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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?
Let's start by understanding what percentiles really mean in everyday terms. When we say Angela is in the 90th percentile, we're saying that \(90\%\) of students scored below her and \(10\%\) scored above her.
Think of it like this: if you line up all students from lowest grade to highest grade, Angela would be near the top with only a few students ahead of her.
Our goal is to find Angela's percentile when we combine both classes together. This means we need to figure out how many total students scored below Angela across both classes.
Process Skill: TRANSLATE - Converting percentile language into concrete numbers of students
In Angela's original class of 80 students, she's in the 90th percentile. Let's figure out exactly how many students scored below her.
If Angela is in the 90th percentile, then \(90\%\) of the 80 students scored below her:
\(90\% \text{ of } 80 = 0.90 \times 80 = 72\) students scored below Angela
This means in her original class:
Let's verify: 7 students above her means she's better than 72 out of 79 other students, which is \(\frac{72}{80} = 0.90 = 90\text{th percentile}\) ✓
Now let's look at the second class of 100 students. We're told that 19 grades were higher than Angela's grade.
This means:
So in the second class, 81 students scored below Angela's level.
Now we can combine both classes to find Angela's overall percentile.
Total students below Angela:
Total students in both classes combined:
\(80 + 100 = 180\) students
Angela's percentile in the combined classes:
Percentile = \(\frac{\text{Number of students below Angela}}{\text{Total number of students}} \times 100\)
Percentile = \(\frac{153}{180} \times 100\)
Let's simplify this fraction:
\(\frac{153}{180} = \frac{153}{180} = \frac{17}{20} = 0.85\)
Therefore: \(0.85 \times 100 = 85\text{th percentile}\)
Process Skill: INFER - Drawing the connection that percentiles are based on the proportion of students scoring below a given level
Angela's percentile in the two classes combined is 85.
Looking at our answer choices: A. 72, B. 80, C. 81, D. 85, E. 92
The answer is D. 85
Many students confuse percentile with percentage score. They might think being in the 90th percentile means Angela scored \(90\%\) on her test, rather than understanding that \(90\%\) of students scored below her. This fundamental misunderstanding derails the entire solution from the start.
Students often misread this as "19 students scored lower than Angela" instead of "19 students scored higher than Angela." This reversal completely changes the calculation for how many students in the second class scored below Angela's level.
When determining how many students scored above or below Angela, students frequently forget that Angela's own score needs to be accounted for separately. They might incorrectly assume that if 72 students scored below Angela in a class of 80, then 8 students scored above her, missing that Angela herself takes up one position.
Students may incorrectly compute \(0.90 \times 80\), getting 8 instead of 72, or make similar multiplication errors. This foundational calculation error propagates through the entire problem.
When calculating \(100 - 19 - 0 = 81\), students might forget to account for the fact that no one has Angela's exact score, leading them to calculate \(100 - 19 = 81\) and get confused about whether this is correct, or they might subtract Angela herself and get \(100 - 19 - 1 = 80\).
Students often struggle with fraction simplification and decimal conversion. They might incorrectly simplify \(\frac{153}{180}\) or make errors when converting to a percentage, potentially getting \(0.85\) but then forgetting to multiply by 100 for the final percentile.
After correctly calculating that \(\frac{153}{180} = 0.85\), students might select this decimal value if it appears among the choices, forgetting that percentiles are expressed as whole numbers rather than decimals.
Students who get confused during the calculation might revert back to Angela's original 90th percentile position and select choice E (92), thinking this is close enough to her original standing, rather than recognizing that combining with a stronger class would lower her relative position.