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12 Days of Christmas GMAT Competition - Day 3: Leonard had each comic

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Bunuel

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12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

Leonard had each comic book in his collection graded by a professional service. Each comic book received one of four different grades. 3 comic books received the lowest grade, 5 comic books received the second-lowest grade, 4 comic books received the second-highest grade, and x comic books received the highest grade. If the four grades were consecutive multiples of 5 between 0 and 100, inclusive, what was the standard deviation of the grades of Leonard's comic books?

(1) x = 2
(2) The lowest grade assigned to the comic books was 35.

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Bunuel

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Bunuel

Leonard had each comic book in his collection graded by a professional service. Each comic book received one of four different grades. 3 comic books received the lowest grade, 5 comic books received the second-lowest grade, 4 comic books received the second-highest grade, and x comic books received the highest grade. If the four grades were consecutive multiples of 5 between 0 and 100, inclusive, what was the standard deviation of the grades of Leonard's comic books?

(1) x = 2
(2) The lowest grade assigned to the comic books was 35.

GMAT Club's Official Explanation:

Leonard had each comic book in his collection graded by a professional service. Each comic book received one of four different grades. 3 comic books received the lowest grade, 5 comic books received the second-lowest grade, 4 comic books received the second-highest grade, and x comic books received the highest grade. If the four grades were consecutive multiples of 5 between 0 and 100, inclusive, what was the standard deviation of the grades of Leonard's comic books?

Let the lowest grade be g. The distribution of grades is as follows:

3 comic books received g.
5 comic books received g + 5.
4 comic books received g + 10.
x comic books received g + 15.

(1) x = 2

With x = 2, the distribution becomes:

{g, g, g, g + 5, g + 5, g + 5, g + 5, g + 5, g + 10, g + 10, g + 10, g + 10, g + 15, g + 15}.

An important property of standard deviation is that adding or subtracting the same constant to every value in a list does not change the standard deviation.

Subtracting g from each term gives:

{0, 0, 0, 5, 5, 5, 5, 5, 10, 10, 10, 10, 15, 15}.

The standard deviation can now be calculated directly using this adjusted list. Sufficient.

(2) The lowest grade assigned to the comic books was 35.

If the lowest grade is 35, then the grades are 35, 40, 45, and 50. However, without knowing the value of x, the number of comic books that received the grade 50, the distribution is incomplete. Different values of x will result in different standard deviations. Not sufficient.

Answer: A.

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Let lowest grade be g, then,

Grade	Number of books
g	3
g + 5	5
g + 10	4
g + 15	x

To calculate standard deviation we need to know the mean of all these numbers and then calculate the distance of these numbers from the mean

Statement 1 : x = 2

This gives us total number of books = 14, but doesn't allow us to calculate the mean as we don't know the value of g.

Insufficient.

Statement 2 : g = 35

Just knowing g, but not the total number of books will not allow us to calculate the mean.

Insufficient.

But combining both these statements will give us values for both the unknown variables and allow us to calculate the mean and hence the resulting standard deviation, so both together should be sufficient.

Answer: C

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IMO C

Given:

3 comic books received the lowest grade.
5 comic books received the second-lowest grade.
4 comic books received the second-highest grade.
x
x comic books received the highest grade.
The grades are consecutive multiples of 5 between 0 and 100, inclusive.
Statement (1):
x
=
2
x=2

Statement (2): The lowest grade assigned to the comic books was 35.

Let's analyze each statement individually and then together.

Analyzing Statement (1) Alone
Statement (1):
x
=
2
x=2

This tells us that 2 comic books received the highest grade, but it does not provide information about the actual grades. Without knowing the specific grades, we cannot calculate the standard deviation. Therefore, Statement (1) alone is not sufficient.

Analyzing Statement (2) Alone
Statement (2): The lowest grade assigned to the comic books was 35.

If the lowest grade is 35 and the grades are consecutive multiples of 5, then the grades are:

Lowest grade: 35
Second-lowest grade: 40
Second-highest grade: 45
Highest grade: 50
We now know the specific grades, but we do not know the number of comic books that received the highest grade (
x
x). Therefore, Statement (2) alone is not sufficient.

Combining Statements (1) and (2)
Combining the information from both statements:

The grades are 35, 40, 45, and 50.
The number of comic books that received each grade is:
3 comic books received 35.
5 comic books received 40.
4 comic books received 45.
2 comic books received 50.

We can find the SD, no need to calculate as it is a DS question

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18 Dec 2024, 09:58

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Bunuel

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We are tasked with determining the standard deviation of the grades assigned to Leonard's comic books.

Key Point : The grades are consecutive multiples of 5,

So The grades are consecutive multiples of 5, so if the lowest grade is g, then the grades are g,g+5,g+10, and g+15.

To find SD we need mean. Without mean we cannot find the SD.

Stmt 1 ; x=2.
So mean = (3g + 5(g+5) + 4(g+10) + 2(g+15))/14 =>

So mean will be in terms of g. Hence we cannot find the correct SD.

Stmt 2: The lowest grade assigned to the comic books was 35.

mean = μ=(3(35)+5(40)+4(45)+x(50))/12+x . Hence we cannot find the correct SD.

But using both we can get the SD , mean =48..

IMO C

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Leonard had each comic book in his collection graded by a professional service. Each comic book received one of four different grades. 3 comic books received the lowest grade, 5 comic books received the second-lowest grade, 4 comic books received the second-highest grade, and x comic books received the highest grade.

If the four grades were consecutive multiples of 5 between 0 and 100, inclusive, what was the standard deviation of the grades of Leonard's comic books?

Grade Description	Number of books	Grade (numeric value)	Total grades
Lowest	3	5k	15k
Second-lowest	5	5k + 5	25k + 25
Second-highest	4	5k + 10	20k + 40
Highest	x	5k + 15	5xk + 15x

Total grades = (60+5x)k + 65 + 15x
Average grade = {(60+5x)k + 65 + 15x}/(12+x)

(1) x = 2

Grade Description	Number of books	Grade (numeric value)	Total grades
Lowest	3	5k	15k
Second-lowest	5	5k + 5	25k + 25
Second-highest	4	5k + 10	20k + 40
Highest	2	5k + 15	10k + 30

Total grades = 70k + 95
Average grade = (70k + 95)/14 = 5k + 95/14

Standard deviation = \sqrt{{3(-95/14)^2 + 5(5-95/14) + 4(10-95/14) + 2(15-95/14)}}/14
Value of standard deviation can be found out
SUFFICIENT

(2) The lowest grade assigned to the comic books was 35.

Grade Description	Number of books	Grade (numeric value)	Total grades
Lowest	3	35	105
Second-lowest	5	40	200
Second-highest	4	45	180
Highest	x	50	50x

Total grades = 485 + 50x
Average grades = (485 + 50x)/(12+x)

Standard deviation will depend on the value of x
NOT SUFFICIENT

IMO A

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To calculate std deviation we need mean and (number of terms)

For mean = {3*(5k) + 5*(5(k+1)) + 4*(5(k+2)) + x*(5(k+3))} / {3+5+4+x}
Clearly 'x' is needed.

Also during std deviation formula calculation the 'k' will not get cancelled.

so knowing both 'k' and 'x' will be needed.
Also the edge case where std dev = 0 (i.e. all the terms are same is not the case)

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(1)
x=2
with given value of x we can find the mean, which when we put in the standard deviation formula, we will get the standard deviation.
And for the grades we can take any 4 consecutive multiples of 5 and check for the standard deviation, the value of standard deviation doesn't get affected by the values of the grades, So any 4 consecutive multiples of 5 would work.
sufficient.

(2)
lowest grade = 35
Just this value of grade is not sufficient, we must know the total number of books, to find the mean. And we cannot find the standard deviation because the variables wont get cancelled in the standard deviation equation when the mean with variables is used.
insufficient.

A

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18 Dec 2024, 10:08

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Let the 4 grades be: 5k, 5k+5, 5k+10, and 5k+15

To calculate Standard Deviation we will need the Sum of Squares of differences between the grade and its mean and the total number of grades (or books)

Statement 1.

x = 2. This gives us total number of grades / books, but doesn't help in calculating the mean until we know k. INSUFFICIENT

Statement 2.

Lowest grade 35 => k = 7. This again won't help us calculate the mean as we don't know the total number of grades. INSUFFICIENT

Both together

Now we have x and k both, we can calculate the mean and get the SD by substituting it with no. of grades in SD formula. SUFFICIENT

Answer C.

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Statement (1): x = 2
This tells us that 2 comic books received the highest grade, but we don’t know the specific grades. Therefore, we cannot calculate the standard deviation. Insufficient.

Statement (2): The lowest grade assigned to the comic books was 35
If the lowest grade is 35, then the grades must be 35, 40, 45, and 50. However, we don’t know how many books received each grade, so we cannot calculate the standard deviation. Insufficient.

Combining both statements:
From statement (2), we know the grades are 35, 40, 45, and 50. From statement (1), we know the number of books for each grade: 3 books at 35, 5 books at 40, 4 books at 45, and 2 books at 50. With this information, we can compute the standard deviation.

Both statements together are sufficient.

Answer: (C) Both statements together are sufficient.

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std deviation is a spread measure and we only care about the spread from the mean which here is 5 points for every grade so starting point doesn't matter only the no of books in different bucket i.e. the value of x

like 5,10,15,20 is indifferent to 35,40,45,50 (mean will change std deviation wont)

Hence only statement (1) is required

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Bunuel

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As four grades were consecutive multiples of 5 between 0 and 100, the starting point will make no difference to the standard deviation. The only thing that we need is the number of x.
In (1) we know x = 2 (sufficient)

In (2) As we don't know the value of x, we are not able to calculate the mean and thus the standard deviation (insufficient)

Therefore the answer is A

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18 Dec 2024, 10:24

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Bunuel

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Let the scores be k, k+5, k+10, k+15.
Average score = (x(k+15) + 4(k+10) + 5(k+5) + 3k ) / 12+x
= (kx+15x+12k+65) / 12+x
= ( k(12+x) +15x+65) / 12+x
= k + (15x+65)/12+x

To calculate SD, we need to find out the distance of each score from the mean,

x[ 15 - (15x+65)/12+x ] + 4[10 - (15x+65)/12+x ] + 5[5- (15x+65)/12+x] +3[ -(15x+65)/12+x]

As, we can see, we only need the value of x in order to calculate the SD,

Statement 1 is clearly sufficient while Statement 2 is not.

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Bunuel

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Step 1: Understand the Problem

Leonard's comic books received grades that are consecutive multiples of 5: g1,g2,g3,g4
Frequencies: f1=3, f2=, f3=4, f4=2.
We need the standard deviation, which requires both the grades and the frequencies.

[hr]
Step 2: Use the Statements
From Statement (1):

x=2x = 2x=2.
Frequencies: f1=3,f2=5,f3=4,f4=2
This is insufficient without the values of the grades.

From Statement (2):

The lowest grade is g1=35g_1 = 35g1=35.
Grades: g1=35,g2=40,g3=45,g4=50
This is insufficient without the value of x.

Combine Statements (1) and (2):

Frequencies: f1=3,f2=5,f3=4,f4=2
Grades: g1=35,g2=40,g3=45,g4=50

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we need to know the values of both the number of books with highest grade and the marks of the grades to calculate the deviation. hence both statements are required.

Bunuel

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18 Dec 2024, 10:37

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Given information :
there are 4 different grades, which are consecutive multiple of 5 between 0 and 100
3 comic books received the lowest grade
5 comic books received the second lowest grade
4 comic book received the second highest grade
x comic books received the highest grade
Lets evaluate each statement:
Statement 1: x = 2
This tells us the number of comic books that received the highest grade . However it doesn't give us information about the actual grades

Statement 2: the lowest grade assigned to the comic books was 35
This gives us the starting point for the grade. Since we know that grade are consecutive multiples of 5, we can determine all four grades : 35, 40, 45, and 50
to calculate the standard deviation, we need both the no of comic books for each grade and the actual grade . By combining both statements
3 comic books received 35
5 comic books received 40
4 comic books received 45
2 comic books received 50(from st. 1)
with this information we can calculate the standard deviation
1. calculate mean (35*3+40*5+45*4+50*2)/(3+4+5+2) = 640/14 = 45.71(approx)
2. calculate the squared difference from the mean, multiply by the frequency, sum them up and divide by the total no of comic books
3. take the square root of the result to get the standard deviation
Therefore combining both the statements gives us sufficient information to calculate the standard deviation of the grade of leonard's comic books

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18 Dec 2024, 10:39

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Bunuel

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Statement 1:
Knowing the value of x (the number of comic books with the highest grade) is not sufficient to determine the exact grades.
For instance, if x = 2, the grades could be 35, 40, 45, and 50, or they could be 20, 25, 30, and 35.Each scenario would result in a different standard deviation.
Statement 2:
Knowing the lowest grade is 35 is also not sufficient. We still don't know the exact value of x or the other grades.
Combining Both Statements:
Even with both statements, we can only determine the grades (35, 40, 45, and 50) and the number of comic books at each grade (3, 5, 4, and 2).With this information, we can calculate the mean, variance, and standard deviation.
Both statements together are sufficient, but neither statement alone is sufficient.

IMO C

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18 Dec 2024, 10:40

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Four grades from low to high: g1, g2, g3, g4

Total grade value = 3*g1 + 4*g2 + 5*g3 + x*g4

To get the S.D we need average of the value and for that we need x and any of the g1 to g4.

So, considering both the statements we can get the required things and get the S.D.

Answer: C

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18 Dec 2024, 10:42

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The answer is C.

We need to find the standard deviation for the grades corresponding to the books.

Statement 1: gives the missing number of comic books for highest grade. We can choose any multiple of 5 and the answer would vary. Not sufficient.

Statement 2 : gives the grade for lowest. Just with this information we cannot find the SD.

Combining both the statements we can find the mean and thus the standard deviation.

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Leonard had each comic book in his collection graded by a professional service. Each comic book received one of four different grades. 3 comic books received the lowest grade, 5 comic books received the second-lowest grade, 4 comic books received the second-highest grade, and x comic books received the highest grade. If the four grades were consecutive multiples of 5 between 0 and 100, inclusive, what was the standard deviation of the grades of Leonard's comic books?

We need to know the grades and x

(1) x = 2
We don't know the grades NS

(2) The lowest grade assigned to the comic books was 35.
We don't know x NS

Combined
We get both grades and x Suff

Ans C