AMC 10 · 2005 · #19
Grade 6 arithmeticOn a certain math exam, 10% of the students got 70 points, 25% got 80 points, 20% got 85 points, 15% got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam?
Pick an answer.
AMC 10 2005 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 math exam, $10\%$ of the students scored $70$, $25\%$ scored $80$, $20\%$ scored $85$, $15\%$ scored $90$, and everyone else scored $95$. Find the difference between the mean score and the median score.
Givens: $10\%$ scored $70$; $25\%$ scored $80$; $20\%$ scored $85$; $15\%$ scored $90$; The rest scored $95$; Answer choices: (A) $0$, (B) $1$, (C) $2$, (D) $4$, (E) $5$
Unknowns: The difference between the mean score and the median score
Understand
Restated: On a math exam, $10\%$ of the students scored $70$, $25\%$ scored $80$, $20\%$ scored $85$, $15\%$ scored $90$, and everyone else scored $95$. Find the difference between the mean score and the median score.
Givens: $10\%$ scored $70$; $25\%$ scored $80$; $20\%$ scored $85$; $15\%$ scored $90$; The rest scored $95$; Answer choices: (A) $0$, (B) $1$, (C) $2$, (D) $4$, (E) $5$
Plan
Primary tool: #9 Solve an Easier Related Problem
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
The problem gives percentages but no head count, so Tool #9 (Solve an Easier Related Problem) pins the count at a convenient $100$ students; then every percentage becomes a plain number of students and the abstract question turns into counting real scores. Tool #7 (Identify Subproblems) splits the work into two independent pieces — find the median, find the mean — and only at the end subtracts them. Tool #3 (Eliminate Possibilities) uses the multiple-choice frame as a safety net: the difference is a small whole number, so a quick estimate can rule out the far-apart choices.
Execute — Answer: B
6.RP.A.3 Step 1 Turn percents into student counts
- The four listed groups add to $10+25+20+15=70$ percent, so the rest is $100-70=30$ percent scoring $95$.
- Because the total number of students is not given, pick a convenient one: $100$ students.
- Then the percentages become head counts directly — $10$ students scored $70$, $25$ scored $80$, $20$ scored $85$, $15$ scored $90$, and $30$ scored $95$.
💡 With no head count given, $100$ students makes each percent an actual number of kids.
6.SP.B.5 Step 2 Locate the median
- Line all $100$ scores up from lowest to highest.
- Positions $1$ to $10$ are the $70$s, positions $11$ to $35$ are the $80$s (that is $10+25$), and positions $36$ to $55$ are the $85$s (that is $35+20$).
- The median of $100$ values sits between the $50$th and $51$st scores.
- Both of those land inside the $85$ block, so the median is $85$.
💡 Stack the scores in order and the middle of the pile is the median.
6.SP.B.5 Step 3 Compute the mean
- Add every student's score, then divide by $100$.
- Multiply each score by how many students got it: $70\times10=700$, $80\times25=2000$, $85\times20=1700$, $90\times15=1350$, $95\times30=2850$.
- The total is $700+2000+1700+1350+2850=8600$, so the mean is $8600\div100=86$.
💡 A weighted average pays each score by how many students earned it, then splits the pot evenly.
6.SP.A.3 Step 4 Subtract to get the difference
- The mean is $86$ and the median is $85$, so the difference is $86-85=1$.
- The two centers land almost on top of each other, which rules out the larger choices $2$, $4$, and $5$; and since the mean sits above the median, the difference is not $0$ either.
- So the difference is $1$, choice (B).
💡 Two measures of center for the same data are close, so their gap is a tiny whole number.
6.RP.A.3 The four listed groups add to $10+25+20+15=70$ percent, so the rest is $100-70=3 6.SP.B.5 Line all $100$ scores up from lowest to highest. Positions $1$ to $10$ are the $ 6.SP.B.5 Add every student's score, then divide by $100$. Multiply each score by how many 6.SP.A.3 The mean is $86$ and the median is $85$, so the difference is $86-85=1$. The two Review
Reasonableness: The mean $86$ and median $85$ both sit inside the $80$-to-$90$ range where almost all the scores are packed, so both numbers are believable. The mean edges above the median because the $30\%$ of high $95$s pull the average up while the median only cares about the middle position — a right-leaning spread, exactly what a difference of $1$ (mean bigger than median) signals. Using $100$ students was a free choice: with $20$ students the counts $2,5,4,3,6$ give the same median $85$ and mean $1720\div20=86$, so the difference stays $1$, confirming it does not depend on the head count.
Alternative: Skip picking a count and work in percents as weights. Median: the running totals $10\%,35\%,55\%$ show the $50\%$ mark falls in the $85$ group, so the median is $85$. Mean: $0.10(70)+0.25(80)+0.20(85)+0.15(90)+0.30(95)=7+20+17+13.5+28.5=86$. The difference is $86-85=1$, again (B).
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning, including percent, to solve problems (Reading each percent as a number of students out of a chosen total of $100$, and finding the missing group as $100\%-70\%=30\%$.)6.SP.B.5Summarize numerical data sets, giving quantitative measures of center (median and mean) (Locating the median as the middle of the ordered scores ($85$) and computing the mean as the weighted average of all scores ($86$).)6.SP.A.3Recognize that a measure of center summarizes a data set with a single number (Comparing the two single-number centers, mean $86$ and median $85$, and subtracting to get the difference $1$.)
⭐ When only percents are given, pretend there are $100$ people so each percent becomes a count; then the median is the middle score in order and the mean is the weighted average.
⭐ When only percents are given, pretend there are $100$ people so each percent becomes a count; then the median is the middle score in order and the mean is the weighted average.
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