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605-655 Level|   Math Related|         
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Yaswanth9696
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 03:07
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Total=240
Pasta=150
Grilled=100
Neither/other=at least 25

The formula Total - None = A+B - both
therefore 240-25=100+150-Both---> both= 35--- So the minimum value of both is 35.

Now for maximum consider Pasta and Grilled, The maximum overlap is 100 between these two,
so 240-100=100+150-Both---> Both=250-140-- 110. the values must be between 35 and 110 inclusive.

So minimum is 35 and maximum is 100 as per the options.
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 03:13
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If we take Venn's diagram logic and mark people who chose both grill and pasta as X and those who chose neither as N, then the total of 240 is:
\(240=150+100-x+n,\) as we double count the X value for both visitor types.
From here, \(x-n=10\) and \(x=n+10\)

Knowing that minimum n is 25, minimum x is 25.
As for maximum, the only case where a huge number of people can choose both dishes is when all grill-lovers also opt for pasta as well.
Then, the maximum value is 100 (and we can't go higher as we have run out of grill-lovers by then).

Therefore, the answer is 35 for min and 100 for max.
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Let the attendees who neither pasta nor grilled dishes be n. So n >= 25. Let the attendees who could have chosen both pasta and grilled dishes be b

240 = 150 + 100 - b + n
240 = 250 - b + n
n - b +10 = 0
n + 10 = b , where n >= 25

Min(b) happens when n = 25. So min (b) = 35

Max(b) happens when the overlapping of both pasta and grilled dishes is maxed. So Max(b) = 100 (grilled dishes)

Answer: Min = 35, max = 100
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Pasta DishesNot Pasta DishesTotal
Grilled Dishes100
Not Grilled Dishes> / = 25140
Total15090240

For the minimum, let the neither pasta nor grilled be 25
So,

Pasta DishesNot Pasta DishesTotal
Grilled Dishes3565100
Not Grilled Dishes11525140
Total15090240

Minimum = 35

For maximum, the neither pasta nor grilled would be 90 (i.e. the lesser total value)

Pasta DishesNot Pasta DishesTotal
Grilled Dishes1000100
Not Grilled Dishes5090140
Total15090240

Maximum = 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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For min:240-25=215=150-x+100-x+x
=>x=35

For max: there is total overlap of 100 grilled dishes under 150 pasta, and 90 are niether

=>x=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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We are tasked with determining the minimum and maximum number of attendees who could have chosen both pasta and grilled dishes, based on the following information:
  1. Total attendees: 240.
  2. Pasta dishes: 150 attendees.
  3. Grilled dishes: 100 attendees.
  4. At least 25 attendees chose neither pasta nor grilled dishes.

Step 1: Calculate the total number of attendees who chose pasta or grilled dishes
If at least 25 attendees chose neither pasta nor grilled dishes, the number of attendees who chose pasta or grilled dishes is:
240−25=215
Thus, the sum of attendees who chose pasta only, grilled only, and both is:
Pasta only+Grilled only+Both=215

Step 2: Determine the relationship between pasta, grilled, and both
Let:
  • A=Pasta only,
  • B=Grilled only,
  • C=Both pasta and grilled dishes
The total number of attendees choosing pasta is:
A+C=150
The total number of attendees choosing grilled dishes is:
B+C=100
From the total, we know:
A+B+C=215
Substitute A=150−C and B=100−C into the total equation:
(150−C)+(100−C)+C=215
Simplify:
250−C=215.
Solve for C:
C=35.
Thus, the minimum number of attendees who chose both pasta and grilled dishes is 35.

Step 3: Calculate the maximum number of attendees who could have chosen both
The maximum occurs when the overlap between pasta and grilled dishes is maximized. The total number of attendees who chose both pasta and grilled cannot exceed the smaller of the two totals (since no more than 100 people could have chosen grilled dishes):
C=100.

Final Answer:
  • Minimum: 35.
  • Maximum: 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.
MinimumMaximum
35
45
90
100
150

(1) Select for Minimum the minimum number of attendees who could have chosen both pasta and grilled dishes

25 attendees chose other food options --- neither pasta nor grilled dishes

then,

Pasta Non-Pasta
Grilled 35 65 100
Non-Grilled 115 25 140
150 90 240


the minimum number of attendees who could have chosen both pasta and grilled dishes = 35


[color=#e82a1f](2) Select for Maximum the minimum number of attendees who could have chosen both pasta and grilled dishes

100 attendees chose both food options --- pasta as well as grilled dishes

then,

Pasta Non-Pasta
Grilled 100 00 100
Non-Grilled 50 90 140
150 90 240


the maximum number of attendees who could have chosen both pasta and grilled dishes = 100[/color]


Minimum = 35; Maximum = 100 is the CORRECT answer
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PastaNo pastaTotal
Grilledx100
Not grilled>=25140
15090240

Minimum -

PastaNo pastaTotal
Grilled3565100
Not grilled11525140
15090240

minimum = 35

Maximum -

PastaNo pastaTotal
Grilled1000100
Not grilled5090140
15090240

Maximum = 100

FINAL ANSWER -

minimum = 35, maximum = 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, 05:36
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we can do this by creating matrix

Grilled not Grilled Total
Pasta Min //Max 150
not pasta 90-25 // 90-90 (25)+ 240-150=90
Total 100 140 240

now in order to get min of Pasta we can have max of not pasta with Grilled which could be when we take not grilled and not pasta to be only 25
which is 90-25=65 and if not pasta is 65 then 35 is min Pasta ,
and for maximum we can have not grilled to be 90 that will make Pasta to be 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.
Show more
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As per the given information , PD (150) + GD(100) - B(both) + N(neither) =240
150+100-B+N=240
B-N=10
Also N>=25
So least B is 35
Highest B is 100 (GD is highest that both dished people could have chose)
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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.
Show more

Total attendees (T) = 240
People who chose Pasta dishes (P) = 150
People who chose Grilled dishes (G) = 100
At least 25 people chose neither Pasta nor Grilled dishes (N) = 25<

We need to find the minimum and maximum number of attendees who could've chosen Pasta or Grilled dishes.
T = P + G + N - both
240 = 150 + 100 + 25 - both
both = 275 - 240
both = 35
So, the minimum possible attendees who chose both is 35.

Maximum possible attendees can only be the least number of attendees among either Pasta or Grilled, in this case Grilled<Pasta
So, the maximum possible attendees is 100.
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Min is possible when we try to put max people as far apart. SIce 25 are none hence we have 240-25=215 people to fit and have 100+150=250 choices so min overlap=250-215=35

Max= If we put all 100 in pasta circle then we have max overlap

Ans 35,100
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Total no. of attendees= 240
Pasta= 150 attendees
Grilled= 100 attendees
At least 25 attendees choose other food opt.

therefor, 240-25= 215 attendees choose either pasta or grilled if we take the least value of people choosing other food opt. as 25.

P+G-B=215
=> 150+100-B=215
=> B= 35
So, min val.= 35

For max val.-
To maximize the number of attendees who chose both dishes, we need to minimize the number of attendees who chose only one of the two options (pasta or grilled). The maximum number of attendees who could have chosen both dishes occurs when all 100 grilled dish attendees are also among the 150 pasta dish attendees. In this case, the number of attendees who chose both dishes would be the maximum possible:

B= min(P,G)= min(150,100)= 100
therefor, Max. val. = 100

Ans- Min= 35, Max= 100.
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Minimum =35
Maximum= 100

Total Attendees- (Neither Pasta nor Grilled dish)= Pasta chooser + Grilled dish Chooser - Both Chooser

240-neither= 150+100-Both
Both= 10+neither

For Both to be minimum, neither be minimum

Neither is least=25
Both= 10+25= 35

Since Grilled lover is 100 so Both grilled and pasta can't exceed 100 so max is 100
Both =100= 10+neither
Neither=90 which is possible

Hence 35, 100 is ans

Thanks!


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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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PastaNo PastaTotal
Grillab100
No Grillcd140
Total15090240

d>=25
according to atleast 25 people choosing no pasta no grill, we get min and max values for a,b,c

Since total people not eating pasta is 90,
d is 25 to 90

b = 0 to 65
c = 50 to 115

Using b and c, we get the min and max values of a

100 - 0 = 100
100 - 65 = 35
150 - 50 = 100
150 - 115 = 35

35 and 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.
Show more
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12 Days of Christmas GMAT Competition - Day 8: At a regional food fest 26 Dec 2024, 07:56
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Total attendees = 240
Attendees who chose pasta = 150
Attendees who chose grilled dishes = 100
At least 25 attendees chose neither pasta nor grilled
Pasta only = 150 - x
Grilled dishes only = 100 - x

Let x be the number of attendees who chose both pasta and grilled dishes

Pasta Only + Grilled Only + x + Neither = 240
(150 − x) + (100 − x) + x + Neither = 240
Simplifying,
Neither = x - 10

Since people who chose neither are >= 25,
x - 10 >= 25
x >= 35

Therefore, the minimum is 35

x cannot be more than 100, because the maximum number of people who chose grilled dishes are 100

Maximum = 100
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Total attendees = 240

Pasta = 150
Grilled = 100
Other('o') >= 25

To find : Minimum of both pasta and grilled, Maximum of both pasta and grilled

Total attendees = Pasta + Grilled - Both pasta and grilled + Other dishes

Let Both pasta and grilled be represented by 'b'

240 = 150 + 100 - b + o

o - b = -10
o = b - 10

o >=25

If o = 25

b = 35

Maximum of both pasta and grilled can be 100(All people who took Pasta also took Grilled)


Min = 35, Max = 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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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.
Show more

Let the number of both pasta and grilled dishes be x

Minimum number
Total = Pasta + Grilled + neither pasta nor grilled - x
240 = 150+100+25-x
240 = 275-x
x = 35
Therefore minimum number is 35

Maximum number
As "at least" 25 attendees chose other food options => can be more than 25
Therefore the maximum number will be the lower of 100 and 150, i.e. 100
Therefore maximum number is 100
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Total attendees: 240
Attendees who chose pasta: 150
Attendees who chose grilled dishes: 100
At least 25 attendees chose other food options, meaning they chose neither pasta nor grilled dishes.

Total number who chose either pasta or grilled = 240 - 25 = 215

We need to find minimum and maximum number who chose both pasta and grilled dishes
So,
Minimum:
Total number who chose either pasta or grilled = 215
Pasta only + Grilled only + Both = 215
150+100+ Both = 215
Both(min) = 35

Maximum:
Highest overlap occurs when as many people as possible are included in both categories. This is constrained by the difference in number of dishes i.e, pasta being 150 and grilled being 100
So, the maximum number of attendees who both pasta and grilled = 100

Minimum number of attendees who both pasta and grilled = 35
Maximum number of attendees who both pasta and grilled = 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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Total attendees (T) = 240
People who chose Pasta dishes (P) = 150
People who chose Grilled dishes (G) = 100
At least 25 people chose neither Pasta nor Grilled dishes (N) = 25<

We need to find the minimum and maximum number of attendees who could've chosen Pasta or Grilled dishes.
T = P + G + N - both
240 = 150 + 100 + 25 - both
both = 275 - 240
both = 35
So, the minimum possible attendees who chose both is 35.

Maximum possible attendees can only be the least number of attendees among either Pasta or Grilled, in this case Grilled<Pasta
So, the maximum possible attendees is 100.
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