Angela's grade was in the 90th percentile out of 80 grades in her class. In another class of 100 students...
GMAT Number Properties : (NP) Questions
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?
- Translate the problem requirements: Understand what "90th percentile" means (how many students are below Angela), determine Angela's position in each class, then find her percentile when both classes are combined
- Determine Angela's position in her original class: Use the 90th percentile information to find how many students scored below Angela in the 80-student class
- Analyze Angela's position in the second class: Use the given information about 19 grades being higher than Angela's to determine how many students scored below her in the 100-student class
- Calculate combined percentile: Add up all students below Angela from both classes and divide by total students to find her percentile in the combined group
Execution of Strategic Approach
1. Translate the problem requirements
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
2. Determine Angela's position in her original class
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:
- 72 students scored below Angela
- Angela has her score (but we're told nobody else has her exact score)
- The remaining students scored above Angela: \(80 - 72 - 1 = 7\) students scored above Angela
Let's verify: 7 students above her means she's better than 72 out of 79 other students, which is \(\frac{72}{80} = 0.90 = 90\text{th percentile}\) ✓
3. Analyze Angela's position in the second class
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:
- 19 students scored above Angela
- 0 students have Angela's exact score (given in the problem)
- The remaining students scored below Angela: \(100 - 19 - 0 = 81\) students scored below Angela
So in the second class, 81 students scored below Angela's level.
4. Calculate combined percentile
Now we can combine both classes to find Angela's overall percentile.
Total students below Angela:
- From her original class: 72 students
- From the second class: 81 students
- Total below Angela: \(72 + 81 = 153\) students
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
Final Answer
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
Common Faltering Points
Errors while devising the approach
1. Misunderstanding what percentile means
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.
2. Incorrectly interpreting "19 grades higher than Angela's"
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.
3. Forgetting that Angela herself is not counted in either "above" or "below" groups
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.
Errors while executing the approach
1. Arithmetic errors when calculating 90% of 80
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.
2. Incorrect subtraction when finding students below Angela in the second class
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 \(100 - 19 - 1 = 80\).
3. Converting the fraction 153/180 incorrectly
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.
Errors while selecting the answer
1. Selecting 0.85 instead of 85
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.
2. Confusing Angela's original percentile with the final answer
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.