Each of the 45 books on a shelf is written either in English or in Spa

Question

Each of the 45 books on a shelf is written either in English or in Spanish, and each of the books is either a hardcover book or a paperback. If a book is to be selected at random from the books on the shelf, is the probability less than 1/2 that the book selected will be a paperback written in Spanish?

(1) Of the books on the shelf, 30 are paperback.

(2) Of the books on the shelf, 15 are written in Spanish.

(1) Of the books on the shelf, 30 are paperback.

(2) Of the books on the shelf, 15 are written in Spanish.

ID: 700283

chetan2u · Accepted Answer

hi,
is overall prob given for paperback+spanish <1/2..
1) 30 out of 45 are paperbeck..
if all 30 are spanish too... prob>1/2..
if none of them is spanish.. prob=0<1/2..
no info ..insuff

2) 15 out of 45 are written inspanish...
if we take all as paperback.. prob =15/45=1/3, which is less than 1/2..
so at the best scenario too, the prob <1/2.. suff
ans B

ENGRTOMBA2018 · Answer

Refer to the attached picture for explanation.

Per the question, 45 books are on a shelf. You need to find whether the probability of picking a Spanish paperback is <1/2 --> In other words, is the number of Spanish paperback books <22.5 (=22 as the books can not 0.5!)

Per statement 1, 30 are paperback. Look at 2nd table in the attached figure. The maximum number of Spanish paperback books can be = 30 , giving us a "no" to the question asked but if the same paperback books are lets say = 10, then you get "yes" for the question asked. Thus this statement is NOT sufficient.

Per statement 2, refer to the 3rd table, you can clearly see that the maximum Spanish paperback books can be 15, thus giving us a definite "yes" to the question asked. Hence this statement is sufficient.

B is the correct answer.

Hope this helps.

MathRevolution · Answer

This is a 2 by 2 question that frequently appears in the GMAT Math.

Looking at the attachment, the question, if we modify, is d/45<1/2?, d<45/2=22.5?. Looking at the condition 1), we get d=15<22.5. The answer is yes and the condition is sufficient. Hence, the correct answer is B.


- Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.

Shenaz · Answer

Is there any further explanation on this? Not convinced at all on the OA.
Please help

amanvermagmat · Answer

Hi Did you go through this solution: https://gmatclub.com/forum/each-of-the- ... l#p1621801 Basically, as per second statement, only 15 books are in Spanish (15 is obviously less than half of 45). Even if all these Spanish books are paperback, then the probability of selecting a paperback Spanish book will be 15/45 = 1/3 which is < 1/2. And if some Spanish books are hardcover also, then the probability of selecing a Spanish paperback will be even lesser than 1/3 (so definitely lesser than 1/2). So, as per second statement, probability of picking a paperback Spanish book will always be < 1/2. That's why the answer is B.

LeenaSai · Answer

Just a small correction , for statement 1) we can’t take none case bcz each book is either written in English or Spanish and each book is either a paperback or hardcover !!

We can take the last numbers like 1,2 instead of taking 0

Posted from my mobile device

BrentGMATPrepNow · Answer

Given : Each of the 45 books on a shelf is written either in English or in Spanish, and each of the books is either a hardcover book or a paperback. A book is to be selected at random from the books on the shelf.

Target question :    Is the probability less than 1/2 that the book selected will be a paperback written in Spanish?  
This is a good candidate for  rephrasing the target question

Let's use the Double Matrix Method to solve this question. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions). 
Here, we have a population of books, and the two characteristics are: 
- English or Spanish
- Hardcover or paperback

We can set up our diagram like this: 
 https://i.imgur.com/L2S8qOz.png

1/2 of 45 = 22.5
So in order for the probability to be less than 1/2, we need 22 or fewer books to be Spanish paperbacks. 
In other words, we need 22 or fewer books in the yellow shaded region. 
Let's use this information to REPHRASE the target question
  REPHRASED target question :    Is the number of books in the yellow shaded region 22 or less?  
 https://i.imgur.com/QEza62V.png

Statement 1 : Of the books on the shelf, 30 are paperback  
If 30 of the 45 books are paperback, then the remaining 15 books must be hardcover. We'll add this to our diagram 
 https://i.imgur.com/cIC8ymw.png

There are many different ways to complete this diagram so that we meet the given conditions. 
For example, we could have the following situation:
 https://i.imgur.com/Z2o0lPz.png 
In this case, the answer to the REPHRASED target question is  YES, the number of books in the yellow shaded region IS 22 or less

Alternatively, we could also have this situation: 
 https://i.imgur.com/cAtHWnH.png 
In this case, the answer to the REPHRASED target question is  NO, the number of books in the yellow shaded region is NOT 22 or less

Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : Of the books on the shelf, 15 are written in Spanish. 
When we add this information to our diagram, we get: 
 https://i.imgur.com/pyP7HgS.png

IMPORTANT: This tells us that the two boxes in the bottom row, must add to 15. 
So it will be  impossible  for the value in the yellow shaded region to be greater than 22.
In other words, the answer to the REPHRASED target question is  YES, the number of books in the yellow shaded region IS 22 or less 
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer: B

This question type is  VERY COMMON  on the GMAT, so be sure to master the technique.

To learn more about the Double Matrix Method, watch this video:
 https://www.youtube.com/watch?v=jK-tiBrrf04

Cheers,
Brent

Anuragjn · Answer

I have a doubt - As per stmt 2, if we say all 45 are paperback, then probability is 1/3; However, if we say 30 are paperback, wouldn't probability be 1/2, or for that matter 15 are paperback, wouldn't it make probability as 1 ?? Why shouldn't the answer be 'C'

CrackverbalGMAT · Answer

Hello Anurag, It seems like you have misinterpreted the question. The question is not about the probability of picking a Spanish paperback OF all the paperbacks. The question asks us if the probability of picking a “Paperback written in Spanish” is less than ½ when ANY book is selected at random. There’s as big a difference between the two as chalk from cheese. This is a very common problem that many GMAT test takers face. They end up solving a drastically different question in their head when they have to actually solve the question they see on the screen. Look at the table given below: 03rd Apr 2020 - Reply 1.jpg The question says “If a BOOK is selected at random”. This means your sample space is 45 (the cell highlighted in Green) and hence should be in the denominator. Whereas, the question that YOU solved in your head is “If a paperback is selected at random”. This means YOUR sample space was the cell highlighted in red, which is of course wrong. I hope that clarifies your doubt. Look at how many Spanish books are there in total, as per Statement II (the cell highlighted in blue). If there are 15 books of Spanish in total, there would be either 15 paperbacks or lesser. (the cell highlighted in yellow) \frac{15}{45} is definitely less than ½. We can answer the question with a definite NO which is an accepted answer on a YES-NO type of DS question. This is another common mistake I have seen test takers making on a YES-No type of DS question – forgetting that a NO is as good as a YES. Since statement II alone is sufficient to give us a NO, the correct answer option is B. Here’s a link to one of our articles on Data Sufficiency for your reference. This will really help you to not make such errors on a DS question. https://bit.ly/348ndXx Hope that helps!

BaskaranK25 · Answer

Each of the 45 books on a shelf is written either in English or in Spanish, and each of the books is either a hardcover book or a paperback. If a book is to be selected at random from the books on the shelf, is the probability less than 1/2 that the book selected will be a paperback written in Spanish?

(1) Of the books on the shelf, 30 are paperback.

(2) Of the books on the shelf, 15 are written in Spanish.

Stem: probability (spanish AND Paperback) = p(Spanish) * p(paperback) < 1/2

S1: paperback = 30
probability (spanish AND Paperback) = p(Spanish) * p(30/45) = p(Spanish) * 0.67 --> NS

S2: spanishbooks = 15
probability (spanish AND Paperback) = p(15/45) * p(paperback) = 0.33 * p(paperback)
now we know p(paperback) is less than 1
probability (spanish AND Paperback) = 0.33 * p(paperback) < 0.33 ---> Sufficient

Hovkial · Answer

OFFICIAL GMAT EXPLANATION

Arithmetic Probability

(1) This indicates that 30 of the 45 books are paperbacks. Of the 30 paperbacks, 25 could be written in Spanish. In this case, the probability of randomly selecting a paperback book written in Spanish is 25/45 > 1/2. On the other hand, it is possible that only 5 of the paperback books are written in Spanish. In this case, the probability of randomly selecting a paperback book written in Spanish is 5/45 < 12; NOT sufficient.

(2) This indicates that 15 of the books are written in Spanish. Then, at most 15 of the 45 books on the shelf are paperbacks written in Spanish, and the probability of randomly selecting a paperback book written in Spanish is at most 15/45 < 1/2; SUFFICIENT.
 
Statement 2 alone is sufficient.

GMATinsight · Answer

Answer: Option B Video solution by GMATinsight https://www.youtube.com/watch?v=qOIiwQ2wplg

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=1Bm0DjnhcPc

AkshayBhushan · Answer

Hello, 
in the second case we are considering all of the 15 books to be paperback, why are we not assuming few books to be hardcover as well?

Bunuel · Answer

We are not assuming that all 15 Spanish books are paperbacks. We are considering the maximum possible number of Spanish paperbacks.

Even if all 15 were paperbacks, 15/45 is still less than 1/2, which gives a YES answer. If some are hardcovers, the number of Spanish paperbacks would be even smaller, so the answer would still be YES.

egmat · Answer

Hi AkshayBhushan,

Good instinct to ask this. You're right that the  15  Spanish books could split into some paperback and some hardcover. The all-paperback case isn't being treated as  the  answer - it's being tested for a specific reason.

Why chetan2u picked the all-paperback case:  in a Yes/No DS question you have to be sure the answer stays the  same  no matter how the books are arranged. So you look for the arrangement that gives the  most  Spanish paperbacks - the worst case, the biggest the count can ever get. Here that's all  15  Spanish books being paperback.

If even that maximum lands below  1/2 , then every other arrangement (some hardcover) can only make the count  smaller , so it stays below  1/2  too. That's exactly what happens:

-  All 15  Spanish books paperback -  15/45  =  1/3 , which is <  1/2  - answer is "yes."
- Say  5  paperback,  10  hardcover -  5/45  =  1/9 , still <  1/2  - answer is "yes."
- Say  0  paperback -  0/45  =  0 , still <  1/2  - answer is "yes."

Every allowed split gives the  same  answer: yes, the probability is less than  1/2 . That's why we don't need to worry about the mixed cases separately - they can't flip the answer. That sameness across all scenarios is precisely what makes Statement ( 2 )  sufficient .

The takeaway:  to prove a statement  is  sufficient, test the extreme that would be most likely to break it. If even the extreme obeys the condition, you're done - no need to check the milder cases, since they can only agree. The ceiling here ( 15 ) is already comfortably under the  22.5  threshold, so the answer is locked in.

Answer: B

Each of the 45 books on a shelf is written either in English or in Spa

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Each of the 45 books on a shelf is written either in English or in Spa

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