GMAT Club Olympics 2024 (Day 8): Emily visited five different

Question

Emily visited five different restaurants in Paris last week, spending some non-zero amounts at each. Did she spend more than the median amount at any of the five restaurants?

(1) She spent the same total amount at any three of the five restaurants.
(2) She did not spend less than the average (arithmetic mean) amount at any of the restaurants.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Bunuel · Accepted Answer

GMAT Club Official Explanation:

Emily visited five different restaurants in Paris last week, spending some non-zero amounts at each. Did she spend more than the median amount at any of the five restaurants?

(1) She spent the same total amount at any three of the five restaurants.

This implies that all amounts must be equal. If this were not true, then at least two of the amounts would have been different, say x < y. In this case, the sum of two amounts plus x would be less than the sum of the same two amounts plus y, making the statement false. Therefore, all amounts must be equal. Consequently, Emily did not spend more than the median amount at any of the five restaurants. Sufficient.

(2) She did not spend less than the average (arithmetic mean) amount at any of the restaurants.

This implies that she also did not spend more than the average amount at any of the restaurants. Therefore, all amounts must be equal. Consequently, Emily did not spend more than the median amount at any of the five restaurants. Sufficient.

Answer: D.

jack5397 · Answer

IMO - B

St:1. She spent the same total amount at any three of the five restaurants.

Case 1 - A,x,x,x,B - Yes 
Case 2 - x,x,x,A,B - Yes 
Case 3 - A,B,x,x,x - No . hence insufficient.

St:2 She did not spend less than the average (arithmetic mean) amount at any of the restaurants.

This simply means all the values are equal and you cannot have any value less than mean.  Hence sufficient -- Gives No as a answer

Fido10 · Answer

(1) She spent the same total amount at any three of the five restaurants. 
This simply means that she spent the same amount an all five restaurants, then the median is the amount she spent.
 Did she spend more than the median amount at any of the five restaurants? Answer is NO 
Sufficient,

(2) She did not spend less than the average (arithmetic mean) amount at any of the restaurants.

If She did not spend less than the average amount at any of the restaurants, then she spends exactly the average at any of the restaurants, because there is no configuratn where she can spent more than the average on some restaus and the exact average on the others, in this case the average won't be what is is actually !

So she spent exactly the average on any restau, then the median is the average
 Did she spend more than the median amount at any of the five restaurants? Answer is NO , Sufficient,

Answer is D

AthulSasi · Answer

(1) She spent the same total amount at any three of the five restaurants.
(2) She did not spend less than the average (arithmetic mean) amount at any of the restaurants.
------------
 Statement 1. 
Suppose the 5 amounts in their ascending orders are
a,b,c,d,e
If some of any three numbers are equal out of these 5, this implies that all the numbers are same. 
Hence Emily did not spend more than the median amount at any restaurant.
Sufficient.

Statement 2. 
Suppose the 5 amounts in their ascending orders are
a,b,c,d,e
If there are no amount less than the average, then there cannot be any amount greater than the average also. This implies that all the amounts are same.
Hence Emily did not spend more than the median amount at any restaurant.
Sufficient.

Hence correct asnwer D

jairovx · Answer

Statement (1):
- Emily spent the same total amount at any three of the five restaurants.
- Let a ≤ b ≤ c = d = e be the amounts spent.
- The median is c.
- a + b + c = a + b + d = a + b + e, so c = d = e.
- a ≤ b < c or a = b = c.
- Emily spent the median amount at least in three restaurants.
- Statement (1) ALONE is sufficient.

Statement (2):
- Emily did not spend less than the average amount at any restaurant.
- Mean = (a1 + a2 + a3 + a4 + a5) / 5.
- a1 >= Mean, a2 >= Mean, ..., a5 >= Mean.
- All amounts must be equal: a1 = a2 = a3 = a4 = a5 = Mean.
- Median amount is equal to any spent amount.
- Statement (2) ALONE is sufficient.

IMO: D

A_Nishith · Answer

To determine if Emily spent more than the median amount at any of the five restaurants, let's analyze the information given in each statement separately and then together.

Statement (1) Analysis 
Emily spent the same total amount at any three of the five restaurants.

Let's denote the amounts Emily spent at the five restaurants as 
A,B,C,D,E such that A≤B≤C≤D≤E. The median amount spent is C.

From statement (1), the total amount spent at any three restaurants is the same. This implies the sum of any three amounts from 
A,B,C,D,E is equal. Let's denote this sum as S. 
Therefore, we can write:
A+B+C=S
A+B+D=S
A+B+E=S
A+C+D=S
A+C+E=S
A+D+E=S
B+C+D=S
B+C+E=S
B+D+E=S
C+D+E=S

From this, we can infer that 
A,B,C,D,E must be equal because if the total of any three amounts is the same, it means each amount must be equal. 
So, A=B=C=D=E.

Since all amounts are equal, the median amount C is equal to any of the amounts Emily spent, so Emily did not spend more than the median amount at any restaurant.

Therefore,  statement (1) alone is sufficient to answer the question.

Statement (2) Analysis 
Emily did not spend less than the average (arithmetic mean) amount at any of the restaurants.
The average amount spent is (A+B+C+D+E)5
​Since Emily did not spend less than the average amount at any restaurant, it implies:

A≥ (A+B+C+D+E)/5
​B≥ (A+B+C+D+E)/5
​C≥ (A+B+C+D+E)/5
D≥ (A+B+C+D+E)/5
E≥ (A+B+C+D+E)/5
For all the above inequalities to hold, all amounts A,B,C,D,E must be equal to the average amount. 
This is because if any one of them were less than the average, the corresponding inequality would not hold.

Since all amounts are equal, the median amount C is equal to any of the amounts Emily spent, so Emily did not spend more than the median amount at any restaurant.

Therefore, statement (2) alone is sufficient to answer the question.

Combining Both Statements
Since both statements alone are sufficient, there is no need to combine them.

The correct answer is: D

Abhijeet24 · Answer

To determine whether Emily spent more than the median amount at any of the five restaurants, we need to analyze the information given in the statements.

Statement (1): 
She spent the same total amount at any three of the five restaurants.

Let the amounts spent at the five restaurants be $ a_1, a_2, a_3, a_4, a_5 $, and assume $ a_1 \leq a_2 \leq a_3 \leq a_4 \leq a_5 $.

Given that the sum of the amounts spent at any three restaurants is the same, we can write:
\ 
\

Given the symmetrical nature of the sum being equal for any combination of three, it implies:
\

Since Emily spends the same amount at each restaurant, each amount is equal to the median. Therefore, she does not spend more than the median amount at any restaurant.

Statement (1) alone is sufficient.

Statement (2): 
She did not spend less than the average (arithmetic mean) amount at any of the restaurants.

The average (arithmetic mean) amount spent is:
\

Given that she did not spend less than the average amount at any restaurant, this implies:
\

For all five amounts to be equal to or greater than the average, it must be the case that:
\

Therefore, each amount is equal to the median. Thus, Emily did not spend more than the median amount at any restaurant.

Statement (2) alone is sufficient.

Conclusion: 
Both statements individually provide sufficient information to determine that Emily did not spend more than the median amount at any of the five restaurants.

The answer is D: Each statement alone is sufficient to answer the question.

HarshaBujji · Answer

Need to find the relation between spend amount and median amount.

(1) She spent the same total amount at any three of the five restaurants. 
This implies all the totals are same;

Hence median and all other spent amounts are the same. Hence she cannot spend more than the median. 
A or D is the perfect choice.

(2) She did not spend less than the average (arithmetic mean) amount at any of the restaurants.

This implies that the average and all other spends are the same.
Hence median and all other spent amounts are the same. Hence she cannot spend more than the median.

So D is the perfect choice.

BGbogoss · Answer

Emily visited five different restaurants in Paris last week, spending some non-zero amounts at each. Did she spend more than the median amount at any of the five restaurants?

(1) She spent the same total amount at any three of the five restaurants. 
This statement means that she payed the same price in every restaurant. If she payed a different price in any of the restaurants, this statement couldn't be true. Since she payed the same prive in every restaurant, she didn't pay more that the median in any of the five restaurant. This statement alone is sufficient.

(2) She did not spend less than the average (arithmetic mean) amount at any of the restaurants.
This statement also means tht she payed the same price in every restaurant because if she payed more than the mean in any of the restaurants, she should have payed less than te mean to counterbalance. 
Thus the mean and the median are equal to the price she payed in every restaurant. Consequently, she didn't pay more that the median in any of the five restaurant. This statement alone is sufficient.

Answer D

xhym · Answer

Let's analyze each statement:

Statement 1

We know that she spent the same amount at any three of the 5 restaurants which means that, for example:

a_1+a_2+a_3 = a_2+a_3+a_4  which means that  a_1 = a_4 .

Similarly,  a_2+a_3+a_4= a_3+a_4+a_5  which means that  a_2 = a_5 .

We can continue doing this and we will find that all of the values are equal to each other! That means she did not spend more than the median amount at any of the restaurants since all the values are the same.

-> Statement 1 is sufficient.

Statement 2

If she did not spend less than the average at any restaurants, it means by definition that she cannot have spent more either as she spent the same amount in each restaurant. If she spent even a dollar less or more at an restaurant, then she would have a variance and this statement would not be possible. That means she did not spend more than the median amount at any of the restaurants since all the values are the same.

-> Statement 2 is sufficient.

The answer is  D. EACH statement ALONE is sufficient to answer the question asked.

Milkyway_28 · Answer

The question states that Emily visited five different restaurants and spent non-zero amounts at each, meaning above 0. The median is defined as the value at the midpoint of a set

Statement (1) "She spent the same total amount at any three of the five restaurants."

" Any  three of the five restaurants" simply means 'all the restaurants'. Therefore Emily spent the same total amount at each restaurant. Therefore, we can answer the question if she spent more than the median amount at any of the five restaurants, which is no; she spent the median.

Statement (2) "She did not spend less than the average (arithmetic mean) amount at any of the restaurants."

If she did not spend less than the average then it follows that she did not spend more than the average. She simply spent the average, which in this case would also be the mean. So we can also answer the question with the information provided in this statement

Each statement is sufficient on its own. Answer is D

Lizaza · Answer

Let's see the answer options to determine the median.

(1)
If  any  three restaurants have the same total, it means  all  five have the same total.
Therefore, all values are the same, and the median is also the same.  Sufficient by itself.

(2)
Not spending less than average anywhere again means that  all  five restaurants have the same total, and the same median.  Sufficient by itself.

The correct asnwer is D.

DG1989 · Answer

Emily visited five different restaurants in Paris last week, spending some non-zero amounts at each. Did she spend more than the median amount at any of the five restaurants?

(1) She spent the same total amount at any three of the five restaurants.
(2) She did not spend less than the average (arithmetic mean) amount at any of the restaurants.

Solution:  Given that,
 
 Emily visited five different restaurants and spent some non-zero amounts at each.

Let's assume
Amount spent at first restaurant = a
Amount spent at second restaurant = b
Amount spent at third restaurant = c
Amount spent at fourth restaurant = d
Amount spent at fifth restaurant = e

We need to find out if she spent more than the median amount.

Statement 1:    She spent the same total amount at any three of the five restaurants. 
This means, assuming the total amount spent is say, X
a + b + c = X
or b + c + d = X
or c + d + e = X
or a + d + e = X
Similarly, the combination of the sum of the amount spent at any 3 restaurants will be X

Hence, a = b = c = d = e
This implies the median amount is also equal to the amount spent at each restaurant. Therefore, Emily did not spend more than the median amount at any restaurant.
  SUFFICIENT

Statement 2:    She did not spend less than the average (arithmetic mean) amount at any of the restaurants. 
The average amount (say Y) spent across all restaurants =  \frac{(a+b+c+d+e)}{5} 
Y =  \frac{(a+b+c+d+e)}{5} 
According to the statement,
a ≥ Y
b ≥ Y
c ≥ Y
d ≥ Y
e ≥ Y

Since, the sum of all the amounts = 5Y
This is possible only when all amounts are equal. Hence, a = b = c = d = e
This implies the median amount is also equal to the amount spent at each restaurant. Therefore, Emily did not spend more than the median amount at any restaurant.
  SUFFICIENT

The right answer is Option D

Urja08 · Answer

From the ques :
is Amt of money in any 5 rest > Median (of money spend in 5 rest)
 From statement 1 : 
if a1,a2,a3,a4,a5= money spend in 5 rest 
a1+a2+a3=a2+a3+a4=a3+a4+a5
For the total to be same every time 
a1=a2=a3=a4=a5
Suff 
 From Statement 2 : 
a(money in any 5 rest)>=avg money spent in any rest
a1>=avg; a2>=avg ; a3>=avg ; a4>=avg  ;a5>=avg  
median =a3
suff 
 Ans D

smile2 · Answer

let's say the amounts spent at five different restaurants in Paris are a, b, c, d, e 
From statement 1, we are given that the total amount is same at any three of the five restaurants
then a+b+c = a+c+d = a+d+c = b+c+d = b+d+e = c+d+e = d+e+a =.....
From these equations, we know that a = d = c = b = e  
So, we know that Emily did not spend more than median amount at any of the five restaurants. 
Statement 1 alone is sufficient.

From statement 2, Emily did not spend less than the average (arithmetic mean) amount at any of the restaurants.
if there is no amount spent less than average, that means there is no amount spent more than average amount. 
Statement 2 alone is sufficient.

Therefore, ans choice is D.

Suboopc · Answer

if a,b,c,d,e are the amount paid in each restaurants

Statement 1:
sum of any 3 amounts are equal
a+b+c = a+b+d
c = d

similarly we can prove that all amounts are equal.
None of the amounts are greater than median
Sufficient

Statement 2:
If none of the amounts are less than average, then none of the amounts are greater than average.
Thus none of the amounts are greater than median.
Sufficient.

D

captain0612 · Answer

Chocie D

Emily visited 5 different restaurants in paris.

Question: Did she spend more than the median at any restaurant?

Deconstruct question: 2 different cases can arise

Case 1: There's atleast 1 restaurant at which she spent less than the median => there's at least 1 restaurant at which she spent more than the median

Case 2 : Oppsotie of case 1 => She spent the equal amount at all 5 restaurants

So, according to the question, we gotta find whether Emily has spent the same amount at each restaurant or not

Statement 1:

The sum of amount spent at any 3 restaurants is same.

let's consider restaurants amounts as : a1, a2, a3, a4, a5, a6 for respective 5 restaurants
Now, consider any 3 restaurants and equate the sum

a1 + a2 + a3 = a1 + a2 + a4 => amount spent at restaurant a4 and a3 are equal
Repeating the above for different sets of restaurants yields a1 = a2 = a3 = a4 = a5

Amount spent at each restaurant is same, and hence she did not spend more than the median amount at any restaurant.  Sufficient

Statement 2:

Amount spent at any restaurant is not less than the average at these 5 restaurants.

Now, let's consider at a4 the amount spent is greater than the average, then there must be another amount to compensate for this increase. And that amount will be lesser than the average.

But, according to this statement, there are no such amounts. => ther amount spent at all the 5 restaurants is equal, and hence she did not spend more than the median amount at any restaurant.  Sufficient

Both choices alone on thier own are sufficient

Raemi · Answer

Am I missing something?

Case 1: 1, 1,   1  , 10, 10 (Spent MORE than Median)

Case 2: 1, 1,   10  , 10, 10 (Did NOT spend more than Median)

#1 alone cannot be sufficient.

Bunuel · Answer

Yes. Your examples do not satisfy the first statement. Statement 1 says Emily spent the same total amount at any three of the five restaurants. This means every combination of three bills must have the same total. For $1, $1, $1, $10, $10, the total of the first three is $3, but the total of the last three is $22. The same with your second example. For Statement 1 to be true, all bills must be the same. The amounts must be equal because of the specific conditions given in each statement: 1. Statement (1): If Emily spent the same total amount at any three of the five restaurants, then all amounts must be equal. If one amount were different (e.g., x < y), then the total of three amounts containing x would be less than the total of three amounts containing y. This would contradict the statement. Hence, all amounts are equal. 2. Statement (2): If Emily did not spend less than the average at any restaurant, then no amount is below or above the average. This is only possible if all amounts are equal because any variation would create values below or above the average. In both cases, equality of amounts is the only scenario consistent with the given conditions. Check similar question here: https://gmatclub.com/forum/m43-433371.html

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