In a certain building, 1/5 of the offices have both a window

Question

In a certain building, 1/5 of the offices have both a window and bookshelves. If the rest of the offices in the building have either a window or bookshelves but not both, what is the ratio of the number of offices with a window but not bookshelves to the number of offices with bookshelves but not a window?

(1) The number of offices with a window is 4/5 the number with bookshelves.

(2) 3/10 of the offices with bookshelves also have a window.

Bunuel - Could you please show me how to solve this problem in 2 set matrix? Many thanks for all your responses so far.

Cheers

Cheers

Bunuel · Accepted Answer

In a certain building, 1/5 of the offices have both a window and bookshelves. If the rest of the offices in the building have either a window or bookshelves but not both, what is the ratio of the number of offices with a window but not bookshelves to the number of offices with bookshelves but not a window?

Say there are total of 300 offices. Then given that:

Attachment:

Stem.png [ 4.53 KiB | Viewed 25940 times ]

We need to find the ratio of the yellow boxes.

(1) The number of offices with a window is 4/5 the number with bookshelves. Say the number of the offices with bookshelves is x and the number of the offices with a window is 4/5*x. Then the number of the offices with bookshelves but not a window is x-60. Also, the total number of the offices with no window is x-60. Thus, 4/5*x+(x-60)=300. We can find x, thus we can find the value of the yellow boxes. Sufficient.

Attachment:

1.png [ 5.68 KiB | Viewed 25918 times ]

(2) 3/10 of the offices with bookshelves also have a window. Again, say the number of the offices with bookshelves is x, then we are given that 60=3/10*x. We can find x (200), thus we can find the number of the offices with both a window and bookshelves (3/10*x=60) and then find the number of the offices with bookshelves but not a window (200-60=140). Also, we can find the total number of the offices with no bookshelves (300-200=100), which is the same as the number of the offices with a window but no bookshelves. So, we have the values of both yellow boxes. Sufficient.

Attachment:

2.png [ 6.23 KiB | Viewed 25897 times ]

Answer: D.

Hope it's clear.

Vips0000 · Answer

Lets assume total number of offices is 10x, with window is W, with Bookshelves is B and with both bookshelf and window -D
From question : 
1/5*10x = D
or D = 2x

Also we know that: W+B+D = 10x (all offices)
=>  W+B = 8x

And we need W/B.

Stem 1: 
W+D = 4/5 * (B+D)
We know D in terms of x, and we have 2 equations now in terms of W and B. So we can find out Wand B in terms of x and therefore ratio W/X. No need to solve further. Sufficient.

Stem 2:
3/10 (B+D) =D
(Note - B+D represent all offices with bookshelves and D represent offices with bookshelves and windows both)

Again we can find B in terms of x and from equation formed with question stem, we cna find W in terms of x and find ratio.
Sufficient.

Note solving any DS question with double matrix would be more time consuming.

hope it helps.

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Overlapping Sets:
advanced-overlapping-sets-problems-144260.html
how-to-draw-a-venn-diagram-for-problems-98036.html

All DS Overlapping Sets Problems to practice: search.php?search_id=tag&tag_id=45
All PS Overlapping Sets Problems to practice: search.php?search_id=tag&tag_id=65

gmatkum · Answer

Let us assume that the number of offices in the building is N. 
Then number of offices containing both bookshelves and window is N/5. 
Also, number of office containing only windows = x and only bookshe.. = y. 
Hence x+y+n/5=n 
or x+y = 4n/5.

From 1)x= (4/5)y. Using this and original equation both x and y can be obtained in terms of n and x/y will be the answer to the original question.

From 2)(3/10)(y+n/5) = n/5. 
Using this and original eq. above we can again solve x and y in terms of n.

hence D is the right answer.

olnata · Answer

GIVEN:
TOTAL = W + B - Both + Neither
1 = W + B - 1/5 + 0
W + B = 6/5

THE QUESTION:
W/B = ?

(1) W = 4/5 B -> Two unknowns and two linear equations we can solve with plug in method -> W + B = 6/5 -> 4/5B + B=6/5 -> B= 2/3-> W= 8/15 ->W/B = 4/5 SUFFICIENT;

(2) 1/5=3/10 B -> B=2/3 -> W+2/3=6/5 -> W = 8/15 -> W/B = 4/5 SUFFICIENT;

ANSWER D.

elainetianfong · Answer

As Both = 1/5 (*)
=> w-only + b-only = 4/5 (**)

(1) The number of offices with a window is 4/5 the number with bookshelves.

=> (w-only + both) / ( b-only + both) = 4 / 5 (***)
From (*), (**) & (***), we surely can find w-only, b-only & both 
=> suff.

(2) 3/10 of the offices with bookshelves also have a window

=> b-only / both = 7/3 (****)
From (*), (**) & (****), we surely can find w-only, b-only & both 
Suff. 
So D is the answer.

Posted from my mobile device

hr1212 · Answer

Let W, NW, B and NB be offices with window, no window, bookshelves and no bookshelves and total offices be z, then we are given,

W 
 NW 
 T 
 
  B 
 24 
 a

NB 
 b 
 0

T

120

Calculate a/b?

Statement 1 => Let TB be x then TW = x*4/5

W 
 NW 
 T 
 
  B 
 24 
  
 x 
 
  NB 
 120-x 
 0 
 120-x 
 
  T 
 4x/5 
 120-4x/5 
 120

144 - x = 4x/5
x=80

W 
 NW 
 T 
 
  B 
 24 
 56 
 80 
 
  NB 
 40 
 0 
 40 
 
  T 
 64 
 56 
 120

a/b = 56/40

Hence, sufficient.

Statement 2 => Let offices with bookshelves be x, then

W 
 NW 
 T 
 
  B 
 3x/10 = 24 
  
 x 
 
  NB 
  
 0

T

120

x = 80

W 
 NW 
 T 
 
  B 
 24 
 56 
 80 
 
  NB 
 40 
 0 
 40 
 
  T 
 64 
 56 
 120

a/b = 56/40

Hence, sufficient.

IMO: D

braddouglas10 · Answer

Why did you use 300? I used 100 and it didn't work!

Bunuel · Answer

Basically any total will work, including 100. With some numbers you’ll just run into fractions. To avoid that, you can set the total as a variable, say t, and the solution will still work fine.

Aashimabhatia · Answer

Since the question is asking for ratio, we can take a smart number: Total offices = 90 & Both = 1/5 of 90 = 18 & 
W + B = 90-18 = 72
We need to find W only/B only = W-Both/B-Both

1. The number of offices with a window is 4/5 the number with bookshelves.
This means Ratio of W/B = 4/5
4x + 5x = 90
From here, we can calculate the values of W and B. We also know that the value of both and neither is 0, so 18 can be be subtracted from W & B to find the values of W only & B only, hence, SUFFICIENT.

2. 3/10 of the offices with bookshelves also have a window.
This statement is simply telling us that 3/10 of B = Both. As we already know the value of both, we can find the value of B. From this we can find the value of W and then W only and B only.
Hence, SUFFICIENT.

ANSWER IS D

olechkakiev · Answer

Sorry, but where are the options?? A, B, ... why I don't see them??

Bunuel · Answer

Hi,

This is a data sufficiency question. Options for DS questions are always the same and usually omitted on the site.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hope this helps.

In a certain building, 1/5 of the offices have both a window

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

In a certain building, 1/5 of the offices have both a window

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards