← Count posts via length divided by spacing · Objects versus Gaps (Fencepost Counting)

Count posts via length divided by spacing · 11 practice problems

3.OA.C.73.OA.A.3

Generated variants — 11

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 18 trees

A straight walking trail is $24$ feet long. Trees are to be planted along both sides of the trail, spaced $3$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 24-foot trail has trees planted every 3 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 24 feet long and straight.
Trees are spaced 3 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 3-foot gaps between trees on one side.

24 \div 3 = 8

Each gap is one 3-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

8 + 1 = 9

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 9 trees, and there are two sides, so multiply by 2.

9 \times 2 = 18

Both sides are identical, so the total is just twice one side's count.

Answer: 18 trees

4 · Reviewdoes it hold up?

9 trees per side over 24 feet (8 gaps of 3 feet = 24 feet) fits exactly, and doubling to 18 for two sides is reasonable. Forgetting the extra end tree would wrongly give 16.

Another way: Solve an easier related problem (tool 9): a 6-foot trail with 3-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 24 by 3 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!

Variant 2 easy answer: 32 trees

A straight walking trail is $30$ feet long. Trees are to be planted along both sides of the trail, spaced $2$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 30-foot trail has trees planted every 2 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 30 feet long and straight.
Trees are spaced 2 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 2-foot gaps between trees on one side.

30 \div 2 = 15

Each gap is one 2-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

15 + 1 = 16

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 16 trees, and there are two sides, so multiply by 2.

16 \times 2 = 32

Both sides are identical, so the total is just twice one side's count.

Answer: 32 trees

4 · Reviewdoes it hold up?

16 trees per side over 30 feet (15 gaps of 2 feet = 30 feet) fits exactly, and doubling to 32 for two sides is reasonable. Forgetting the extra end tree would wrongly give 30.

Another way: Solve an easier related problem (tool 9): a 4-foot trail with 2-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 30 by 2 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!

Variant 3 easy answer: 20 trees

A straight walking trail is $36$ feet long. Trees are to be planted along both sides of the trail, spaced $4$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 36-foot trail has trees planted every 4 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 36 feet long and straight.
Trees are spaced 4 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 4-foot gaps between trees on one side.

36 \div 4 = 9

Each gap is one 4-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

9 + 1 = 10

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 10 trees, and there are two sides, so multiply by 2.

10 \times 2 = 20

Both sides are identical, so the total is just twice one side's count.

Answer: 20 trees

4 · Reviewdoes it hold up?

10 trees per side over 36 feet (9 gaps of 4 feet = 36 feet) fits exactly, and doubling to 20 for two sides is reasonable. Forgetting the extra end tree would wrongly give 18.

Another way: Solve an easier related problem (tool 9): a 8-foot trail with 4-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 36 by 4 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!

Variant 4 medium answer: 18 trees

A straight walking trail is $40$ feet long. Trees are to be planted along both sides of the trail, spaced $5$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 40-foot trail has trees planted every 5 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 40 feet long and straight.
Trees are spaced 5 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 5-foot gaps between trees on one side.

40 \div 5 = 8

Each gap is one 5-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

8 + 1 = 9

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 9 trees, and there are two sides, so multiply by 2.

9 \times 2 = 18

Both sides are identical, so the total is just twice one side's count.

Answer: 18 trees

4 · Reviewdoes it hold up?

9 trees per side over 40 feet (8 gaps of 5 feet = 40 feet) fits exactly, and doubling to 18 for two sides is reasonable. Forgetting the extra end tree would wrongly give 16.

Another way: Solve an easier related problem (tool 9): a 10-foot trail with 5-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 40 by 5 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!

Variant 5 medium answer: 18 trees

A straight walking trail is $48$ feet long. Trees are to be planted along both sides of the trail, spaced $6$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 48-foot trail has trees planted every 6 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 48 feet long and straight.
Trees are spaced 6 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 6-foot gaps between trees on one side.

48 \div 6 = 8

Each gap is one 6-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

8 + 1 = 9

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 9 trees, and there are two sides, so multiply by 2.

9 \times 2 = 18

Both sides are identical, so the total is just twice one side's count.

Answer: 18 trees

4 · Reviewdoes it hold up?

9 trees per side over 48 feet (8 gaps of 6 feet = 48 feet) fits exactly, and doubling to 18 for two sides is reasonable. Forgetting the extra end tree would wrongly give 16.

Another way: Solve an easier related problem (tool 9): a 12-foot trail with 6-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 48 by 6 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!

Variant 6 medium answer: 22 trees

A straight walking trail is $50$ feet long. Trees are to be planted along both sides of the trail, spaced $5$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 50-foot trail has trees planted every 5 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 50 feet long and straight.
Trees are spaced 5 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 5-foot gaps between trees on one side.

50 \div 5 = 10

Each gap is one 5-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

10 + 1 = 11

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 11 trees, and there are two sides, so multiply by 2.

11 \times 2 = 22

Both sides are identical, so the total is just twice one side's count.

Answer: 22 trees

4 · Reviewdoes it hold up?

11 trees per side over 50 feet (10 gaps of 5 feet = 50 feet) fits exactly, and doubling to 22 for two sides is reasonable. Forgetting the extra end tree would wrongly give 20.

Another way: Solve an easier related problem (tool 9): a 10-foot trail with 5-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 50 by 5 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!

Variant 7 medium answer: 22 trees

A straight walking trail is $60$ feet long. Trees are to be planted along both sides of the trail, spaced $6$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 60-foot trail has trees planted every 6 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 60 feet long and straight.
Trees are spaced 6 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 6-foot gaps between trees on one side.

60 \div 6 = 10

Each gap is one 6-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

10 + 1 = 11

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 11 trees, and there are two sides, so multiply by 2.

11 \times 2 = 22

Both sides are identical, so the total is just twice one side's count.

Answer: 22 trees

4 · Reviewdoes it hold up?

11 trees per side over 60 feet (10 gaps of 6 feet = 60 feet) fits exactly, and doubling to 22 for two sides is reasonable. Forgetting the extra end tree would wrongly give 20.

Another way: Solve an easier related problem (tool 9): a 12-foot trail with 6-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 60 by 6 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!

Variant 8 medium answer: 26 trees

A straight walking trail is $84$ feet long. Trees are to be planted along both sides of the trail, spaced $7$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 84-foot trail has trees planted every 7 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 84 feet long and straight.
Trees are spaced 7 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 7-foot gaps between trees on one side.

84 \div 7 = 12

Each gap is one 7-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

12 + 1 = 13

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 13 trees, and there are two sides, so multiply by 2.

13 \times 2 = 26

Both sides are identical, so the total is just twice one side's count.

Answer: 26 trees

4 · Reviewdoes it hold up?

13 trees per side over 84 feet (12 gaps of 7 feet = 84 feet) fits exactly, and doubling to 26 for two sides is reasonable. Forgetting the extra end tree would wrongly give 24.

Another way: Solve an easier related problem (tool 9): a 14-foot trail with 7-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 84 by 7 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!

Variant 9 hard answer: 22 trees

A straight walking trail is $90$ feet long. Trees are to be planted along both sides of the trail, spaced $9$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 90-foot trail has trees planted every 9 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 90 feet long and straight.
Trees are spaced 9 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 9-foot gaps between trees on one side.

90 \div 9 = 10

Each gap is one 9-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

10 + 1 = 11

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 11 trees, and there are two sides, so multiply by 2.

11 \times 2 = 22

Both sides are identical, so the total is just twice one side's count.

Answer: 22 trees

4 · Reviewdoes it hold up?

11 trees per side over 90 feet (10 gaps of 9 feet = 90 feet) fits exactly, and doubling to 22 for two sides is reasonable. Forgetting the extra end tree would wrongly give 20.

Another way: Solve an easier related problem (tool 9): a 18-foot trail with 9-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 90 by 9 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!

Variant 10 hard answer: 30 trees

A straight walking trail is $98$ feet long. Trees are to be planted along both sides of the trail, spaced $7$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 98-foot trail has trees planted every 7 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 98 feet long and straight.
Trees are spaced 7 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 7-foot gaps between trees on one side.

98 \div 7 = 14

Each gap is one 7-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

14 + 1 = 15

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 15 trees, and there are two sides, so multiply by 2.

15 \times 2 = 30

Both sides are identical, so the total is just twice one side's count.

Answer: 30 trees

4 · Reviewdoes it hold up?

15 trees per side over 98 feet (14 gaps of 7 feet = 98 feet) fits exactly, and doubling to 30 for two sides is reasonable. Forgetting the extra end tree would wrongly give 28.

Another way: Solve an easier related problem (tool 9): a 14-foot trail with 7-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 98 by 7 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!

Variant 11 hard answer: 24 trees

A straight walking trail is $99$ feet long. Trees are to be planted along both sides of the trail, spaced $9$ feet apart. If trees are planted from the very start of the trail to the very end, how many trees are needed in all? (Ignore the thickness of each tree.)

Show solution

1 · Understandwhat's really being asked

A straight 99-foot trail has trees planted every 9 feet along both sides, from start to end. We must count the total number of trees on both sides.

Givens

The trail is 99 feet long and straight.
Trees are spaced 9 feet apart.
Trees are planted along both sides.
Trees go from the very start to the very end of the trail.

Unknowns

The total number of trees on both sides combined.

Constraints

Both endpoints of each side get a tree (planting includes start and end).
Tree thickness is ignored.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern

Sketch a row of posts and gaps to see that a straight line with both ends planted has one more tree than gaps. Find the count per side, then double it for two sides.

3 · Execute3 carry out the plan

1Count the gaps along one side

#1 Draw a Diagram 3.OA.C.7

Dividing the trail length by the spacing gives the number of 9-foot gaps between trees on one side.

99 \div 9 = 11

Each gap is one 9-foot step, so the length divided by the step gives the gap count.

2Add one for the posts-versus-gaps rule

#5 Look for a Pattern 3.OA.A.3

On a straight line with a tree at both ends, the number of trees is one more than the number of gaps.

11 + 1 = 12

Posts include both endpoints, so there is always one extra post beyond the gaps between them.

3Double for both sides

#1 Draw a Diagram 3.OA.A.3

Each side of the trail has 12 trees, and there are two sides, so multiply by 2.

12 \times 2 = 24

Both sides are identical, so the total is just twice one side's count.

Answer: 24 trees

4 · Reviewdoes it hold up?

12 trees per side over 99 feet (11 gaps of 9 feet = 99 feet) fits exactly, and doubling to 24 for two sides is reasonable. Forgetting the extra end tree would wrongly give 22.

Another way: Solve an easier related problem (tool 9): a 18-foot trail with 9-foot spacing has 2 gaps and 3 trees per side, confirming the gaps-plus-one rule before scaling up.

Standardsmin grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Dividing 99 by 9 to find the number of gaps on one side.
3.OA.A.3 Solve multiplication and division word problems within 100 — Adding one for the end tree and doubling for both sides.

💡Takeaway. This only needs Grade 3 division: count the gaps, add one for the end post, then double for both sides!