Competition · AMC preparation · step 4 of 4

AMC 10 · 2012A · #3

Grade 6 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum
interval-arithmeticabsolute-value physical-representationidentify-subproblems ↑ Prerequisites: multi-digit-arithmetic
📏 Medium solution 💡 1 insight
Problem
A bug on a number line starts at -2, crawls to -6, then turns around and crawls to 5. Find the total number of units it travels along the way — the full path length, not how far it ends from where it began.

Pick an answer.

(A)
9
(B)
11
(C)
13
(D)
14
(E)
15

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

How to solve
Strategy Draw a Diagram

The bug moves along positions on a number line, so the natural first move is Tool #1: sketch the line and mark -2, -6, and 5. The picture instantly shows the trip has two straight legs in opposite directions. Tool #7 then splits the whole crawl into those two subproblems — leg one from -2 to -6, leg two from -6 to 5 — each an easy distance on its own. Tool #8 keeps the focus on "units of distance," reminding us that a distance is always the positive gap between two points (never a negative), so the two legs simply add up.

1STEP 1

Draw the number line

Sketch a number line and mark the start -2, the turning point -6, and the end 5.

-6 longleftarrow -2 then -6 ⟶ 5
2STEP 2

Length of leg one

Leg one runs from -2 to -6, so its length is the absolute value of the difference: 4 units.

|-6 - (-2)| = |-4| = 4
3STEP 3

Length of leg two

Leg two runs from -6 to 5, and the same absolute value of the difference gives 11 units.

|5 - (-6)| = |11| = 11
4STEP 4

Add the two legs

The total crawl is the sum of the two legs: 4 + 11 = 15 units.

4 + 11 = 15 → (E)
Answer
15
The total must be larger than the longer single leg of 11 units, because the bug also crawled the 4-unit backtrack before it — and 15 ＞ 11, so that fits. Notice the choice 11 is the trap for anyone who counts only the long leg and forgets the first crawl. The full trip runs from -6 up to 5 (a span of 11) plus the extra 4 units the bug went left first, giving 15 units, so (E) is consistent.
💡Key takeaway

Distance is always the positive gap between two points, so when something turns around you add each leg separately: 4 + 11 = 15 units.

  • Draw the number line
  • Length of leg one
  • Length of leg two
  • Add the two legs

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