AMC 10 · 2012 · #8

Grade 6 algebra
Archetype Pair-Sum Invariant └ Sub-archetype Universe Total Minus Known
systems-of-equations convert-to-algebra ↑ Prerequisites: linear-equations-two-var
📏 Short solution 💡 2 insights
Problem
Three whole numbers are added two at a time, giving the pair sums 12, 17, and 19. When the three numbers are put in order from smallest to largest, find the middle number.

Pick an answer.

(A)
4
(B)
5
(C)
6
(D)
7
(E)
8

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Tool #4 (Introduce a Variable) turns the words into three clean pair-sum equations by naming the numbers a ≤ b ≤ c. Tool #15 (Organize Information in More Ways) is what makes it easy: order the pair sums so each sum is matched to the pair it came from. Then Tool #7 (Identify Subproblems) splits the work into two small steps — first find the total of all three numbers, then peel off the middle one — instead of solving a full three-way system.

1STEP 1

Name the numbers, match the sums

Call the numbers a ≤ b ≤ c; the smallest pair gives the smallest sum, so a+b=12, a+c=17, b+c=19.

a+b=12, a+c=17, b+c=19
2STEP 2

Add all three pair sums

Add all three equations: every number appears twice, so 2(a+b+c)=48 and the grand total is a+b+c=24.

(a+b)+(a+c)+(b+c)=2(a+b+c)=12+17+19=48 → a+b+c=24
3STEP 3

Peel off the middle number

The pair a+c=17 leaves b out, so b = 24 - 17 = 7, choice (D); the numbers are 5, 7, 12.

b = 24 - 17 = 7 → (D)
Answer
7
The recovered numbers 5, 7, 12 are whole numbers and their pair sums are 5+7=12, 5+12=17, 7+12=19 — exactly the three sums given, so nothing was invented. The middle value is 7, and it sits between 5 and 12 as a middle number should. A tempting trap is 5 (choice B), which is the smallest number, not the middle one; the question asks for the middle, so 7 is correct.
💡Key takeaway

Add all the pair sums to get double the total, then subtract the pair that skips the middle number to find it.

  • Name the numbers, match the sums
  • Add all three pair sums
  • Peel off the middle number

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