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605-655 (Medium)|   Math Related|         
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OmerKor
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 25 Dec 2024, 12:29
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Hi everyone :)

We are dealing with overlap question (I will use double matrix method).

PastaNoPastaTotal
Grill ?100
NoGrillNN=>25140
Total15090240

Min:
PastaNoPastaTotal
Grill3565100
NoGrill25140
Total15090240

Max:
PastaNoPastaTotal
Grill1000100
NoGrill90140
Total15090240

Our Answers are: Min=35 Max=100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 25 Dec 2024, 13:11
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first let neither=25

240=100+150-both+25
both=35

Max both can be when none eats only grilled. So both =100 max

Answer 35 & 100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 25 Dec 2024, 14:05
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The maximum number of attendees who chose both could be the minimum of both groups min(100,150) = 100
The minimum number of attendees who chose both could be given by: 240 = 150 + 100 -overlaps +25 ->
35 = overlaps

Thus min = 35
max = 100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 25 Dec 2024, 14:16
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x=attendees who chose both pasta and grilled dishes
n=attendees who chose other food options

240 = 150+100-x+n
x=n+10

The maximum is when all the attendees who chose grilled dishes, also chose pasta dishes: 100 (and n would be 90>25).

The minimum is when n=25 and x=35.

minimum=35 and maximum=100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 25 Dec 2024, 14:43
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T = G1 + G2+ ... + GN - OVERLAPS + NONE
240 = 150 + 100 -X +25

So here the minimum number of overlaps can be the number of people involved in two activies so that the equation is reasonable ->

35 = x

Maximum, on the other hand is more of a logic-based answer. If every member from G1 is also involved in G2, then the duplicates are maximum, so it will be 100, since 150 can't be it.

x = 100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 25 Dec 2024, 15:07
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There must be at least 10 attendees who chose both pasta and grilled dishes because the number of attendees is 240 and 150+100=250.
Moreover, as the number of attendees who chose other food options is at least 25, we have to add 25 to 10 to calculate the minimum:
10+25=35

The maximum is reached when all the 100 attendees who chose grilled dishes also chose pasta dishes.

Answers: Minimum 35, Maximum 100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 25 Dec 2024, 16:42
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For minimum number of attendees who could have chosen both pasta and grilled dishes:

Let's keep the number of people who like neither pasta nor grilled dishes at it's least so that we can maximize the number of people who like Pasta but not like grilled dishes.
Upon calculating, we find that the minimum number of attendees who could have chosen both pasta and grilled dishes is 35 (reference table below):
PastaNot PastaTotal
Grilled3565100
Not Grilled11525140
Total15090240

For maximum number of attendees who could have chosen both pasta and grilled dishes:

Let's keep the number of people who like neither pasta nor grilled dishes at it's maximum so that we can minimize the number of people who don't like Pasta but like grilled dishes.
Upon calculating, we find that the maximum number of attendees who could have chosen both pasta and grilled dishes is 100 (reference table below):
PastaNot PastaTotal
Grilled1000100
Not Grilled5090140
Total15090240

Final Answer : Minimum overlap: 35 and Maximum overlap: 100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 25 Dec 2024, 19:43
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T=240; P = 150; G = 100; Neither Pasta nor Grilled (O) >= 25;
Now, in the first observation, the minimum value of O could be 25 and the maximum could be 240 - 150 = 90; (Since most people choose pasta)

240 = 150 + 100 - B + 25
B = 35
240 = 150 +100 - B + 90
B = 100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 25 Dec 2024, 22:53
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Understanding the Problem
  • Total Attendees: 240
  • Chose Pasta: 150
  • Chose Grilled Dishes: 100
  • Chose Other Options: At least 25 (neither pasta nor grilled)

Finding Minimum and Maximum Overlap (Both Pasta and Grilled)
Let’s find out how many people chose both pasta and grilled dishes.

Step 1: Calculate People Who Chose Either Pasta or Grilled or Both
Since at least 25 chose other options:
People who chose pasta or grilled or both = 240 − 25 = 215
Step 2: Use the Inclusion-Exclusion Principle
Total (Pasta ∪ Grilled)=Pasta+Grilled−Both
215 = 150 + 100 − Both
Both = 150 + 100 − 215 = 35

Minimum number of people who chose both:35

Step 3: Determine Maximum Possible Overlap
The maximum number who could have chosen both is limited by the smaller group (Grilled dishes):
Maximum Both = 100
(Everyone who chose grilled dishes also chose pasta.)

Maximum number of people who chose both:100

Final Answer

  • Minimum: 35
  • Maximum: 100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 25 Dec 2024, 23:06
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Problem Breakdown

We need to determine:

1. The minimum number of attendees who could have chosen both pasta and grilled dishes.


2. The maximum number of attendees who could have chosen both.



Information provided:

Total attendees = 240

Attendees who chose pasta = 150

Attendees who chose grilled dishes = 100

Attendees who chose neither = at least 25


Formula:

Using the principle of inclusion and exclusion for sets:

Total attendees = Pasta only + Grilled only + Both + Neither

Let the number of attendees who chose both be x .

1. From the formula:



240 = (150 - x) + (100 - x) + x + Neither

240 = 250 - x + Neither

240 = 250 - x + 25 or more.

Minimum :

For x to be minimum, the value of neither must be maximum (>=25). Assuming :

240 = 250 - x + 25

240 = 275 - x

x = 35

Maximum :

For x to be maximum, if we assume that the maximum number of people who had both pasta & grilled dishes are equal to the number of people who had grilled dishes which is equal to 100 then the number of people who had pasta dishes only can be assumed to be (150-100=50 people). Thus 150 people have either pasta or grilled dishes & hence the number of people who have neither dishes would be equal to 240-150=90 people which satisfies our condition of their being at least 25 attendees chose other food options—neither pasta nor grilled dishes. Hence max number of people who have both dishes is equal to 100.


Final Answer:

Minimum : 35

Maximum : 100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 00:19
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total= 240
25 neither choose paste or grilled, so either or both= 215
215=150+100-min
min=35
max=100 (limited by smaller set)
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 00:22
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Please find the attached solution.
Attachments

TPA24.jpg
TPA24.jpg [ 297.98 KiB | Viewed 811 times ]

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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 00:47
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Total attendees = 240
150 attendees chose pasta dishes.
100 attendees chose grilled dishes
Attendees who chose neither pasta nor grilled dishes = at least 25

So, P U G = 240 - 25 = 215

People who choose only 1 dish = 150+100 = 250

Minimum people who could have chosen both pasta and grilled dishes = 250-215 = 35
Maximum people who could have chosen both pasta and grilled dishes = lower of 100 and 150 = 100



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At a regional food festival, there were 240 attendees in total.

  • 150 attendees chose pasta dishes.
  • 100 attendees chose grilled dishes.
  • At least 25 attendees chose other food options—neither pasta nor grilled dishes.

Select for Minimum the minimum number of attendees who could have chosen both pasta and grilled dishes, and select for Maximum the maximum number of attendees who could have chosen both. Make only two selections, one in each column.
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 00:53
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I would say (35,100)
P: Number of attendees who chose pasta.

G: Number of attendees who chose grilled dishes.

O: Number of attendees who chose other food options (neither pasta nor grilled).
From the problem:

P
=
150
,
G
=
100
,
O
≥
25.
P=150,G=100,O≥25.
The relationship between these sets is:

P
+
G
−
B
+
O
=
240
,
P+G−B+O=240,
where B is the number of attendees who chose both pasta and grilled dishes.

Rearranging:

B=P+G+O−240.
Substitute
P
=
150
P=150,
G
=
100
G=100, and
O
≥
25
O≥25:

B=150+100+O−240⟹B=10+O.

Allow to determine min and max
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 01:02
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Hi All,

for minium people who have chose pasta and grilled dishes we can

240-25=215 ( total people - those who ate other dishes)

this 215 will also be having those who ate both,

now taking out people who ate grilled from this 215-100=115.

these people will be only pasta eaters,

but total pasta eaters are 150, there fore 150-115=35

therefore minimum is 35


for maximum,

we can assume all those who ate Grilled food also ate pasta (we cannot do other way around a no of pasta eaters are more than grilled food eaters)

therefore 100 people ate both grilled food and pasta.

hence maximum is 100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 01:24
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At a regional food festival, there were 240 attendees in total.

150 attendees chose pasta dishes.
100 attendees chose grilled dishes.
At least 25 attendees chose other food options—neither pasta nor grilled dishes.

Select for Minimum the minimum number of attendees who could have chosen both pasta and grilled dishes, and select for Maximum the maximum number of attendees who could have chosen both. Make only two selections, one in each column.



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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 01:41
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At a regional food festival, there were 240 attendees in total.

  • 150 attendees chose pasta dishes.
  • 100 attendees chose grilled dishes.
  • At least 25 attendees chose other food options—neither pasta nor grilled dishes.

Select for Minimum the minimum number of attendees who could have chosen both pasta and grilled dishes, and select for Maximum the maximum number of attendees who could have chosen both. Make only two selections, one in each column.
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Ans: Minimum 35, Max = 100

Total 240 , neither = 25 so remaining total for P and G = 215
out of 215 we know 150 choose P and 100 Chose G
so for minimum both = 100+150 -215 = 35
Max is the minimum of those 2 categories = 100
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 02:13
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The maximum number is 100 (if all 100 attendees chose grilled dishes also choose pasta).

The minimum number is:

We have: 150+100−x=215. ==> x = 35
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At a regional food festival, there were 240 attendees in total.

  • 150 attendees chose pasta dishes.
  • 100 attendees chose grilled dishes.
  • At least 25 attendees chose other food options—neither pasta nor grilled dishes.

Select for Minimum the minimum number of attendees who could have chosen both pasta and grilled dishes, and select for Maximum the maximum number of attendees who could have chosen both. Make only two selections, one in each column.
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 03:01
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Minimum = 35 and maximum = 100

To solve for minimum: 240 - 25 = 215 choosing pasta, grilled, or both
150 + 100 = 250 --> 250 - 215 = 35
115 chose only pasta, 65 chose only grilled, 35 chose both, and 25 chose neither.

To solve for maximum: at most 100 can choose both since 100 chose grilled, and since attendees choosing neither just has to be greater than 25, this is possible. If 100 chose both, then 50 chose only pasta, 0 chose only grilled, and 90 chose neither.
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 03:01
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Using the two set formula,

Total = Pasta + Grilled - Both + Neither
240 = 150 + 100 - Both + 25
Both = 35 (Minimum)

Since the question says atleast 25 chose other, this number could be higher upto 100 (since that would imply all those who liked grilled also like pasta).

Correct Answer - Minimum: 35 , Maximum: 100
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This question was provided by GMAT Club
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At a regional food festival, there were 240 attendees in total.

  • 150 attendees chose pasta dishes.
  • 100 attendees chose grilled dishes.
  • At least 25 attendees chose other food options—neither pasta nor grilled dishes.

Select for Minimum the minimum number of attendees who could have chosen both pasta and grilled dishes, and select for Maximum the maximum number of attendees who could have chosen both. Make only two selections, one in each column.
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