12 Days of Christmas GMAT Competition - Day 10: A library has both

Question

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

A library has both fiction and nonfiction books. Of the fiction books, 4/5 are hardcover, and 3/4 of these hardcover books are authored by local writers. If the number of fiction books is three times the number of nonfiction books, what is the least total number of books in the library?

A. 15
B. 20
C. 40
D. 60
E. 80

F=3N
F+N= 4N+ Total is a multiple of 4 so reject A

Lets test 4N=20, then N=5, then F=15 and 3/5*15=9 is an integer hence it is possible

Ans B

A. 15
B. 20
C. 40
D. 60
E. 80

Bunuel · Accepted Answer

GMAT Club's Official Explanation:

Assuming the number of fiction books is F, then F * 4/5 * 3/4 = F * 3/5 must be an integer. This implies that F must be a multiple of 5. Additionally, since the number of fiction books is three times the number of nonfiction books, F must also be a multiple of 3. The least common multiple of 5 and 3 is 15, so the least number of fiction books is 15. This makes the least number of nonfiction books 5, and the least total number of books is 15 + 5 = 20.

Answer: B.

siddhantvarma · Answer

Let a = fiction, b = non-fiction
Of the fiction books, 4/5 are hardcover => 4/5a 
and 3/4 of  these  hardcover books are authored by local writers => 3/4 * 4/5a => 3/5a

If the number of fiction books is three times the number of nonfiction books => a = 3*b
We need to find the minimum value of total number of books (a+b)
To minimise a+b, we'll minimise a
We know both a and b are integers and 3/5a should also be an integer => a should be a multiple of 5
But since a=3*b => b = a/3 => a must also be a multiple of 3

Smallest number multiple of both 3 and 5 = 15 => a is 15
If a is 15, b = a/3 => 5
Hence minimum value is 15+5 =  B. 20

hr1212 · Answer

Let non fiction books be x then fiction books would be 3x.

Hardcover books = 3x * 4/5 = 12x/5
 Hardcover books written by local writers = (12x/5)*(3/4) = 9x/5

For this number to be an integer, x should be a multiple of 5.

Let's try assuming minimum value for x that is 5.

Then,

Non fiction books = 5
Fiction books = 15
Hardcover books = 12
 Hardcover books written by local writers = 9

Satisfies all the given constraints, hence min total = 5 + 15 = 20

Answer: B

Kinshook · Answer

A library has both fiction and nonfiction books. Of the fiction books, 4/5 are hardcover, and 3/4 of these hardcover books are authored by local writers.

If the number of fiction books is three times the number of nonfiction books, what is the least total number of books in the library?

Hardcover books = 4/5 * Fiction books
Books authored by local writers = 3/4 * hardcover books = 3/5 * Fiction books
Fiction Books = 3*Nonfiction books

Let Fiction books = 5k
Hardcover books = 4k
Books authorised by local writers = 3k 
Nonfiction books = 5k/3; Minimum k = 3
Fiction books = 15
Nonfiction books = 5
The least number of total books = Fiction books + Nonfiction books = 15 + 5 = 20

IMO B

Oppenheimer1945 · Answer

F,NF
F=15, NF=5

Hardcover Fiction=(4/5)(15)=12, (3/4)(12)=9 are local authored

Hence 20 is min
B)

Krunaal · Answer

No. of fiction books = 3 * No. of non-fiction books

Total no. of books is divisible by 4  as no. of fiction books + no. of non-fictions books = 4 * no. of non-fiction books.

Thus, (A) 15 can't be the answer.

Testing for (B) 20 => we get 15 fiction books, in which 12 are hard-covered of which 9 are authored by local writers, and 5 non-fiction books. This satisfies all the conditions.

Answer B.

TDTGiang · Answer

As per given, fiction is 3 times non fiction.So:
Non fiction: x 
=> Fiction: 3x
Total: 4x

Hardcover authorized by local = 4/5 * 3/4 * 3x = 9x/5 
=> x is a multiple of 5
To make this 9x/5 smallest, x must be the smallest multiple of 5 (x must be different than 0 because otherwise the total numbers of books would be 0) => x = 5 
=> Total number of books = 4x = 20 
Matching again with the options, we can see that 20 give the smallest integer numbers in all categories (hardcover/hardcover authorized by local writers) 
=> IMO, B

Yaswanth9696 · Answer

Let x be the no of non fiction books, that makes Fiction books as 3x, as there is an information fiction books is three times the number of nonfiction books.

Total = F+N.F---> x+3x= 4x= total. The total no of books must be a multiple of 4. Eliminate A

Start with least value- 20
Total=4x=20 --> x=5
F=15 and N.F=5
4/5 of Fiction books are hardcover---> (4/5)(15)==12
and 3/4 of these Hardcover books are authored by local writers.. (3/4)(12)---9 are local authored and 3 are others. Every value has become an integer and thus Option B is the answer.

AnuK2222 · Answer

IMO answer is B 
no. of fiction must be multiple of 5 for no. of fiction hardcover books to be integer
smallest possible value of fiction books is 15 and 5 for non fiction
no of fiction hardcover local author= (4/5)*(3/4)*15=9
total books 15+5= 20

nishantswaft · Answer

The stem tells us that;  
 
  A library has both fiction and nonfiction books   
  Of the fiction books, 4/5 are hardcover   
  3/4 of these hardcover books are authored by local writers.   
  the number of fiction books is three times the number of nonfiction books,

And we have to select;  
 
  the least total number of books in the library

Let the number of nonfiction books be x
Since the number of fiction books is three times the number of nonfiction books, the number of fiction books is 3x.

4/5 of the fiction books are hardcover. Thus, the number of hardcover fiction books is:4/5*3x=12x/5

3/4 of the hardcover fiction books are authored by local writers. Thus, the number of hardcover fiction books by local writers is:3/4*12x/5=36x/20=9x/5

For all the calculations to be valid, x must make 3x, 12x/5, and 9x/5 integers. 
I.e. x must be a multiple of 5

The total number of books is the sum of fiction and nonfiction books:
Total books=3x+x=4x.
Since x must be a multiple of 5, the smallest value for x is 5. Substituting x=5
Total books=4x=4⋅5=20.

the total number of books is 20

Final Answer:
 B. 20

Navya442001 · Answer

We are tasked with determining the  least total number of books  in the library, given the following information:
 
 The number of  fiction books  is three times the number of  nonfiction books . 
 4/5 of the fiction books are hardcover. 
 3/4 of these hardcover fiction books are authored by local writers.  
The total number of books in the library must be an integer. Let's solve this step-by-step.

Define Variables:
Let:
 
 N = the number of nonfiction books, 
 F = the number of fiction books.  
From the first condition:
F=3N

Hardcover Fiction Books:
The number of hardcover fiction books is:
(4/5)F

Hardcover Fiction Books by Local Writers:
The number of hardcover fiction books authored by local writers is:
(3/4)×(4/5)F=(12/20)F=(3/5)F

Total Number of Books:
The total number of books in the library is:
T=F+N
Substitute F=3N:
T=3N+N=4N.

Least Total Number of Books:
Since the fractions (4/5)F (hardcover fiction books) and (3/5)F (local writer hardcover fiction books) must represent whole numbers, F must be divisible by 5. Additionally, F=3N, so F must also be divisible by 3.
The least common multiple of 5 and 3 is 15. Therefore:
F=15,N=(F/3)=5
The minimum total number of books is:
T=F+N=15+5=20

Elite097 · Answer

A library has both fiction and nonfiction books. Of the fiction books, 4/5 are hardcover, and 3/4 of these hardcover books are authored by local writers. If the number of fiction books is three times the number of nonfiction books, what is the least total number of books in the library?

A. 15
B. 20
C. 40
D. 60
E. 80

F=3N
F+N= 4N+ Total is a multiple of 4 so reject A

Lets test 4N=20, then N=5, then F=15 and 3/5*15=9 is an integer hence it is possible

Ans B

A. 15
B. 20
C. 40
D. 60
E. 80

F=3N
F+N= 4N+ Total is a multiple of 4 so reject A

Lets test 4N=20, then N=5, then F=15 and 3/5*15=9 is an integer hence it is possible

Ans B

SunnyPoddar · Answer

If x = NF books
So, F Books become 3x.

Solving for F

Total = 3x
HC F books = 4/5 * 3x
Hard cover book by local writers = (3/4) * 4/5 * 3x = 9x/5

for this fraction to be an integer x must be multiple of 5.

Let us substitute the value of x by choosing the smallest multiple of 5 first. smallest multiple is 5.

Total NF books = 5
F books = 5*3 = 15.

total number of books = 20

On further substituting the value of x we get 
 The number of hardcover fiction books is 4/5×15=12. 
 The number of hardcover fiction books by local writers is 3/4×12=9, which is an integer, confirming that our choice of x=5 Is correct.

HansikaSachdeva · Answer

Nonfiction books = x
Fiction books = 3x

Hardcover fiction books = 4/5 * 3x = 12x/5
Hardcover fiction books by local authors = 3/4 * 12x/5 = 9x/5

Since the number of books must be integers, x has to be a multiple of 5

Total number of books = x + 3x = 4x

For the least x, 
Let it be 5 
Nonfiction books = 5
Fiction books = 3 * 5 = 15

Hardcover fiction books = 4/5 * 15 = 12 
Hardcover fiction books by local authors = 3/4 * 12 = 9

Therefore, the total number of books is 5 + 15 = 20

sushanth21 · Answer

B

Fiction books are thrice of the numebr of non fictional books

Let F be the number of fiction books
 Hardcover fiction books are 4/5 of F
 Local Writer Hardcover Fiction books are 3/4 of Hardcover fiction books
Let n be noumber of non fiction books

F = 3N

Hardcover fiction books = 4/5 * F
Local writer hardcover fiction books = 3/4 * 4/5 * F

Total books = F + N => 3N + N = 4N
Since F (and therefore N) must allow all the fractions in the problem to resolve to integers, F must be a multiple of 5 (to accommodate 4/5) and a multiple of 5 again for 3/5. Therefore, F must be a multiple of the least common multiple (LCM) of 5, which is 5. Since F=3N, N must also be a multiple of 5.

The smallest multiple of 5 for N is 5. Then;
F= 3N = 3*15 = 15

Total books is 4N = 20

Prakhar9802 · Answer

Hardcover(H) 
 not hardcover

local author(L) 
 (3/4)*(4F/5) = 3F/5

not local author 
 F/5

4F/5 
 F/5 
 Fiction (F)

Let's first make the table for fictional books hardcover.

From the above table, we can see that F is definitely a multiple of 5.

Fictional (F) 
 non fictional

Hardcover(H)  
 4F/5

not hardcover 
 F/5

F 
 T-F 
 Total (T)

It is also given that,

F = 3(T-F)
F = 3T - 3F
4F = 3T
T = 4F/3

From this we can see that F is also a multiple of 3.

So, F is a multiple of 5 and 3 both, that means F is a multiple of 15.

If F is 15 (lowest multiple of 15) then,

T = 4*15/3
T = 20.

Therefore, FINAL ANSWER - Option B

depjsr31 · Answer

Let the number of fiction books be x and non-fiction be y.

Hardcover fiction books =  \frac{4}{5}  x

Hardcover fiction books by Local Writers =  \frac{3}{4}  *  \frac{4}{5}  * x =  \frac{3}{5}  * x

Also given, x= 3y

To get the total # of books in the library, it will x + y =  \frac{4}{3}  * x

For the # of all the books to be integer, we can see that

1. For  "Hardcover fiction books by Local Writers"  to be integer, it has to be multiple of 5 (  \frac{3}{5}  * x)

2. For  total # of books in the library,  it has to be multiple of 3 (  \frac{4}{3}  * x)

So, for getting the minimum count of books, fiction book has to be a multiple of 15.
Considering 15 as the minimum number of fiction books, we get # of non-fiction books to be 5.

Total books = 15+5 = 20

The answer is (B)

jkkamau · Answer

Ration of fiction to non fiction is 3:1. Looking at the other information, we just need a number that is divisible by both 5 and 4 which are the denominators of the fractions derived from the equations

That number is the Lcm of 4 and 5 which is 20
B

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12 Days of Christmas GMAT Competition - Day 10: A library has both

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	Hardcover(H)	not hardcover
local author(L)	(3/4)*(4F/5) = 3F/5
not local author	F/5
	4F/5	F/5	Fiction (F)

	Fictional (F)	non fictional
Hardcover(H)	4F/5
not hardcover	F/5
	F	T-F	Total (T)

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