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GMAT Club Olympics 2025 (Day 7): Jacob, Yuki, and Stephen each took

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Total=90 , avg=90/3=30
is the median 30?
(1) Yuki's score was 20 greater than Stephen’s score.
let S=x, Y=x+20
So, 2x+20+J=90
2x+J=70
If , the case is x=20,then Y=40 this makes J=30
if , x=15,Y=35,J=40
Not sufficient
(2) Jacob's score was 30.
We know one of the score is 30 , so median is 30
Sufficient
IMO:B

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Statement (1) alone is insufficient. The difference in scores mentioned is not helpful.
Y = 30, J = 70 - 20 = 50
Scores: 10, 30, 50 -> median = 30

But,
Y = 45, J = 70 - 50 = 20
Scores: 20, 25, 45 -> median = 25

2 - J=30, Y + S = 60
Between Y and S, one will always be greater than 30, and the other smaller than 30.
Ex -
Y = 20, S = 40 -> Scores: 20, 30, 40 -> median = 30

Y = 40, S = 20 -> Scores: 20, 30, 40 -> median = 30

Statement (2) alone is sufficient
(B)

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Total score given = 90

So average = 30

Question: is average = median?
Statement 1:
Y = S+20
We have insufficient info regarding Jacob's score
2S + 20 + J = 90
2S + J = 70
We may have 30 as median in this or may not have

Statement 2:
J = 30
Y+S = 60
In this case, 30 will always be median
because maximum Y and S can be is 30
So if Y increases, S decreases and vice versa but 30 will always be median

hence this is sufficient
B is answer

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Avg is not equal to median, we need both the statements together for the answer, each alone is not sufficient.Hence Point C is the answer.

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From the info in the question, we know:
J + Y + S = 90
Avg = 30
The question asks: Avg = median = 30?

Condition 1:
Y = S + 20
J + 2S + 20 = 90
J = 70 - 2S
Assume:
If S = 10, Y = 30, then J = 50, condition fits
If S = 5, Y = 25, J = 60, condition doesn't fit

Condition 1 is not sufficient

Condition 2:
J = 30
Y + S = 60
Y = 5, S = 55, J = 30
Y = 29, S = 31, J = 30
Y = 31, S = 29, J = 30
This means no matter what,
J will always be in the middle
J = avg = median. Condition fits

Condition 2 is sufficient

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T=90
Avg=30
Is Avg=Median?

S1
Yuki's score was 20 greater than Stephen’s score.
We have no info on Jacob's score
Scores can be 10,30,50 or 15,35,40
Insufficient

S2
J=30
Y+S=90-30=60
If they were to split either both of them will have a score of 30 or one will have a score below 30 and other will have a score above 30. Either ways, we know Avg=Median
Sufficient

Answer B

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The total of the three scores is 90, so the average is always 30. The question is whether the median is also 30. From statement (1), Yuki’s score is 20 more than Stephen’s, but without knowing Jacob’s score, the median could vary; for example, if Stephen scored 25, then Yuki scored 45 and Jacob scored 20, giving scores 20, 25, 45 with median 25, which is not 30, so statement (1) alone is not sufficient. From statement (2), Jacob’s score is 30, so the other two scores must add to 60; no matter what those two scores are, the median will always be 30 because 30 lies between the other two scores or is equal to one, ensuring median equals average, making statement (2) alone sufficient. Combining both statements, Jacob is 30 and Yuki is 20 more than Stephen, so the three scores are fixed as 20, 30, and 40, with median 30 equal to the average 30, confirming sufficiency. Therefore, only statement (2) alone is sufficient to answer the question.

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Notes:
3 students.
total of 3 tests was 90
Average of 3 scores = 90/3 = 30

Ask:
Was the average = to median

Looking for:
The median

1) Yuki score was 20 greater than Stephans score
Y + (y-20) + J = 90
Plug and play and we can get 20, 30, 40 or 15,35,40 Not sufficient

2) Jacobs score was 30
Y + S +30 = 90
Y + S = 60
Plug and play- 10,50 -good 20,40 - good, 25,35- good, Sufficient

2 alone is sufficient. B

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J+Y+S= 90

First Statament
Y= S+20

If Y= 0 S= 20 J=70 Average= 30 Median= 20 Different
If Y= 10 S=30 J= 50 Average= 30 Median = 30 Same

We don't have a solution with Stament 1

Statement 2

If J= 30 (Y+S)= 60 Y could be Y=0 and S= 60 Average= 30 Median 30
or J= 30 Y= 10 S= 50 Average= 30 Median= 30

So Statement 2 is enough, the answer is B

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Here's a brief explanation:

Average Score: Given the total score of 90 for 3 people, the average score is 90/3=30.

Question: Is the median score equal to 30?

Statement (1): Yuki's score was 20 greater than Stephen’s score.

This gives a relationship between Y and S.

Example 1: S=10, Y=30, J=50. Scores: 10, 30, 50. Median = 30. (Yes)

Example 2: S=25, Y=45, J=20. Scores: 20, 25, 45. Median = 25. (No)

Since it can be both "Yes" and "No", Statement (1) is not sufficient.

Statement (2): Jacob's score was 30.

If Jacob's score (J) is 30, and the total is 90, then Y + S = 60.

For the median of three numbers to be equal to their average (which is 30), one of the numbers must be 30, and the other two must sum to twice the average (60).

Since J = 30 and Y + S = 60, the scores will always be 30 and two numbers that "balance" around 30 (e.g., 20 and 40, or 10 and 50, or 30 and 30). When sorted, 30 will always be the middle score.

Therefore, the median will always be 30.

Statement (2) is sufficient.

The final answer is B

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First, notice that their three scores add up to 90, so the average score is 90 /3 = 30. The question asks whether the middle score (the median) is also 30.
Statement (1) tells us that Yuki scored 20 points more than Stephen, so if Stephen scored S, then Yuki scored S + 20 and Jacob scored whatever makes the total 90. There are many ways to pick S and Jacob’s score so that the numbers add to 90, and in some of those cases the middle score is not 30. Thus (1) alone does not guarantee the median is 30.
Statement (2) tells us directly that Jacob’s score is 30. Since the other two scores must then total 60, one of them must be above 30 and the other below 30 so Jacob’s score of 30 falls right in the middle. That means the median is 30, matching the average. Statement (2) alone is enough to answer the question.
Therefore, the correct answer is that statement (2) alone is sufficient, but statement (1) is not.

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Bunuel

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(1) Yuki's score was 20 greater than Stephen’s score.

In the first case, we can have multiple possible scenarios in which 30 can either be at the median or we can have another scenario where 30 is not the median. Hence, the statement alone is not sufficient.

(2) Jacob's score was 30.

If Jacob's score was 30, then we can have two possibility. Someone has scored above Jacob. If that's the case, someone should also score below Jacob so as to keep the mean at 30.

In this case, Jacob's score will be at the median.

In other case, everyone's score is at 30. Even in this case, mean = median.

Option B

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Average score $= \frac{90}{3} = 30$
Median of 3 terms $= [\frac{(3+1)}{2}]$th term $=$ 2nd term
We need to know if the 2nd term is equal to 30

(1) Y = 20 + S
If J $= 0$, then Y $= 55$, and S $= 35$
=> 2nd term $≠ 30$ ($0$, $35$, $55$)

If J $= 30$, then Y $= 40$, and S $= 20$
=> 2nd term $= 30$ ($20$, $30$, $40$)

Not sufficient. Options A and D are out.

(2) J = 30
=> Y $+$ S $= 90 - 30 = 60$
If Y $=$ S $= 30$, then 2nd term $= 30$ ($30$, $30$, $30$)

If Y $≠ 30$, then either Y$>$S or S$>$Y and Y $+$ S $= 60$
=> 2nd term $= 30$ (S, $30$, Y OR Y, $30$, S)

Sufficient. The answer is B.

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Given conditions,

Jacob(J), Yuki(Y), and Stephen(S) each took the same science test.
total of their scores on the test = 90
J+Y+S = 90--->condition 1
average of (J,Y,S) = 90/3 = 30

was the average of the 3 scores equal to the median of the 3 scores [x1 <= x2 <= x3]?
J+Y+S/3 = median(x2)
30 = x2 ?

Lets consider statement (1) Yuki's score was 20 greater than Stephen’s score.
Y = S + 20
J + Y + S = 90 => J + (S+20) + S = 90 => J + 2S = 70.
If S=10, Y=30, J=50. Scores: {10, 30, 50}. Median=30. Yes.
If S=25, Y=45, J=20. Scores: {20, 25, 45}. Median=25. No.
statement (1) is Insufficient.

Lets consider statement (2) Jacob's score was 30.
J = 30.
J + Y + S = 90 => 30 + Y + S = 90 => Y + S = 60.
Let the three scores be x1, x2, x3
When ordered, the median is x2
The mean is (x1 + x2 + x3 )/3.
If one of the scores is equal to the mean, let's say J=(J+Y+S)/3.
This implies 3J=J+Y+S, so 2J=Y+S.
Since J=30, this means 2(30)=Y+S, so Y+S=60. This is consistent with what we derived from statement (2).
If one score is equal to the mean, then the other two scores must be one above and one below the mean, or both equal to the mean.
Case 1: Y=S=30. Scores are {30, 30, 30}. Median = 30.
Case 2: Y < 30 and S > 30 (or vice versa).
Example: Y=20, S=40. Scores are {20, 30, 40}. Median = 30.
Example: Y=10, S=50. Scores are {10, 30, 50}. Median = 30.
In any of these cases, 30 is the middle value when sorted. So 30 is the median.
statement (2) is Sufficient

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We need to see if Median = 90 (the same as average)

Statement 1 is not sufficient because median can be any number,
let's say S=70, Y=90, J =110 (median = average in this case as we don't know the maximum score for the test)
but if we take S=50, Y=70, J=150 (Med <> Average)

Statement 2 is also not sufficient as we do not know the values of S and Y.

Combining the 2 statements would be sufficient as,
S+20+S+30=270
2S=250
S=125
Y=150

Therefore, median is not equal average. Please Note that No is also an answer, hence, C is sufficient.

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If the total of their score is 90, then the average is $\frac{90}{3}=30$
So the question is, "Is the median 30?"

Statement 1:
This statement does not give enough information, the scores can be 1,21, 68, or it can be 20,40,30
So, Statement 1 is not enough.

Statement 2:
If one of the scores is 30, then we have one score that is equal to the average. So we have to find out if this score is also the median.
Based on the total of the scores, we can say that Yuki's score and Stephen's score have a total of 60. These 2 scores can be equal to 30, or one of them less than 30, and one more than 30, in a way that their sum is 60. In any of these ways, 30 would still remain the median, because they can't both be more than 30 simultaneously, or they can't both be less than 30 simultaneously, because their sum should equal 60 no matter what.
So, Statement 2 is enough.

The answer is B.

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(B) is sufficient

We know that the average of 3 scores is 30.
If Jacob's score is the same as the average score, we can conclude that the other 2 scores lie above and below 30. Hence, 30 is also the median.

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We can easily calculate the mean here, which is 90/3=30
We need to find if the median of J, Y, and S is 30

Statement 1:
Yuki's score was 20 points greater than Stephen’s score.

This just tells us that, Y = S + 20.

We could have the following order, S>Y>J OR J>S>Y OR S>J>Y

In all these cases, the median (the middle value) could be different. So, insufficient.

Statement 2:
Jacob's score was 30.

This says that J=30,

If we use it in our equation, J+S+Y=90, we get, S+Y=60

Now, for any value of S and Y that satisfies this equation, there will be 3 scenarios:

Scenario 1: S>30 and Y<30 ---> Median will be 30
Scenario 2: S<30 and Y>30 ---> Median will be 30
Scneraio 3: S=30 and Y=30 ---> Median will be 30

Since in all cases the median is 30, this is sufficient.

Answer B.

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Jacob, Yuki, and Stephen each took the same science test. If the total of their scores on the test was 90, was the average (arithmetic mean) of the 3 scores equal to the median of the 3 scores?

(1) Yuki's score was 20 greater than Stephen’s score.
(2) Jacob's score was 30.

Jacob + Yuki + Stephen = 90, Average = 90/3 = 30 is Average = Median ?
(1) Yuki's score was 20 greater than Stephen’s score.
Stephen = X, and Yuki = X+20, let's say Jacob = y

then X+X+20+y = 90
X = (70-Y)/2
If y = 10, then X = 30 and x+20 = 50, Median = X
If Y = 30, then X = 20 and X +20 = 40, Median = Y
Not Sufficient

2) Jacob's score was 30.
Jacob + Yuki + Stephen = 90
Yuki + Stephen = 60. If both same, then Yuki and Stephen = 30 each, Median = 30
If not, then in any case one will be greater than or 1 less than 30, Jacob will be in mid as one is lower than 30 and one is greater than 30
Median is 30 = Average = B is Sufficient

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Bunuel

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Let the scores of Jacob, Yuki and Stephen be J, Y, and S respectively.

Total scores = 90. Thus the average scores = (J+Y+S)/3 = 90/3 = 30.

Was 30 = median of 3 scores ?

Statement 1:

(1) Yuki's score was 20 greater than Stephen’s score.

Y = 20+ S

J+Y+S = J + 20+ 2S = 90

J + 2S = 70. There are multiple combinations for J and S.

if J = 0, S = 35, Y = 55. Median = 35 > Average =30. So, Is Median = 30? No

if J =10, S = 30, Y = 50. Median = 30 = Average =30. So, Is Median = 30? Yes.

if J= 20, S = 25, Y = 45. Median = 25 < Average =30. So, Is Median = 30 ? No.

Hence, Insufficient

Statement 2:

(2) Jacob's score was 30.

J+ S+ Y = 90

J = 30 , then Y+ S = 60

The possible values of Y and S are (0,60); (5,55); (20,40); (30,30); (40,20); (60,0).

In all the cases the values are (0, 30, 60 ) OR (5,30,55) OR (30,30,30) OR (20,30,40) etc .

The median value is always 30. Is Median = 30? The answer is YES.

Hence Sufficient

Option B

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