AMC 10 · 2018 · #6

Grade 8 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
mean-median-mode-rangesystems-of-equationslinear-equations-two-varinterval-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: mean-median-mode-rangelinear-equations-two-var
📏 Medium solution 💡 3 insights
Problem
Two positive whole numbers m and n satisfy that m plus 10 is less than n plus 1. The six numbers m, m plus 4, m plus 10, n plus 1, n plus 2, and 2n have a mean equal to n, and their median is also equal to n. Find the value of m plus n.

Pick an answer.

(A)
20
(B)
21
(C)
22
(D)
23
(E)
24
How to solve
Strategy Convert to Algebra

The problem hands over two separate facts about the same six numbers: the mean is n and the median is n. Each fact turns into one equation in the two letters m and n, so the natural move is to write both equations and solve them together. Before either equation can be written, the six numbers must be put in order, because the median depends on position; the given inequality m+10 ＜ n+1 does exactly that work. Treating the median condition and the mean condition as two separate subproblems keeps the algebra short, and plugging the final numbers back into the original set checks the whole chain.

1STEP 1

Sort the six numbers

The inequality sorts them completely.

m ＜ m+4 ＜ m+10 ＜ n+1 ＜ n+2 ＜ 2n
2STEP 2

Turn the median into an equation

The median averages the middle two.

(m+10)+(n+1)/2=n ⟹ m+n+11=2n ⟹ n=m+11
3STEP 3

Turn the mean into an equation

The mean gives a second equation.

(3m+4n+17)/6=n ⟹ 3m+4n+17=6n ⟹ 3m+17=2n
4STEP 4

Solve the two equations together

Solving both together fixes the numbers.

3m+17=2(m+11)=2m+22 ⟹ m=5, n=5+11=16
5STEP 5

Check the set and answer

Their sum is 21.

{5,9,15,17,18,32}: 96/6=16, (15+17)/2=16, m+n=21
Answer
21
The found values obey every stated condition: m=5 and n=16 are positive integers, m+10=15 is less than n+1=17, and the set {5,9,15,17,18,32} has mean 16 and median 16. The answer can also be squeezed from the choices. The median condition alone forces n=m+11, so m+n=2m+11 is always odd, which rules out 20, 22, and 24 immediately. That leaves 21 and 23. The value 23 would mean m=6 and n=17, giving the set {6,10,16,18,19,34} with sum 103; since 103 divided by 6 is not 17, choice (D) fails and 21 is the only survivor.
💡Key takeaway

When a list has an even number of entries, its median is the average of the two middle ones, so a mean condition and a median condition become two equations you can solve together.

  • Sort the six numbers
  • Turn the median into an equation
  • Turn the mean into an equation
  • Solve the two equations together
  • Check the set and answer

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