← Overlaps equal tape count minus one · Overlap Reduces the Total

Overlaps equal tape count minus one · 12 practice problems

5.NBT.B.7

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Generated variants — 12

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

Variant 1 easy

Twelve strips of colored tape, each $6.5\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $0.5\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

12 strips of tape, each 6.5 cm, are glued in a row. Every join overlaps by 0.5 cm. We need the length of the finished band.

Givens

12 strips, each 6.5 cm long.
Each neighbouring pair overlaps by 0.5 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 12 strips in a row there are 11 gaps between neighbours.

12 - 1 = 11

11 overlaps, not 12.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 12 strips laid end to end without any glue.

12 \times 6.5 = 78

78 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 11 overlaps hides 0.5 cm.

11 \times 0.5 = 5.5

5.5 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

78 - 5.5 = 72.5

The joined tape is 72.5 cm long.

Answer: 72.5 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 6.5 cm, and each of the other 11 adds only 6 cm. That gives 6.5 + 11 x 6 = 72.5 cm.

Another way: Counting 12 overlaps instead of 11 would take off 0.5 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 2 easy

Eleven strips of colored tape, each $7.2\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $0.8\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

11 strips of tape, each 7.2 cm, are glued in a row. Every join overlaps by 0.8 cm. We need the length of the finished band.

Givens

11 strips, each 7.2 cm long.
Each neighbouring pair overlaps by 0.8 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 11 strips in a row there are 10 gaps between neighbours.

11 - 1 = 10

10 overlaps, not 11.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 11 strips laid end to end without any glue.

11 \times 7.2 = 79.2

79.2 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 10 overlaps hides 0.8 cm.

10 \times 0.8 = 8

8 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

79.2 - 8 = 71.2

The joined tape is 71.2 cm long.

Answer: 71.2 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 7.2 cm, and each of the other 10 adds only 6.4 cm. That gives 7.2 + 10 x 6.4 = 71.2 cm.

Another way: Counting 11 overlaps instead of 10 would take off 0.8 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 3 easy

Six strips of colored tape, each $8.4\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $1.2\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

6 strips of tape, each 8.4 cm, are glued in a row. Every join overlaps by 1.2 cm. We need the length of the finished band.

Givens

6 strips, each 8.4 cm long.
Each neighbouring pair overlaps by 1.2 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 6 strips in a row there are 5 gaps between neighbours.

6 - 1 = 5

5 overlaps, not 6.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 6 strips laid end to end without any glue.

6 \times 8.4 = 50.4

50.4 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 5 overlaps hides 1.2 cm.

5 \times 1.2 = 6

6 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

50.4 - 6 = 44.4

The joined tape is 44.4 cm long.

Answer: 44.4 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 8.4 cm, and each of the other 5 adds only 7.2 cm. That gives 8.4 + 5 x 7.2 = 44.4 cm.

Another way: Counting 6 overlaps instead of 5 would take off 1.2 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 4 easy

Ten strips of colored tape, each $8.8\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $1.4\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

10 strips of tape, each 8.8 cm, are glued in a row. Every join overlaps by 1.4 cm. We need the length of the finished band.

Givens

10 strips, each 8.8 cm long.
Each neighbouring pair overlaps by 1.4 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 10 strips in a row there are 9 gaps between neighbours.

10 - 1 = 9

9 overlaps, not 10.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 10 strips laid end to end without any glue.

10 \times 8.8 = 88

88 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 9 overlaps hides 1.4 cm.

9 \times 1.4 = 12.6

12.6 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

88 - 12.6 = 75.4

The joined tape is 75.4 cm long.

Answer: 75.4 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 8.8 cm, and each of the other 9 adds only 7.4 cm. That gives 8.8 + 9 x 7.4 = 75.4 cm.

Another way: Counting 10 overlaps instead of 9 would take off 1.4 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 5 medium

Ten strips of colored tape, each $9.5\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $1.5\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

10 strips of tape, each 9.5 cm, are glued in a row. Every join overlaps by 1.5 cm. We need the length of the finished band.

Givens

10 strips, each 9.5 cm long.
Each neighbouring pair overlaps by 1.5 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 10 strips in a row there are 9 gaps between neighbours.

10 - 1 = 9

9 overlaps, not 10.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 10 strips laid end to end without any glue.

10 \times 9.5 = 95

95 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 9 overlaps hides 1.5 cm.

9 \times 1.5 = 13.5

13.5 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

95 - 13.5 = 81.5

The joined tape is 81.5 cm long.

Answer: 81.5 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 9.5 cm, and each of the other 9 adds only 8 cm. That gives 9.5 + 9 x 8 = 81.5 cm.

Another way: Counting 10 overlaps instead of 9 would take off 1.5 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 6 medium

Eight strips of colored tape, each $9.6\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $1.6\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

8 strips of tape, each 9.6 cm, are glued in a row. Every join overlaps by 1.6 cm. We need the length of the finished band.

Givens

8 strips, each 9.6 cm long.
Each neighbouring pair overlaps by 1.6 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 8 strips in a row there are 7 gaps between neighbours.

8 - 1 = 7

7 overlaps, not 8.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 8 strips laid end to end without any glue.

8 \times 9.6 = 76.8

76.8 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 7 overlaps hides 1.6 cm.

7 \times 1.6 = 11.2

11.2 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

76.8 - 11.2 = 65.6

The joined tape is 65.6 cm long.

Answer: 65.6 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 9.6 cm, and each of the other 7 adds only 8 cm. That gives 9.6 + 7 x 8 = 65.6 cm.

Another way: Counting 8 overlaps instead of 7 would take off 1.6 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 7 medium

Nine strips of colored tape, each $10.8\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $1.8\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

9 strips of tape, each 10.8 cm, are glued in a row. Every join overlaps by 1.8 cm. We need the length of the finished band.

Givens

9 strips, each 10.8 cm long.
Each neighbouring pair overlaps by 1.8 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 9 strips in a row there are 8 gaps between neighbours.

9 - 1 = 8

8 overlaps, not 9.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 9 strips laid end to end without any glue.

9 \times 10.8 = 97.2

97.2 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 8 overlaps hides 1.8 cm.

8 \times 1.8 = 14.4

14.4 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

97.2 - 14.4 = 82.8

The joined tape is 82.8 cm long.

Answer: 82.8 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 10.8 cm, and each of the other 8 adds only 9 cm. That gives 10.8 + 8 x 9 = 82.8 cm.

Another way: Counting 9 overlaps instead of 8 would take off 1.8 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 8 medium

Seven strips of colored tape, each $11\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $2.2\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

7 strips of tape, each 11 cm, are glued in a row. Every join overlaps by 2.2 cm. We need the length of the finished band.

Givens

7 strips, each 11 cm long.
Each neighbouring pair overlaps by 2.2 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 7 strips in a row there are 6 gaps between neighbours.

7 - 1 = 6

6 overlaps, not 7.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 7 strips laid end to end without any glue.

7 \times 11 = 77

77 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 6 overlaps hides 2.2 cm.

6 \times 2.2 = 13.2

13.2 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

77 - 13.2 = 63.8

The joined tape is 63.8 cm long.

Answer: 63.8 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 11 cm, and each of the other 6 adds only 8.8 cm. That gives 11 + 6 x 8.8 = 63.8 cm.

Another way: Counting 7 overlaps instead of 6 would take off 2.2 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 9 hard

Eight strips of colored tape, each $12\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $2.5\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

8 strips of tape, each 12 cm, are glued in a row. Every join overlaps by 2.5 cm. We need the length of the finished band.

Givens

8 strips, each 12 cm long.
Each neighbouring pair overlaps by 2.5 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 8 strips in a row there are 7 gaps between neighbours.

8 - 1 = 7

7 overlaps, not 8.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 8 strips laid end to end without any glue.

8 \times 12 = 96

96 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 7 overlaps hides 2.5 cm.

7 \times 2.5 = 17.5

17.5 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

96 - 17.5 = 78.5

The joined tape is 78.5 cm long.

Answer: 78.5 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 12 cm, and each of the other 7 adds only 9.5 cm. That gives 12 + 7 x 9.5 = 78.5 cm.

Another way: Counting 8 overlaps instead of 7 would take off 2.5 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 10 hard

Six strips of colored tape, each $13.5\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $3.5\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

6 strips of tape, each 13.5 cm, are glued in a row. Every join overlaps by 3.5 cm. We need the length of the finished band.

Givens

6 strips, each 13.5 cm long.
Each neighbouring pair overlaps by 3.5 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 6 strips in a row there are 5 gaps between neighbours.

6 - 1 = 5

5 overlaps, not 6.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 6 strips laid end to end without any glue.

6 \times 13.5 = 81

81 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 5 overlaps hides 3.5 cm.

5 \times 3.5 = 17.5

17.5 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

81 - 17.5 = 63.5

The joined tape is 63.5 cm long.

Answer: 63.5 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 13.5 cm, and each of the other 5 adds only 10 cm. That gives 13.5 + 5 x 10 = 63.5 cm.

Another way: Counting 6 overlaps instead of 5 would take off 3.5 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 11 hard

Seven strips of colored tape, each $15\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $3\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

7 strips of tape, each 15 cm, are glued in a row. Every join overlaps by 3 cm. We need the length of the finished band.

Givens

7 strips, each 15 cm long.
Each neighbouring pair overlaps by 3 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 7 strips in a row there are 6 gaps between neighbours.

7 - 1 = 6

6 overlaps, not 7.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 7 strips laid end to end without any glue.

7 \times 15 = 105

105 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 6 overlaps hides 3 cm.

6 \times 3 = 18

18 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

105 - 18 = 87

The joined tape is 87 cm long.

Answer: 87 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 15 cm, and each of the other 6 adds only 12 cm. That gives 15 + 6 x 12 = 87 cm.

Another way: Counting 7 overlaps instead of 6 would take off 3 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Variant 12 hard

Five strips of colored tape, each $20\ \text{cm}$ long, are glued end to end so that each pair of neighboring strips overlaps by $4.5\ \text{cm}$ , as shown in the figure. What is the total length, in $\text{cm}$ , of the joined tape?

Your answer

Show solution

1 · Understandwhat's really being asked

5 strips of tape, each 20 cm, are glued in a row. Every join overlaps by 4.5 cm. We need the length of the finished band.

Givens

5 strips, each 20 cm long.
Each neighbouring pair overlaps by 4.5 cm.

Unknowns

The total length of the joined tape.

Constraints

Only neighbouring strips overlap, and each join overlaps by the same amount.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #5 Look for a Pattern#7 Identify Subproblems

Lay the strips out end to end first, then take back what the overlaps hide. The only thing to be careful about is how many overlaps there are, which the picture settles.

3 · Execute4 carry out the plan

1Count the overlaps from a diagram

#1 Draw a Diagram 5.NBT.B.7

Overlaps happen between strips, not on them. With 5 strips in a row there are 4 gaps between neighbours.

5 - 1 = 4

4 overlaps, not 5.

2Total length if there were no overlaps

#7 Identify Subproblems 5.NBT.B.7

All 5 strips laid end to end without any glue.

5 \times 20 = 100

100 cm of tape in total.

3Total length lost to overlaps

#7 Identify Subproblems 5.NBT.B.7

Each of the 4 overlaps hides 4.5 cm.

4 \times 4.5 = 18

18 cm disappears into the joins.

4Subtract the overlap from the total

#5 Look for a Pattern 5.NBT.B.7

What is left is the length of the band as glued.

100 - 18 = 82

The joined tape is 82 cm long.

Answer: 82 cm

4 · Reviewdoes it hold up?

Another way to see it: the first strip contributes its whole 20 cm, and each of the other 4 adds only 15.5 cm. That gives 20 + 4 x 15.5 = 82 cm.

Another way: Counting 5 overlaps instead of 4 would take off 4.5 cm too much -- the classic slip, and what the diagram is there to prevent.

Standardsmin grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Both decimal multiplications and the closing subtraction.

💡Takeaway. Overlaps sit between strips, so there is always one fewer overlap than strips.

Where next?

← Back to the problem: Overlaps equal tape count minus one