A group of 90 students took a certain test. All of the students either

Question

A group of 90 students took a certain test. All of the students either scored 800 points or 500 points. If the average score for the entire group was 560 points, how many students scored 800 points?
 
(A)15
(B)18
(C)20
(D)24
(E)36

GMATWhizTeam · Answer

A straight forward question on weighted averages which can be solved easily by drawing the Alligation diagram.

The total number of students = 90.

All students either scored 800 points or 500 points – these are the two averages

Average score for the entire group was 560 points – this is the weighted average for all the 90 students.

The alligation diagram, when drawn, should look something like this:

Attachment:

19th Apr 2022 - Post 8.png [ 5.87 KiB | Viewed 4804 times ]

From the bottom half of the diagram, it is clear that the ratio in which the 90 students are to be distributed among the two categories is 1 : 4.
Therefore, number of students who scored 800 points = \(\frac{1 }{ (1+4) }\)* 90 = \(\frac{1 }{ 5}\) * 90 = 18.

The correct answer option is B.

ashishmishra06 · Answer

Sorry couldnt understand the 1:4 part could you please explain

GMATWhizTeam · Answer

Hello Ashishmishra06, 
Happy to answer your question.

The alligation diagram is derived by rearranging the weighted average equation,

A =  \frac{A_1 * N_1 + A_2 * N_2 }{ N_1 + N_2}

where,  A_1  = Higher average,  A_2  = Lower average,  N_1  = Weightage of the higher average and  N_2  = Weightage of the lower average

You can reorganize the equation above to get the ratio of  N_1  :  N_2 . Once this is done, when drawn in the form of a diagram, we get the Alligation diagram.

In the alligation diagram, the top half (including the middle value) represents the averages, while the bottom half represents the weightages (or in other words, number of values).

Another point to note is that the weightages should always be expressed as a ratio. 60 and 240 are in the ratio of 1:4. 
What does this mean? This means that, if you want to achieve a mixture whose quality is 560, by mixing two components of qualities 800 and 500, you will have to mix them in the ratio of 1:4 respectively.

Does this all make more sense now? Happy to answer any other question you may have on this.

nasgmatbag · Answer

Simple approach: use statistic concepts, unit digit and answer choices

Analysis stem: total of 90 students, scored either 800 or 500 points, average = 560, how many scored 800?

S = (#)(A) -> (sum) = (amount of numbers)(average)
In this question we would get: (90)(560) = (90)(560)

800x + 500y = sum (a.k.a. (90)(560))
Notice also that x represents the students who scored 800, and y the students who scored 500

Divide by 100 to get: 8x + 5y = (9)(56) which is 504
From here on out just use the answer choices to your advantage. A very quick approach is to use the unit digit, we need the 8 to end in a 4, since the 5, in this scenario, will only be a round number given our answer choices

(A) 15
If x = 15, then (8)(15) = 120 (doesn't end in a 4, and also we can't get 5y to 504 with this result)
(B) 18
If x = 18, then (8)(18) = 144 (ends in a 4, and we can get 5y to 504, where y would be 72)
(C) 20
If x = 20, then (8)(20) = 160 (same as A)
(D) 24
If x = 24, then (8)(24) = 192 (same as A)
(E) 36
If x = 36, then (8)(36) = 288 (same as A)

Hence B is the answer

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A group of 90 students took a certain test. All of the students either

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