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22 Jul 2020, 08:55
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The chart above gives the average (arithmetic mean) of the results, broken down by class and gender, of a graduation test administered to 12th graders at Central High School. If all students have taken the test, how many of the 12th grade classes have more female students than male students?
(A) 1
(B) 2
(C) 3
(D) 4
(E) It cannot be determined from the information given.
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22 Jul 2020, 09:30
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Theory behind the problem is that the Average of the class will be closer to the Average of the Group with More Number of Students
So, in all 4 cases we need to find out the the flat average of Average of Male and Female and then see which side is Average of all Students more inclined towards.
Mr. Anderson -> Average of Male and Female = \(\frac{83+91}{2}\) = 87
Now, if Class Average is Between 83 to 87 then Inclined towards Male => More Male students as compared to Female
if Class Average is Between 87 to 91 then Inclined towards Female => More Female students as compared to Male
if Class Average = 87 then Equal Number of Male and Female Students
But Average of all students is 88 => More inclined towards Female => More number of females in the class.
Ms. Baker -> Average of Male and Female = \(\frac{87+83}{2}\) = 85
But Average of all students is 85 => Equal Number of Male and Female Students
Mr. Chavez -> Average of Male and Female = \(\frac{93+90}{2}\) = 91.5
But Average of all students is 92 => More inclined towards Male=> More number of Males in the class.
Ms. Do -> Average of Male and Female = \(\frac{91+86}{2}\) = 88.5
But Average of all students is 89 => More inclined towards Male => More number of females in the class.
So, Answer will be A (1)
Hope it helps!
Algebra for Average of the class will be closer to the Average of the Group with More Number of Students
Let number of Male students be M and Number of Female be F => Total students = M + F
Average of all student =\( \frac{Sum Of All Students }{ Total Number Of Students}\)
Sum of all students = Sum of Male + Sum of Female = Average of Male * M + Average of Female *F
=> Average of all student =\( \frac{Average of Male * M + Average of Female *F}{ M+F}\)
=> Average of all student =\( \frac{Average of Male * M}{M+F}\) + \( \frac{Average of Female *F}{ M+F}\)
=> Average of all students is nothing but the weighted Average of Average of Male and Average of Female.
As, M+ F will be same for both the values so Average of all students will be more inclined towards the value which has more weight
=> If M > F then Average of all students will be more inclined towards Male
=> If M < F then Average of all students will be more inclined towards Female
=> If M = F then Average of all students will be equal to average of Male and Female
Check out THIS POST if you are looking to Theory on Weighted Mean!
So, in all 4 cases we need to find out the the flat average of Average of Male and Female and then see which side is Average of all Students more inclined towards.
Mr. Anderson -> Average of Male and Female = \(\frac{83+91}{2}\) = 87
Now, if Class Average is Between 83 to 87 then Inclined towards Male => More Male students as compared to Female
if Class Average is Between 87 to 91 then Inclined towards Female => More Female students as compared to Male
if Class Average = 87 then Equal Number of Male and Female Students
But Average of all students is 88 => More inclined towards Female => More number of females in the class.
Ms. Baker -> Average of Male and Female = \(\frac{87+83}{2}\) = 85
But Average of all students is 85 => Equal Number of Male and Female Students
Mr. Chavez -> Average of Male and Female = \(\frac{93+90}{2}\) = 91.5
But Average of all students is 92 => More inclined towards Male=> More number of Males in the class.
Ms. Do -> Average of Male and Female = \(\frac{91+86}{2}\) = 88.5
But Average of all students is 89 => More inclined towards Male => More number of females in the class.
So, Answer will be A (1)
Hope it helps!
Algebra for Average of the class will be closer to the Average of the Group with More Number of Students
Let number of Male students be M and Number of Female be F => Total students = M + F
Average of all student =\( \frac{Sum Of All Students }{ Total Number Of Students}\)
Sum of all students = Sum of Male + Sum of Female = Average of Male * M + Average of Female *F
=> Average of all student =\( \frac{Average of Male * M + Average of Female *F}{ M+F}\)
=> Average of all student =\( \frac{Average of Male * M}{M+F}\) + \( \frac{Average of Female *F}{ M+F}\)
=> Average of all students is nothing but the weighted Average of Average of Male and Average of Female.
As, M+ F will be same for both the values so Average of all students will be more inclined towards the value which has more weight
=> If M > F then Average of all students will be more inclined towards Male
=> If M < F then Average of all students will be more inclined towards Female
=> If M = F then Average of all students will be equal to average of Male and Female
Check out THIS POST if you are looking to Theory on Weighted Mean!
General Discussion
22 Jul 2020, 22:41
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BrushMyQuant
I answered E, because of Ms. Baker's class where the average was in between the male and female averages. Was it wrong to assume we couldn't answer the question because it is never stated if the number is rounded?
