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In a survey of 600 students, 75% indicated they like orange juice, 40%

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In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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In a survey of 600 students, 75% indicated they like orange juice, 40% indicated they like apple juice, and 30% indicated they like grape juice. If all of the survey participants like at least one of these juices and 35% of them like exactly two of these juices, how many survey participants like only one of these juices?

(A) 210
(B) 250
(C) 260
(D) 360
(E) 390
[Reveal] Spoiler: OA

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Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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New post 20 Mar 2018, 23:16
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Bunuel wrote:
In a survey of 600 students, 75% indicated they like orange juice, 40% indicated they like apple juice, and 30% indicated they like grape juice. If all of the survey participants like at least one of these juices and 35% of them like exactly two of these juices, how many survey participants like only one of these juices?

(A) 210
(B) 250
(C) 260
(D) 360
(E) 390


19. Overlapping Sets



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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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New post 21 Mar 2018, 00:44
Bunuel wrote:
In a survey of 600 students, 75% indicated they like orange juice, 40% indicated they like apple juice, and 30% indicated they like grape juice. If all of the survey participants like at least one of these juices and 35% of them like exactly two of these juices, how many survey participants like only one of these juices?

(A) 210
(B) 250
(C) 260
(D) 360
(E) 390


IMO D.

This is a bit tricky and I always like to solve three overlapping set problems with a diagram. ( Diagram is attached)

We want to find a + b + c

x + y + z = 210. (35% of 600 like exactly two juices liked ) ...(1)
a + d + x + z = 450 ( 75% of 600 like orange ) ...(2)
b + d + x + y = 240 ( 40% of 600 like apple ) ...(3)
c + d + y + z = 180 ( 30% of 600 like grape ) ...(4)

add (4) + (2) + (3) equations.
(a + b + c) + 3*d + 2*(x + y + z) = 450 + 240 + 180 = 870
a + b + c + 3*d = 870 - 2*(210) = 870 - 420 = 450 ...(5)

Also since everyone likes at least one juice... sum of all variables is 600.

a + b + c + d + x + y + z = 600
a + b + c + d = 600 - 210 = 390 ...(6)

subtract (6) from (5)

2d = 450 - 390 = 60
d = 30 ... (7)

Put value of d back in either (5) or (6)

to obtain (a + b + c) = 360

Hence D.

Please give kudos if you liked the explanation... :-)

Best,
Gladi
Attachments

overlapping sets.jpg
overlapping sets.jpg [ 1004.24 KiB | Viewed 891 times ]

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Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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New post 21 Mar 2018, 02:33
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Bunuel wrote:
In a survey of 600 students, 75% indicated they like orange juice, 40% indicated they like apple juice, and 30% indicated they like grape juice. If all of the survey participants like at least one of these juices and 35% of them like exactly two of these juices, how many survey participants like only one of these juices?

(A) 210
(B) 250
(C) 260
(D) 360
(E) 390


Let x be the total number. We know that,

\(x = 0.75x + 0.4x + 0.3x - 0.35x - 2(All Three)\)

All Three = 0.05x

Only One = Total - Exactly Two - Exactly Three = x - 0.35x - 0.05x = 0.6x = 360.

Hence, D

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Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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New post 22 Mar 2018, 11:06
rahul16singh28 wrote:
Bunuel wrote:
In a survey of 600 students, 75% indicated they like orange juice, 40% indicated they like apple juice, and 30% indicated they like grape juice. If all of the survey participants like at least one of these juices and 35% of them like exactly two of these juices, how many survey participants like only one of these juices?

(A) 210
(B) 250
(C) 260
(D) 360
(E) 390


Let x be the total number. We know that,

\(x = 0.75x + 0.4x + 0.3x - 0.35x - 2(All Three)\)

All Three = 0.05x

Only One = Total - Exactly Two - Exactly Three = x - 0.35x - 0.05x = 0.6x = 360.

Hence, D


Can you explain in detail how you got all three?
I would like to know short method venn diagram is taking time :(

Posted from my mobile device
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Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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New post 22 Mar 2018, 11:11
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akshata19 wrote:
Can you explain in detail how you got all three?
I would like to know short method venn diagram is taking time :(


Hey akshata19 ,

I think if you learn the theory from this link, you will be able to solve these questions within a fraction of a minute.

The formula is :

Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither

Does that make sense?
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Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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New post 22 Mar 2018, 11:25
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akshata19 wrote:
rahul16singh28 wrote:
Bunuel wrote:
In a survey of 600 students, 75% indicated they like orange juice, 40% indicated they like apple juice, and 30% indicated they like grape juice. If all of the survey participants like at least one of these juices and 35% of them like exactly two of these juices, how many survey participants like only one of these juices?

(A) 210
(B) 250
(C) 260
(D) 360
(E) 390


Let x be the total number. We know that,

\(x = 0.75x + 0.4x + 0.3x - 0.35x - 2(All Three)\)

All Three = 0.05x

Only One = Total - Exactly Two - Exactly Three = x - 0.35x - 0.05x = 0.6x = 360.

Hence, D


Can you explain in detail how you got all three?
I would like to know short method venn diagram is taking time :(

Posted from my mobile device


Hi akshata19

We have the below formula for overlapping sets -

Total=A+B+C−(sum of EXACTLY 2−group overlaps)−2∗(all three)+Neither. (You can know more on this here - https://gmatclub.com/forum/advanced-ove ... 44260.html).

Hope, it Helps.
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Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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New post 22 Mar 2018, 21:48
Thank you for the link
It is definitely quick than Venn

Kudos for help :)

[size=80][b][i]Posted from my mobile device[/i][/b][/size]

[size=80][b][i]Posted from my mobile device[/i][/b][/size]
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Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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New post 26 Mar 2018, 09:58

Solution:



Given:

    • Total students in the survey = 600 = 100% (Assume)

    • Students who like orange juice = 75%

    • Students who like apple juice = 40%

    • Students who like grape juice = 30%

    • Students who like exactly two juices = 35%

Working out:

We need to find the number of students who like only one type of juices.

Let us denote the number of students who like exactly 1 type of juice by I, who like exactly 2 types of juices by II and students who like exactly three types of juices by III.

Since the question mentions that all of the students like at least one type of juice, we can say: I +II + III = 100%.

Now, let us represent this information via a venn diagram.
Image

    • Students who like apple juice = 40% = a+d+g+e

    • Students who like orange juice =75% = b+d+g+f

    • Students who like grape juice = 30% = e+g+f+b

    • Students liking at least one juices = 40% + 75% + 30% = a+b+c + 2*(d+e+f) + 3(g)
As discussed earlier, a+b+c = I, d+e+f = II, g = III

    • Thus, I + 2II + 3III = 145%___(i)

    • Also, I + II + III = 100%_______(ii)

    • Subtracting (ii) from (i), we have: II + 2III = 45%

II= 35% (Given in the question)

Thus, 2III = 10%, Or, III = 5%.

Putting the values of II and III in equation (ii), we get:

    • 100% = I + 35% + 5%

    • Or, I = 60%.

    • And this is equal to 60% of 600 = 360

Answer: Option D
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Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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New post 28 Mar 2018, 10:33
Bunuel wrote:
In a survey of 600 students, 75% indicated they like orange juice, 40% indicated they like apple juice, and 30% indicated they like grape juice. If all of the survey participants like at least one of these juices and 35% of them like exactly two of these juices, how many survey participants like only one of these juices?

(A) 210
(B) 250
(C) 260
(D) 360
(E) 390


We can use the formula:

Total = orange juice + apple juice + grape juice - double - 2(triple) + neither

600 = 0.75(600) + 0.4(600) + 0.3(600) - 0.35(600) - 2t + 0

600 = 450 + 240 + 180 - 210 - 2t

600 = 660 - 2t

-60 = -2t

t = 30

So, 600 - 210 - 30 = 360 like exactly one juice.

Answer: D
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Re: In a survey of 600 students, 75% indicated they like orange juice, 40%   [#permalink] 28 Mar 2018, 10:33
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