y/N Question : if lower shelf > 30/60 pages => T3 on lower shelf ?

Statement (1)
T2 and T4 on lower shelf => 26/60 < 30/60
=> so another book needs to be on the lower shelf T1, T3 or T5 => NS

Statement (2)
T1 and T5 on upper shelf = 20/30
If the lower shelf has to have > 30/60 pages => it has to have at least T3 and T4
=> Sufficient

Answer B.

Harika2024

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10 Jul 2025, 15:07

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Given data,
T1 = 11/60 , T2 = 2/15 , T3 = 7/30 , T4 = 3/10 , T5 = 3/20
Lets round up the denominator, based on LCM = 60

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

Given condition, 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 are on the lower shelf.

T2 and T4 on the lower shelf = 8/60 + 18/60 = 26/60 < 30/60, to meet given condition least one more book (T1, T3, or T5) must also be on the lower shelf.

if we consider T3 on the lower shelf along with T2 and T4 = 26/60 + 14/60 = 40/60 > 30/60, T3 on the lower shelf

if we consider T3 not on the lower shelf = T1 is also on the lower shelf with T2 and T4 = 26/60 + 11/60 = 37/60 > 30/60, T3 not on the lower shelf

Statement (1) is not sufficient

Statement (2): T1 and T5 are on the upper shelf.

T1 and T5 on the upper shelf = 11/60 + 9/60 = 20/60 < 30/60

if we consider T3 is on the upper shelf along with T1 and T5 = 20/60+ 14/60 = 34/60 > 30/60, the lower shelf has more than half

as T3 on the upper shelf makes the condition impossible, T3 cannot be on the upper shelf. So, If T3 cannot be on the upper shelf, it must be on the lower shelf.

Statement (2) is sufficient.

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10 Jul 2025, 15:08

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We know that more than 1/2 of the total pages are on the lower shelf. We have to check if T3 is on the lower shelf.

Total pages = 60
T1 = 11, T2 = 8, T3 = 14, T4 = 18, T5 = 9

Statement 1: T2 and T4 are on lower shelf
-> 8 + 18 = 26 pages
This is less than half. So we don’t know if T3 is on lower shelf
Insufficient

Statement 2: T1 and T5 are on upper shelf
-> 11 + 9 = 20
T3 can't be on upper shelf because then the upper shelf would contain more than half the pages. It has to be on lower shelf
Sufficient

Answer: B