A number of people were asked whether they liked drinks of orange, lem

Question

A number of people were asked whether they liked drinks of orange, lemon or grape flavor. The responses are as follows- 85 liked orange, 45 liked orange and lemon, 65 liked grapes, 40 liked lemon and grape, 90 liked lemon, 30 liked orange and grape, 15 liked all three and 25 liked none. Find the number of people who liked only orange and how many people were?

(A) 10 and 165
(B) 25 and 165
(C) 5 and 170
(D) 10 and 175
(E) 0 and 180

Show Answer

Official Answer

B

(A) 10 and 165
(B) 25 and 165
(C) 5 and 170
(D) 10 and 175
(E) 0 and 180

Kinshook · Answer

Given: A number of people were asked whether they liked drinks of orange, lemon or grape flavor. The responses are as follows- 85 liked orange, 45 liked orange and lemon, 65 liked grapes, 40 liked lemon and grape, 90 liked lemon, 30 liked orange and grape, 15 liked all three and 25 liked none.

Asked: Find the number of people who liked only orange and how many people were?

Attachment:

Screenshot 2021-12-15 at 8.42.44 PM.png [ 78.52 KiB | Viewed 9112 times ]

The number of people who liked only orange = 25

The number of people who were interviewed / asked = 165

WizakoMani · Answer

The safer approach to solving a problem with only one variable is by using the formulaic method. Fewer avenues for mistakes and lesser chances of you computing an unwanted data.

Let O be the number of people who like orange
Let L be the number of people who like lemon
Let G be the number of people who like Grape

Number of people = n(O U L U G) + n(O U L U G)'
n(O U L U G) = n(O) + n(L) + n(G) - n(O ∩ L) - n(L ∩ G) - n(O ∩ G) + n(O ∩ L ∩ G).
n(O U L U G) = 85 + 90 + 65 - 45 - 40 - 30 + 15
n(O U L U G) = 140

Total Number of people = n(O U L U G) + n(O U L U G)'
Total Number of people = 140 + 25 = 165.

Number of people who liked only orange = n(O) - n(O ∩ L) - n(O ∩ G) + n(O ∩ L ∩ G)
Number of people who liked only orange = 85 - 45 - 30 + 15 = 25.

sfsdfs · Answer

GMATNinja  would love your help here! i'm trying out your 'grid' method on youtube but am not sure how to make it work in this example... i'd really appreciate it if you could help guide me? thank you so much

GMATNinja · Answer

The "grid" method works with two-way overlapping sets. For example, it works if you have a group of people who fall into one of these 4 categories: 1) drink beer and sake, 2) drink beer but not sake, 3) drink sake but not beer, 4) drink neither beer nor sake.

The grid won't work here because it's a three-way overlapping set, as illustrated by the Venn diagram earlier in the thread.

In this  youtube video , we go through an example of a three-way overlapping set  starting at around the 10:50 mark . Check it out and then give this one another shot? And of course if you're still stumped, let us know.

Su1206 · Answer

GMATNinja 
can u please tell us how to use your video method to solve this. I am not able to apply your method to get the correct answer here. Thank you.

Krunaal · Answer

Alternatively, you can also use the formula = T - n = A + B + C - (All_2) + All_3 to find the total number of people, and for Only Orange = Total Orange - (All_2 that include orange) + All_3

(1) T - n = A + B + C - (All_2) + All_3

=> T - 25 = 240 - 115 + 15
=>  T = 165

(2) Only Orange = Total Orange - (All_2 that include orange) + All_3

=> Only Orange = 85 - (45 + 30) + 15
=>  Only Orange = 25

GMATNinja · Answer

You're right: this is actually not like the one in our video, sorry! Here we need to figure out the # of people who only like 1 or 2 flavors. Let's start with the number of people who only like 2 flavors (but not all 3).

We know that 45 like orange and lemon, 30 like orange and grape, and 40 like lemon and grape. But those numbers also include people who like all three flavors (if you like orange, lemon, and grape, then of course you also like each of the three pairs).

So you have to subtract 15 (the # of people who like all three) to find the number of people who only like two flavors:

orange and lemon: 45-15 = 30
 
 orange and grape: 30-15 = 15
 
 lemon and grape: 40-15 = 25

That's 70 people total who like ONLY two flavors (and not all three). Now you have to figure out how many people like ONLY one flavor. Let's try orange first:

We know that 85 people like orange, but that includes people who (1) ONLY like orange, (2) like only orange and lemon (30 people), (3) like only orange and grape (15 people), (4) like all three flavors (15).
 
 To find the people who only like orange, you have to subtract (2), (3), and (4) from 85: 85-30-15-15 =  25  (the first half of the answer).

Using similar logic, the # of people who only like grape is 65-15-25-15 = 10

And the # of people who only like lemon is 90-30-25-15 = 20

So the # of people who only like 1 flavor is 25 (only orange) + 10 (only grape) + 20 (only lemon) = 55.

Adding everything up, we get 25 (don't like any) + 55 (like 1 flavor) + 70 (like 2 flavors) + 15 (like all 3) =  165  total people.

A number of people were asked whether they liked drinks of orange, lem

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A number of people were asked whether they liked drinks of orange, lem

Prep Toolkit

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

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From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards