AMC 8 · 2013 · #5

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

mean-median-mode-rangemental-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Short solution 💡 2 insights

Problem

Hammie weighs 106 pounds and his four quadruplet baby sisters weigh 5, 5, 6, and 8 pounds. For these five children, decide whether the mean weight or the median weight is larger, and by how many pounds.

Pick an answer.

(A)

median, by 60

(B)

median, by 20

(C)

average, by 5

(D)

average, by 15

(E)

average, by 20

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

The question bundles two independent summary statistics into one comparison, so Tool #7 (Identify Subproblems) splits it into three clean pieces: find the median, find the mean, then compare. Tool #2 (Make a Systematic List) does the heavy lifting for the median — sort the five weights from smallest to largest and read off the 3rd entry. The median needs no arithmetic at all once the list is in order, which makes the contrast with the mean (a sum-and-divide computation) very visible: one giant outlier (106) doesn't move the middle slot, but it nearly triples the average.

1STEP 1

Sort the five weights smallest to largest; once ordered, the median is just the value at position 3.

sorted: {5, 5, 6, 8, 106}

2STEP 2

Subproblem 1: with 5 values the middle is the 3rd sorted entry, so the median weight is 6 pounds.

median = (3rd of 5) = 6 lb

3STEP 3

Subproblem 2: add the five weights to get the total, 130 pounds, then divide by 5.

sum = 5 + 5 + 6 + 8 + 106 = 130

4STEP 4

Divide the total by the 5 children to get the mean, 26 pounds.

mean =

\frac{130}{5}

= 26 lb

5STEP 5

Subproblem 3: the mean (26) beats the median (6); subtracting gives a gap of 20 pounds → (E).

mean - median = 26 - 6 = 20 → average, by 20 → (E)

Answer

average, by 20

Four of the five weights are tiny (5, 5, 6, 8), all clustered near 6, so the median should be tiny too — and it is, exactly 6. The fifth weight (106) is huge and pulls the average up; spreading the extra 106 - 6 = 100 pounds across 5 children adds 100 / 5 = 20 pounds to a baseline of 6, giving a mean of 26. That's exactly the 20-pound gap the answer claims, and it lines up with (E).

💡Key takeaway

This AMC 8 problem only needs Grade 6 mean-vs-median ideas you already know — and one big outlier (106 pounds!) shifts the mean way more than the median.

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