AMC 8 · 2001 · #6

Grade 4 arithmeticpattern

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

equal-spacingratio-proportionmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: equal-spacingratio-proportion

📏 Short solution 💡 2 insights

Problem

Six trees stand in a straight line, equally spaced. From the first tree to the fourth tree the distance is 60 feet. How far is it from the first tree to the last (sixth) tree?

Pick an answer.

(A)

(B)

100

(C)

105

(D)

120

(E)

140

AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Break Into Subproblems

The trap is counting trees instead of the spaces between them — the classic "fence-post" mistake. Tool #1 (Draw a Picture) clears that up: sketch six dots in a row and mark the gaps, and you can see at a glance that going from tree 1 to tree 4 crosses 3 gaps, not 4. Tool #7 (Break Into Subproblems) then splits the work into two clean steps: (a) use the 60 feet to find one gap length, (b) multiply by the number of gaps from tree 1 to tree 6.

1STEP 1

Sketch six dots ● ● ● ● ● ●: tree 1 to tree 4 crosses gaps 1-2, 2-3, 3-4, so there are 3 gaps, not 4 (in general b − a).

gaps from tree 1 to tree 4 = 4 - 1 = 3

2STEP 2

Split the 60 feet evenly among the 3 gaps: each gap is 60 ÷ 3 = 20 feet.

one gap =

\frac{60}{3}

= 20 ft

3STEP 3

Tree 1 to tree 6 spans 6 − 1 = 5 gaps, each 20 feet, so the distance is 5 × 20 = 100 feet → (B).

distance = 5 × 20 = 100 ft → (B)

Answer

100

Cross-check with a ratio. Tree 1 to tree 4 covers 3 gaps and is 60 ft. Tree 1 to tree 6 covers 5 gaps, so the distance should be

\frac{5}{3}

× 60 = 100 ft. Same answer, no need to find the gap length explicitly. Also a quick sanity check on the choices: each gap must be a whole-feeling number, and

\frac{60}{3}

= 20 gives nice round distances like 100 — choice (B) lines up.

💡Key takeaway

When things are lined up evenly, count the spaces between them, not the things themselves. Three spaces give 60 feet, so one space is 20 feet — and five spaces from tree 1 to tree 6 is 5 × 20 = 100 feet, answer (B).

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