AMC 10 · 2014 · #5
Grade 6 arithmeticOn an algebra quiz, 10% of the students scored 70 points, 35% scored 80 points, 30% scored 90 points, and the rest scored 100 points. What is the difference between the mean and median score of the students' scores on this quiz?
Pick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: On a quiz, $10\%$ of students scored $70$, $35\%$ scored $80$, $30\%$ scored $90$, and everyone else scored $100$. Find the difference between the mean score and the median score.
Givens: $10\%$ scored $70$ points; $35\%$ scored $80$ points; $30\%$ scored $90$ points; The rest scored $100$ points; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$
Unknowns: The mean of all the scores; The median of all the scores; The (positive) difference between the mean and the median
Understand
Restated: On a quiz, $10\%$ of students scored $70$, $35\%$ scored $80$, $30\%$ scored $90$, and everyone else scored $100$. Find the difference between the mean score and the median score.
Givens: $10\%$ scored $70$ points; $35\%$ scored $80$ points; $30\%$ scored $90$ points; The rest scored $100$ points; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$
Plan
Primary tool: #15 Organize Information in More Ways
Secondary: #16 Change Focus / Count the Complement, #4 Introduce a Variable
The scores come as a distribution (percent of students at each score), not a plain list. Tool #15 says: lay the data out in an organized table — score, share of students, and running total — because that single table hands you both answers. The mean is the weighted sum straight from the shares; the median is wherever the running total first passes the halfway mark. First, Tool #16 fills the gap: the four groups must total $100\%$, so the unstated group is the complement of the others. Then Tool #4 makes the counting concrete — instead of juggling percentages, pick one convenient class size so every group becomes a whole number of students and the median is easy to point at.
Execute — Answer: C
6.RP.A.3 Step 1 Find the missing group
- The four groups have to account for all the students, so their percentages add to $100\%$.
- Subtract the three known shares to get the last one.
💡 "The rest" always means whatever is left over after the named parts — the complement up to the whole.
6.RP.A.3 Step 2 Pick a convenient class size
- Nothing depends on the actual head count, so choose a number that turns every percentage into a whole number of students.
- A class of $20$ works: $10\%$, $35\%$, $30\%$, $25\%$ of $20$ are all whole.
💡 Turning percents into a real count of $20$ students makes the middle student easy to locate.
6.SP.B.5 Step 3 Compute the mean
- The mean is the total of all $20$ scores divided by $20$.
- Multiply each score by how many students got it, add, then divide.
💡 A weighted average just counts each score as many times as it actually occurred.
6.SP.B.5 Step 4 Find the median
- Line the $20$ scores up from lowest to highest.
- The median is the average of the $10$th and $11$th scores.
- Using the running total: positions $1$–$2$ are $70$, positions $3$–$9$ are $80$, positions $10$–$15$ are $90$.
- So both the $10$th and $11$th scores are $90$.
💡 The median sits where the running count first crosses the halfway point — here, inside the $90$ group.
6.NS.B.3 Step 5 Take the difference
- Subtract the smaller center from the larger.
- The mean is $87$ and the median is $90$, so their difference is $90 - 87 = 3$.
💡 The low $70$ scores drag the mean below the median, so the gap between them is what we report.
6.RP.A.3 The four groups have to account for all the students, so their percentages add t 6.RP.A.3 Nothing depends on the actual head count, so choose a number that turns every pe 6.SP.B.5 The mean is the total of all $20$ scores divided by $20$. Multiply each score by 6.SP.B.5 Line the $20$ scores up from lowest to highest. The median is the average of the 6.NS.B.3 Subtract the smaller center from the larger. The mean is $87$ and the median is Review
Reasonableness: The result must not depend on the class size, and it doesn't: using percentages directly gives the same mean $0.10\cdot70 + 0.35\cdot80 + 0.30\cdot90 + 0.25\cdot100 = 7 + 28 + 27 + 25 = 87$, and the median is still $90$ because $45\%$ of students scored below $90$ while only $25\%$ scored above it, so the halfway student sits in the $90$ group. Mean $87$, median $90$, difference $3$. The mean landing below the median also makes sense — the small but heavy tail of low $70$s pulls the average down. The answer $3$ is choice (C).
Alternative: Skip choosing a class size and reason in percentages alone (Tool #16 on the median): the bottom $10\% + 35\% = 45\%$ scored at most $80$, and adding the $90$ group reaches $75\%$, so the $50\%$ mark lands inside the $90$ group — median $90$. The mean is the percentage-weighted sum $87$. Same difference, $3$, with no need to invent a head count.
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Finding the missing $25\%$ group as the complement to $100\%$, and converting each percentage into a whole number of students out of $20$.)6.SP.B.5Summarize numerical data sets by reporting number of observations and measures (Computing the mean as a weighted average and locating the median from the ordered scores / running totals.)6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals (Carrying out the weighted-sum arithmetic and the final subtraction $90 - 87 = 3$.)
⭐ A pile of percentages is really an ordered data list — build the table once and the mean pops out as a weighted average while the median is just the score sitting at the halfway mark.
⭐ A pile of percentages is really an ordered data list — build the table once and the mean pops out as a weighted average while the median is just the score sitting at the halfway mark.
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