Angela’s grade was in the 90th percentile out of 80 grades in her class. In another class of 100 students there were 19 grades higher than Angela’s. If nobody had Angela’s grade, then Angela was what percentile of the two classes combined?

A. 72
B. 80
C. 81
D. 85
E. 92

She is in 90th percentile in one class and 81st in the other. We need to find her 'average percentile'. The average will lie between 81 and 90 so it has to be 85!

If we had closer answer options e.g. 84, 86 etc, we would need to calculate the weighted average.
Using weights as 80 and 100 respectively, you can find their average using weighted average formula which is as given below:
\(C_{avg} = \frac{C_1*W_1 + C_2 * W_2}{W_1 + W_2}\)

\(C_{avg} = \frac{90*80 + 81*100}{180}\) = 85

In case you are not comfortable with weighted averages, check this discussion:
http://gmatclub.com/forum/tough-ds-105651.html#p828579
and get back if there is a doubt.
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Thanx a ton to Bunuel & Karishma for such detailed explanations.

Special thanx to Bunuel for editing the question part i missed n also to Karishma for the helpful link on weighted averages.

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE


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90th Percentile means she has scored more then 90% of the people.
==> (90/100)*80 = 72 ( she scored more then 72 people)

In the other class of 100, we are given that 19 people scored more then Angela
==> 100 - 19 = 81 ( she scored more then 81 people)

Since we are told to find her percentile in the combined class, we need to find how many people did she score over in a combined class of 100 + 80 = 180.

(81+72)/180 = 85 %

Answer is D.

One could also use the logic that if in the first class her percentile was 90 and in the second her percentile was 81, the average would be greater then 81 and less than 90. Hence the Answer has to be D

Bunuel
I think it would have been simpler to calculate: if we add up people ahead of her 8+19 = 27, therefore 27/180*100 = 15% ahead of her, therefore 85% is percentile of both classes.

dave13

The 19 students in other class are those who have scored higher than Angela, so if we are calculating percentile of angela's we need to know how many students did angela outscored in the other class, which is 100 - 19 = 81 students. In other class angela outscored 81 students.

To calculate percentile of two classes combined, we need to calculate how many students have been outscored by angela out of total number of students.
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Alternate approach (similar to Karishma´s nice explanations above!):

\(?\,\,\,\,:\,\,\,{\text{Angela}}\,\,{\text{percentile}}\)

The approximations presented are validated a posteriori, because there are no alternative choices "near" the correct one!

\(\left\{ \begin{gathered}
\,80\,\,{\text{people}}\,\,\,:\,\,\,90\% \,\,{\text{below}} \hfill \\
\,10\boxed1\,\,\,{\text{people}}\,\,\,:\,\,\,\, \cong 80\% \,\,{\text{below}}\,\,\, \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\, \cong \,\,\,\,\,\,\,\left\{ \begin{gathered}
\,\,4\,\,{\text{people}}\,\,\,:\,\,\,90\% \,\,{\text{below}} \hfill \\
\,\,5\,\,\,{\text{people}}\,\,\,:\,\,\,80\% \,\,{\text{below}}\,\, \hfill \\
\end{gathered} \right.\)

\(\boxed1\,\,\,:\,\,\,{\text{if}}\,\,{\text{Angela}}\,\,{\text{there!}}\)


\(?\,\,\, \cong \,\,\,\frac{{4 \cdot 90\% + 5 \cdot 80\% }}{9}\,\,\, \cong \,\,\,\frac{{4 \cdot 90\% + 5 \cdot 81\% }}{9}\,\,\, = \,\,\,4 \cdot 10\% + 5 \cdot 9\% \,\,\, = \,\,\,85\%\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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In the first class, Angela scored 90th Percentile of 80 that means you calculate 90% of which is 72

Second class of 100, only 19 grades were higher than her, this means she scored 81 out of 100.

Combined percentile = 72 + 81/180 *100% = 153/180 *100% = 85%

Answer = D