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 are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees?

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: There are $6$ trees in a row; Consecutive trees are the same distance apart; Distance from tree $1$ to tree $4$ is $60$ feet; Answer choices: (A) $90$, (B) $100$, (C) $105$, (D) $120$, (E) $140$ (feet)

Unknowns: Distance from tree $1$ to tree $6$ (in feet)

Understand

Restated: 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?

Plan

Primary tool: #7 Break Into Subproblems

Secondary: #1 Draw a Picture

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 $6$.

Execute — Answer: B

#1 Draw a Picture 4.OA.A.3 Step 1

Draw the row of trees and count gaps.
Picture six dots: $\bullet\;\bullet\;\bullet\;\bullet\;\bullet\;\bullet$.
From tree $1$ to tree $4$ you step over the gaps between $1$-$2$, $2$-$3$, and $3$-$4$ — that is $3$ gaps.
In general, going from tree $a$ to tree $b$ uses $b - a$ gaps.

$$\text{gaps from tree } 1 \text{ to tree } 4 = 4 - 1 = 3$$

💡 The Grade 4 "interpret a word problem" move: read carefully and notice $4 - 1 = 3$ gaps, not $4$.

#7 Break Into Subproblems 3.OA.A.3 Step 2

Subproblem 1: find the length of one gap.
The $60$ feet from tree $1$ to tree $4$ is divided evenly among the $3$ equal gaps, so each gap is $60 \div 3$ feet.

$$\text{one gap} = \dfrac{60}{3} = 20 \text{ ft}$$

💡 Splitting $60$ feet equally into $3$ gaps is Grade 3 equal-sharing division.

#7 Break Into Subproblems 3.OA.A.3 Step 3

Subproblem 2: count gaps from tree $1$ to tree $6$, then multiply by the gap length.
From tree $1$ to tree $6$ there are $6 - 1 = 5$ gaps, and each gap is $20$ feet.

$$\text{distance} = 5 \times 20 = 100 \text{ ft} \;\Rightarrow\; \textbf{(B)}$$

💡 Five equal jumps of $20$ feet is Grade 3 "equal groups" multiplication.

[1] #1 4.OA.A.3 Draw the row of trees and count gaps. Picture six dots: $\bullet\;\bullet\;\bull

[2] #7 3.OA.A.3 Subproblem 1: find the length of one gap. The $60$ feet from tree $1$ to tree $4

[3] #7 3.OA.A.3 Subproblem 2: count gaps from tree $1$ to tree $6$, then multiply by the gap len

Review

Reasonableness: 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 $\dfrac{5}{3} \times 60 = 100$ 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 $\tfrac{60}{3} = 20$ gives nice round distances like $100$ — choice (B) lines up.

Alternative: Tool #5 (Look for a Pattern) on cumulative distances from tree $1$. Tree $1$ to tree $2$: $20$ ft. Tree $1$ to tree $3$: $40$ ft. Tree $1$ to tree $4$: $60$ ft (matches the given). Tree $1$ to tree $5$: $80$ ft. Tree $1$ to tree $6$: $100$ ft. The pattern "$20$ ft per extra tree past the first" lands directly on $100$ ft for tree $6$.

CCSS standards used (min grade 4)

4.OA.A.3 Solve multistep word problems with the four operations, including problems with whole-number remainders (Reading the problem carefully to see that going from tree $1$ to tree $4$ crosses $4 - 1 = 3$ gaps (not $4$), which is the key multistep insight.)
3.OA.A.3 Use multiplication and division within $100$ to solve word problems in equal groups (Dividing $60 \div 3 = 20$ to get one gap length, then multiplying $5 \times 20 = 100$ to get the distance from tree $1$ to tree $6$.)

⭐ 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 $5 \times 20 = 100$ feet, answer (B).

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 4)

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