Lesson: Double Matrix Method

Comment on Double Matrix Method

Euro's picture

Another method shows the matrix a 3X3 to leave room for the totals, in the 2X2 you write out the number anyway.
gmat-admin's picture

That method is exactly the same. The only difference is a small change in how we show the totals for each row and column.

For the motel ques below, the 2/3rd statement is a bit confusing. Read it as 2/3 of rented, not 2/3 of AC rooms. How can I avoid mistakes like this - the question writing is a bit vague if you ask me

One night a certain motel rented 3/4 th of its rooms, including 2/3rd of its air-conditioned rooms. If 3/5th of its rooms were air-conditioned, what percent of the rooms that were not rented were air-conditioned?
gmat-admin's picture

You're referring to this question: http://www.beatthegmat.com/motel-t271938.html

This is an official GMAT question, and I think the wording is fine. We're told that 2/3 of the air-conditioned rooms were rented, so I'm not sure what could be changed to make it more readable.

Hi, Brent! I used the double Matrix Method for the following question. But I got different result than the answer choice. I got 34%. After compared with the explanation, I think the major confusion that I have is how to interpret "75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts". This to me means out of all interns who received fewer than 6 hours of sleep, there are 75% report feeling tired during their shifts. If I used 100 as total intern number, the grid of less than 6 hours and tired intern should be 0.75 X 80 = 60. But the answer states that there are 75 people in this grid. Could you please help me out?

At a certain hospital, 75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts. At the same time, 70% of the interns who receive 6 or more hours of sleep report no feelings of tiredness. If 80% of the interns receive fewer than 6 hours of sleep, what percent of the interns report no feelings of tiredness during their shifts?

A) 6
B) 14
C) 19
D) 20
E) 81

Answer C
gmat-admin's picture

Hi Jingyjess,

The key word here is AND: "At a certain hospital, 75% of the interns receive fewer than 6 hours of sleep AND report feeling tired during their shifts."

If the test-makers wanted to convey your interpretation, they'd write "OF THE interns who received fewer than 6 hours of sleep, 75% report feeling tired during their shift." (Notice that's there's no AND here. Instead, we have OF THE)

Does that help?


Thank you for the feedback!I understand where you are coming from. I just still need take some time to digest and prepare better strategy for tricky wordings like this in the future!

Hey Brent,
Please help me with the following question posted on Beat the GMAT. I am extremely confused wrt using the double matrix method for this.

At least 100 students at a certain high school study Japanese. If 4 percent of the students at the school who study French also study Japanese, do more students at the school study French than Japanese?
(1) 16 students at the school study both French and Japanese.
(2) 10 percent of the students at the school who study Japanese also study French.

Thanks in advance,
gmat-admin's picture

Hi Aashay,

I'm happy to help.
You'll find my solution here: http://www.beatthegmat.com/at-least-100-students-at-a-certain-high-schoo...


On a certain road 10 percent of the motorists exceed the posted speed limit and receive speeding tickets, but 20 percent of the motorists who exceed the posted speed limit do not receive speeding tickets. What percent of the motorists on the road exceed the posted speed limit?

My concern is: .1x + .2x = x, but this is not right. Why is 10% of the speeders who get a ticket 10, but those who speed and not get a ticket not 20, but .2x? I can't see why you don't add .1x of some unknown to .2x of some unknown to get a total of x?
gmat-admin's picture

Hi Bertyy,

The question tells us that "10 percent of the motorists exceed the posted speed limit and receive speeding tickets"

So, if there are 100 motorists in TOTAL, then 10 of them 10 percent of the motorists exceed the posted speed limit AND receive speeding tickets.

We're also told that "20 percent of the motorists who exceed the posted speed limit do not receive speeding tickets."

Here, we are limiting the discussion to ONLY those motorists who speed. AMONG those speeders, 20% receive a ticket. In this case, we do not know how many speeders there are. So, if we let x = the number of SPEEDERS, then 0.2x = the number of SPEEDERS who receive a ticket.

Does that help?

Here's my full solution: https://gmatclub.com/forum/on-a-certain-road-10-of-the-motorists-exceed-...


Thanks Brent,

Starting to get it.

So, the inclusion of "And" (everything in the Double Matrix Box) makes it an "Absolute" if you have a total number, like 100 in the problem. And everything on the periphery on the Double Matrix is "Only", so no "Absolute" and thus you can't draw from the total?
gmat-admin's picture

Not quite.

"On a certain road 10 percent of the motorists exceed the..."

Here we are talking about 10% of ALL of the motorists. We know this, because there are no words that restrict the KIND of motorists we are examining.

"20 percent of the motorists WHO EXCEED the posted speed limit do not..."

Here the words "WHO EXCEED" restrict the discussion to those who exceed the speed limit. So, we are no longer talking about all motorists; we are talking about just those motorists WHO EXCEED the speed limit.

Does that help?


Hi Brent,

The question below seems to be bizarrely worded, maybe not..maybe it's just me. Anyways. Here it is:

Of the applicants passes a certain test, 15 applied to both college X and Y. If 20 % of the applicants who applied college X and 25% of the applicants who applied college Y applied both college X and Y, how many applicants applied only college X or college Y?
(A) 135
(B) 120
(C) 115
(D) 105
(E) 90
gmat-admin's picture

The wording isn't great. I'd definitely change the last part to read "how many applicants applied to college X or college Y BUT NOT BOTH?"

There's a nice solution (with double matrix charts) by mevicks here: http://www.beatthegmat.com/of-the-applicants-passes-a-certain-test-15-t2...

Let me know if you need any clarification.



Was wondering if you could help with the following question from 2017 Official Guide For GMAT:

14. Of the 84 parents who attended a meeting at a school, 35 volunteered to supervise children during the school picnic and 11 volunteered both to supervise children during the picnic and to bring refreshments to the picnic. If the number of parents who volunteered to bring refreshments was 1.5 times the number of parents who neither volunteered to supervise children during the picnic nor volunteered to bring refreshments, how many of the parents volunteered to bring refreshments?

(A) 25
(B) 36
(C) 38
(D) 42
(E) 45
gmat-admin's picture

I'm happy to help.

Here's my full solution: https://gmatclub.com/forum/of-the-84-parents-who-attended-a-meeting-at-a...


HI Brent,

Can you post the solution of the below question using double matrix method

In a group of x students, w students are taking Chemistry but not French, y students are taking French but not Chemistry, and z students are NOT taking French. Which of the following represents the number of students who are taking Chemistry?

sir please explain

Hi Brent,

My answer to the below question is E but the OA says A.
Need your help.

gmat-admin's picture

Hi Brent,

Need your help:

gmat-admin's picture

Hi Brent,

Looking for a simple approach to the handle this one:

gmat-admin's picture

Question link: https://gmatclub.com/forum/200-people-responded-to-a-survey-that-asked-t...

To answer this question, we can use the same approach I used for the above question.
Give it a try, and see how you do.


Thanks, I was able to solve it easily with the approach described by you.

Hi Brent,

The question is from gmat prep test 2 and I fail to understand why the answer is not C but E.

Each of the students in a certain class received a single grade of P, F, or I. What percent of the students in the class were females?

(1) Of those who received a P, 40 percent were females.
(2) Of those who received either an F or I, 80 percent were males
gmat-admin's picture

Every object in a box is either a sphere or a cube, and every object in the box is either red or green. How many objects are in the box?

(1) There are six cubes and 5 green objects in the box.
(2) There are two red spheres in the box.

What makes this an overlapping question? How would you know instantly (this is a time-starved exam) that the matrix method should be used? Normally, the clear hint is when the questions says that some number is in either X category or Y category or both. But this question doesn't mention any of this terminology.

That said, I am comfortable with deriving the answer correctly once I knew it was an overlapping question.
gmat-admin's picture

In order to apply the Double Matrix Method, we need a population in which each member has exactly two characteristics associated with it.

With your question, we have a population of objects, and the two characteristics are:
- sphere or cube
- red or green

Given this, we SHOULD be able to apply the Double Matrix Method.

Here's my full solution: https://gmatclub.com/forum/every-object-in-a-box-is-either-a-sphere-or-a...



Hi Brent!

Questions like these are really warping my brain. I understand that we assume there are 100 people etc. But why do we need 'x', if we are assuming number of people, wouldn't 100 just suffice?
gmat-admin's picture

Question link: https://gmatclub.com/forum/at-the-opening-night-performance-of-a-new-pla...

The only thing we're assuming is that the TOTAL population is 100. We can do this, because the rest of the information is in terms of percent, AND we're asked to find a certain percent.

Once we've assigned a nice value for the total number of people, we have to use the GIVEN INFORMATION to complete the rest of the matrix.

Since we aren't told the total number of patrons who are late, we can assign a variable to this value, and then go from there to create an equation, which we can solve for x.

Does that help?


Hi Brent!

I think I've figured it out! I re-did a lot of the questions you've listed above and read through your explanations while solving it step by step so that really helped! Thanks for your explanation above as well, it definitely clears things up. What is the probability of Matrix questions appearing on the GMAT? I honestly feel a bit on the unprepared side when it comes to them.
gmat-admin's picture

It's quite likely that you'll see a Double Matrix question on test day. In fact, you might even see 2 such questions.


Hey Brent, I would appreciate your clarification in regarding this Q:


If I take both statements together I have: 50=A+B-AB+neither

In the solutions people have +AB instead of -AB and I don´t understand why? I mean, that way, we count some owners twice?

Thanks in advance,

gmat-admin's picture

Question link: https://gmatclub.com/forum/of-a-group-of-50-households-how-many-have-at-...

Aside: I'm not a big fan of formulas for overlapping sets questions.

Those that are stating 50 = A + B + AB + neither are saying that:
A = # with at least one dog BUT no cats
B = # with at least one cat BUT no dogs

So, they're essentially noting each of the 4 parts of the Double Matrix.

Does that help?


Thanks, I can see it now.

I'll definitely use this *double matrix* method - very useful! Is there a *triple matrix* technique? If so, could you please provide an example? Thanks!

BTW, in answering this question, I immediately eliminated answers 12 and 14. Since there are 30 small toys (of a total of 40), only a maximum of 10 toys could be large. Likewise, 10 is a suspect, because it appears that not all 10 would be blue or green.
gmat-admin's picture

Yes there is a strategy for 3 overlapping sets. The lesson can be found here: https://www.gmatprepnow.com/module/gmat-word-problems/video/921

This question type is quite rare on the GMAT, primarily because these questions can be a huge time killer.

I just saw it: the 3-circle Venn diagram. Thanks.


I had taken GMAT prep course from elsewhere before signing up for yours. I’m super glad that I registered fro your course.

Where I used to take classes they had taught Venn Diagrams using these formulas below:
Exactly 1 of 2: A + B - 2K
At least 1 of 2: Total - Neither = A + B - K
Exactly 1 of 3: [(A - P - R - K) + (B - P - Q - K) + (C - Q - R-K)]
At least 1 of 3: Total - Neither = A + B + C - P - Q - R - 2K
Exactly 2 of 3: P + Q + R
At least 2 of 3: P + Q + R + K
Note: I don’t know if there were even more than above mentioned formulas.

My concern is that I understood the Venn Diagram your way ie Double Matrix method (& it’s fabulous). So, I just wanted to know IF These formulas are an alternative method to solve Venn Diagram questions? OR ELSE I need to learn those formulas too for GMAT EXAM?
gmat-admin's picture

If you have 2 overlapping sets, the Double Matrix technique is all you need. No formulas necessary.
That said, the formulas will work (as long as they're used correctly).

Hey Brent!
For the problem associated with this link: https://gmatclub.com/forum/in-a-class-of-120-students-80-enrolled-for-a-mathematics-seminar-326020.html

Is there a reason for the 80 in the top right box in your solution or is that just an error?
gmat-admin's picture

Good catch! That number should be 48.
I've added to my response: https://gmatclub.com/forum/in-a-class-of-120-students-80-enrolled-for-a-...

Hi Brent,

Can you pls solve this?

Qs 1: https://gmatclub.com/forum/the-table-above-shows-the-responses-of-100-consumers-to-a-taste-test-310621.html

Qs 2: https://gmatclub.com/forum/the-table-above-shows-the-results-of-a-survey-of-100-voters-who-each-89187.html

I'm consfused on what approach should be followed for such questions.


gmat-admin's picture

Hi Brent,

Can you solve this please? - https://gmatclub.com/forum/m30-186700.html
gmat-admin's picture

Here's my full solution: https://gmatclub.com/forum/m30-186700.html#p2741268
In the future, please post your questions under the related video lesson.
In this case, your question would fit nicely under the lesson on absolute value equations.



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