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guerrero: Master | Next Rank: 500 Posts; Posts: 227; Joined: Sun Apr 08, 2012 4:53 amFollowed by: 3 members; Thanked: 12 times

by guerrero » Mon Apr 01, 2013 11:25 am

A school has 3 classes, math class has 14 students. English class has 10 students, PE class has 11 students. There are 20 students taking only one class, 3 students are taking all three classes. How many students are taking two classes?

A) 3
B) 6
C) 9
D) 18
E)20

could any one show me the solution using 3 overlapping groups formula .Mitch's way ..

T = A + B + C - (AB + AC + BC) - 2(ABC) ... I am getting a wrong answer

OA A

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Source: Beat The GMAT — Problem Solving | Mon Apr 01, 2013 11:25 am

hemant_rajput: Master | Next Rank: 500 Posts; Posts: 447; Joined: Sun Apr 22, 2012 7:13 amFollowed by: 13 members; Thanked: 46 times; GMAT Score:700

by hemant_rajput » Mon Apr 01, 2013 11:47 am

guerrero wrote:A school has 3 classes, math class has 14 students. English class has 10 students, PE class has 11 students. There are 20 students taking only one class, 3 students are taking all three classes. How many students are taking two classes?

A) 3
B) 6
C) 9
D) 18
E)20

could any one show me the solution using 3 overlapping groups formula .Mitch's way ..

T = A + B + C - (AB + AC + BC) - 2(ABC) ... I am getting a wrong answer

OA A

I don't know how Mitch solve this type of question but you can have look at my approach, may be you get how to do it in Mitch's way by yourself.

a,b and c - students only in English, only in maths and only in PE resp.
d,e and f - both English and maths, maths and PE, and PE and english
g - students in all there - English, math and PE

now, given - a+d+e+g =14
b+e+f+g = 10
c+d+f+g = 11

a+b+c = 20(only one class)

g = 3(all three classes)

now,

a+b+c+2(d+e+f)+3g = 35

we know a+b+c and g values.

20+2(d+e+f)+(3*3) = 35

2(d+e+f) = 35 - 29
2(d+e+f) = 6
d+e+f = 3(students taking two classes only)

hope this helps

Kudos,
Hemant

I'm no expert, just trying to work on my skills. If I've made any mistakes please bear with me.

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srcc25anu: Master | Next Rank: 500 Posts; Posts: 423; Joined: Fri Jun 11, 2010 7:59 amFollowed by: 2 members; Location: Seattle, WA; Thanked: 86 times

by srcc25anu » Mon Apr 01, 2013 12:01 pm

i found it easier to plug in answer choices for this one.

if we plug 3 for 2 class students, and if we divide it as shown in the diagram above, we can calculate the number of students for only maths, only english and only PE which should sum to 20 as its given in the question stem that students taking a single class total 20.

hence A is our answer.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Apr 01, 2013 1:48 pm

guerrero wrote:A school has 3 classes, math class has 14 students. English class has 10 students, PE class has 11 students. There are 20 students taking only one class, 3 students are taking all three classes. How many students are taking two classes?

A) 3
B) 6
C) 9
D) 18
E)20

could any one show me the solution using 3 overlapping groups formula .Mitch's way ..

T = A + B + C - (AB + AC + BC) - 2(ABC) ... I am getting a wrong answer

OA A

Here is the formula for 3 overlapping groups:

T = A + B + C - (AB + AC + BC) - 2(ABC)

The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in A, everyone in B, and everyone in C:
Those in exactly 2 of groups (AB+AC+BC) are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups (ABC) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.

In the problem above:
A = Math = 14.
B = English = 10.
C = PE = 11.
Since the number of students taking exactly 2 classes is unknown, AB + AC + BC = x.
Since 3 students take all three classes, ABC = 3.

Plugging these values into the formula, we get:
T = 14 + 10 + 11 - x - 2(3)
T = 29 - x.

The following must also be true:
T = (students taking exactly 1 class) + (students taking exactly 2 classes) + (students taking all 3 classes).
Since 20 students are taking exactly 1 class, x students are taking exactly 2 classes, and 3 students are taking all 3 classes, we get:
T = 20 + x + 3.
T = 23 + x

Since T = 29 - x and T = 23 + x, we get:
29 - x = 23 + x
6 = 2x
x = 3.

The correct answer is A.

Check here for another problem about triple-overlapping groups:

https://www.beatthegmat.com/og-13-178-vi ... 11188.html

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guerrero: Master | Next Rank: 500 Posts; Posts: 227; Joined: Sun Apr 08, 2012 4:53 amFollowed by: 3 members; Thanked: 12 times

by guerrero » Wed Apr 03, 2013 10:16 am

GMATGuruNY wrote:
guerrero wrote:
The following must also be true:
T = (students taking exactly 1 class) + (students taking exactly 2 classes) + (students taking all 3 classes).
Since 20 students are taking exactly 1 class, x students are taking exactly 2 classes, and 3 students are taking all 3 classes, we get:
T = 20 + x + 3.
T = 23 + x

Since T = 29 - x and T = 23 + x, we get:
29 - x = 23 + x
6 = 2x
x = 3.

The correct answer is A.

Check here for another problem about triple-overlapping groups:

https://www.beatthegmat.com/og-13-178-vi ... 11188.html

Hi Mitch , thanks for the help . Could you please help me understand the second step a little better ? Why did you calculate - (students taking exactly 1 class) in the the second step .

shouldn't the total be 35 ?

35 = 20 +x -2(3)?.. There is something I am missing

thanks !

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed Apr 03, 2013 11:11 am

guerrero wrote:
GMATGuruNY wrote:
guerrero wrote:
The following must also be true:
T = (students taking exactly 1 class) + (students taking exactly 2 classes) + (students taking all 3 classes).
Since 20 students are taking exactly 1 class, x students are taking exactly 2 classes, and 3 students are taking all 3 classes, we get:
T = 20 + x + 3.
T = 23 + x

Since T = 29 - x and T = 23 + x, we get:
29 - x = 23 + x
6 = 2x
x = 3.

The correct answer is A.

Check here for another problem about triple-overlapping groups:

https://www.beatthegmat.com/og-13-178-vi ... 11188.html

Hi Mitch , thanks for the help . Could you please help me understand the second step a little better ? Why did you calculate - (students taking exactly 1 class) in the the second step .

shouldn't the total be 35 ?

35 = 20 +x -2(3)?.. There is something I am missing

thanks !
Here's the GENERAL FORM of the second question:

T = (Only A + Only B + Only C) + (AB + AC + BC) + (ABC).

In this equation, three are NO OVERLAPS:
Red = everyone in EXACTLY ONE group.
Blue = everyone in EXACTLY TWO groups.
Green = everyone in ALL 3 groups.

Since there are no overlaps, the TOTAL is simply the SUM of all of the colored values.

In the problem above, red = 20, blue = x, and green = 3.
Thus:
T = 20 + x + 3.

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guerrero: Master | Next Rank: 500 Posts; Posts: 227; Joined: Sun Apr 08, 2012 4:53 amFollowed by: 3 members; Thanked: 12 times

by guerrero » Wed Apr 03, 2013 12:48 pm

makes sense now

. Thank you so much !

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rairavig: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Wed Mar 20, 2013 2:22 am; Thanked: 2 times

by rairavig » Wed Apr 03, 2013 10:20 pm

since 3 students are attending all three classes
student left in each class
math- 14-3=11
English- 10-3=7
PE- 11-3=8
total attendance = 26
student attending only one class= 20
attendance for students attending 2 classes = 6
number of students attending 2 calsses = attendance / classes
= 6/2
= 3

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