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

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Math Expert
Joined: 02 Sep 2009
Posts: 47037
In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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20 Mar 2018, 23:15
2
2
00:00

Difficulty:

35% (medium)

Question Stats:

64% (03:03) correct 36% (02:10) wrong based on 56 sessions

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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

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Math Expert
Joined: 02 Sep 2009
Posts: 47037
Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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20 Mar 2018, 23:16
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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Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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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,
Attachments

overlapping sets.jpg [ 1004.24 KiB | Viewed 1280 times ]

Senior Manager
Joined: 31 Jul 2017
Posts: 374
Location: Malaysia
WE: Consulting (Energy and Utilities)
Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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21 Mar 2018, 02:33
3
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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Manager
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Posts: 117
Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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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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22 Mar 2018, 11:11
1
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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Senior Manager
Joined: 31 Jul 2017
Posts: 374
Location: Malaysia
WE: Consulting (Energy and Utilities)
Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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22 Mar 2018, 11:25
1
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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Manager
Joined: 22 May 2017
Posts: 117
Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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22 Mar 2018, 21:48
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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Posts: 1755
Re: In a survey of 600 students, 75% indicated they like orange juice, 40% [#permalink]

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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.

• 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

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

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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.

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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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