AMC 8 · 2001 · #6
Easy mode Grade 4Problem
Picture 6 trees in a straight row along the side of a road. The trees are spaced evenly, so the gap between any two next-door trees is the same.
The distance from the 1st tree to the 4th tree is 60 feet.
What is the distance from the 1st tree to the 6th (last) tree, in feet?
Pick an answer.
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.
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?
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)
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
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.
💡 The Grade 4 "interpret a word problem" move: read carefully and notice $4 - 1 = 3$ gaps, not $4$.
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.
💡 Splitting $60$ feet equally into $3$ gaps is Grade 3 equal-sharing division.
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.
💡 Five equal jumps of $20$ feet is Grade 3 "equal groups" multiplication.
4.OA.A.3 Draw the row of trees and count gaps. Picture six dots: $\bullet\;\bullet\;\bull 3.OA.A.3 Subproblem 1: find the length of one gap. The $60$ feet from tree $1$ to tree $4 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.3Solve 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.3Use 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).
⭐ 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).
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