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abhasjha: Master | Next Rank: 500 Posts; Posts: 379; Joined: Tue Sep 30, 2008 7:17 amFollowed by: 11 members; Location: NY; Thanked: 28 times

by abhasjha » Mon Aug 25, 2014 10:47 am

A teacher gave the same test to three history classes: A, B, and C. The average (arithmetic mean) scores for the three classes were 65, 80, and 77, respectively. The ratio of the numbers of students in each class who took the test was 4 to 6 to 5, respectively. What was the average score for the three classes combined?

A. 74
B. 75
C. 76
D. 77
E. 78

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Source: Beat The GMAT — Problem Solving | Mon Aug 25, 2014 10:47 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Aug 25, 2014 10:55 am

abhasjha wrote:A teacher gave the same test to three history classes: A, B, and C. The average (arithmetic mean) scores for the three classes were 65, 80, and 77, respectively. The ratio of the numbers of students in each class who took the test was 4 to 6 to 5, respectively. What was the average score for the three classes combined?

A. 74
B. 75
C. 76
D. 77
E. 78

Let:
Class A have 4 students, each scoring 65 points, for a total of 260 points.
Class B have 6 students, each scoring 80 points, for a total of 480 points.
Class C have 5 students, each scoring 77 points, for a total of 385 points.
Average score for all 15 students = (260+480+385)/15 = 75.

The correct answer is B.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Aug 25, 2014 11:07 am

abhasjha wrote:A teacher gave the same test to three history classes: A, B, and C. The average (arithmetic mean) scores for the three classes were 65, 80, and 77, respectively. The ratio of the numbers of students in each class who took the test was 4 to 6 to 5, respectively. What was the average score for the three classes combined?

A. 74
B. 75
C. 76
D. 77
E. 78

A slightly different approach:

This is a weighted averages question, so we can use the following formula:

Weighted average of groups combined = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...

For more information on weighted averages, you can watch this free GMAT Prep Now video: https://www.gmatprepnow.com/module/gmat- ... ics?id=805

Ratio of classes A, B, C = 4 : 6 : 5, respectively
4 + 6 + 5 = 15
So, class A represents 4/15 of the TOTAL population
Class B represents 6/15 of the TOTAL population
Class C represents 5/15 of the TOTAL population

So, weighted average of 3 groups combined = (4/15)(65) + (6/15)(80) + (5/15)(77)
= 260/15 + 480/15 + 385/15
= 1125/15
= 75
= B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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abhasjha: Master | Next Rank: 500 Posts; Posts: 379; Joined: Tue Sep 30, 2008 7:17 amFollowed by: 11 members; Location: NY; Thanked: 28 times

by abhasjha » Mon Aug 25, 2014 11:35 am

Mitch ,

since this is a weighted avg problem - is it possible to solve the question using method of mxiture and alligation too?

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by [email protected] » Mon Aug 25, 2014 7:39 pm

Hi abhasjha,

Mitch's solution is likely the easiest way for most Test Takers to go about answering this question (and it's exactly how I would solve it). You can use the answer choices against the prompt though and use some algebra "concepts" to figure out which is correct.

Since we're calculating an average, the "net above average" must "cancel out" the "net below average" of the numbers.

eg. the average of 1, 2 and 6 is 3.

The 1 is "2 below" the average,
the 2 is "1 below" the average, and
the 6 is 3 "above" the average.

The net "3 below" cancels out the net "3 above" and we end up with the average.

With the info in the prompt, we know that we essentially have 4 "65s", 6 "80s" and 5 "77s"

If we start with answer C: 76....

The 65s are each "11 below" --> 4(11) = 44 below
The 80s are each "4 above" --> 6(4) = 24 above
The 77s are each "1 above" --> 5(1) = 5 above
Here we have 44 below and 29 above; these numbers don't cancel out, so 76 CANNOT be the average. Since we have more "below", the average must be SMALLER.

With answer B: 75....

The 65s are each "10 below" --> 4(10) = 40 below
The 80s are each "5 above" --> 6(5) = 30 above
The 77s are each "2 above" --> 5(2) = 10 above
Here, the 40 below cancels out the 40 above, so 75 MUST be the average.

Final Answer: B

To be fair, I wouldn't go about solving this problem in this fashion, but it does serve as an interesting concept for how groups of numbers can affect an average.

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Rich

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Aug 29, 2014 6:57 am

abhasjha wrote:Mitch ,

since this is a weighted avg problem - is it possible to solve the question using method of mxiture and alligation too?

Alligation is a great approach for determining the ratio of two ingredients in a mixture composed solely of these two ingredients.
Alligation should not be used for a mixture composed of 3 ingredients.
For the reason, check my post here:
https://www.beatthegmat.com/allegation-t69552.html

For those unfamiliar with alligation, check my post here:
https://www.beatthegmat.com/ratios-fract ... 15365.html

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