Practice · Overlaps are one fewer than the strips · Sensim Math

Variant 1 easy

You want to join $5$ strips of tape, each $26 \text{ cm}$ long, by overlapping them by the same amount as shown in the figure, to make one long tape that is $1 \text{ m } 10 \text{ cm}$ long. By how many $\text{cm}$ should each overlap be?

Your answer

Show solution

1 · Understandwhat's really being asked

5 tape strips, each 26 cm long, are joined into one long row by overlapping each neighboring pair by the same amount. The finished tape is 1 m 10 cm (110 cm) long. We must find how many cm each overlap is.

Givens

5 strips, each 26 cm long
Strips overlap by the same amount between neighbors
Final joined length is 1 m 10 cm = 110 cm

Unknowns

The length of each overlap in cm

Constraints

All overlaps are equal
With 5 strips in a row there are 5 - 1 = 4 overlap regions

2 · Planchoose the strategy

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

The figure shows that joining strips makes them overlap, and the key pattern is that the number of overlaps is one fewer than the number of strips. The total length lost equals the combined overlaps, which we then split equally.

3 · Execute4 carry out the plan

1Total length of the separate strips

#7 Identify Subproblems 3.OA.A.3

Add up all 5 strips before any overlapping: 5 strips of 26 cm.

5 \times 26 = 130 \text{ cm}

If they were laid end to end with no overlap, the lengths simply add.

2Find what was lost

#1 Draw a Diagram 3.MD.D.8

The finished tape is 1 m 10 cm = 110 cm. Overlapping makes the tape shorter, so the amount lost is the separate total minus the final length.

130 - 110 = 20 \text{ cm}

Every overlap hides some tape; the missing 20 cm is all the hidden overlap together.

3Count the overlaps

#5 Look for a Pattern 3.OA.D.9

The overlap regions sit between neighboring strips. With 5 strips there are 4 such overlaps (one fewer than the number of strips).

5 - 1 = 4

Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.

4Find each overlap

#7 Identify Subproblems 3.OA.A.3

The 20 cm of hidden tape is shared equally among the 4 overlaps.

20 \div 4 = 5 \text{ cm}

Equal overlaps means equal sharing, so divide the total overlap by how many there are.

Answer: 5 cm

4 · Reviewdoes it hold up?

With 5 cm overlaps: total 130 cm minus 4 overlaps times 5 cm = 130 - 20 = 110 cm = 1 m 10 cm, matching the target. An overlap of 5 cm is smaller than a 26 cm strip, which is sensible.

Another way: Build up by addition: first strip is the full length, then each added strip contributes the strip length minus the overlap; setting 26 + 4 times (26 - overlap) = 110 also gives overlap = 5 cm.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 5 by 26 and dividing the lost length by the number of overlaps
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Reasoning about combined lengths and the length lost to overlapping
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the number of overlaps is one fewer than the number of strips

💡Takeaway. When you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!

Variant 2 easy

You want to join $7$ strips of tape, each $12 \text{ cm}$ long, by overlapping them by the same amount as shown in the figure, to make one long tape that is $60 \text{ cm}$ long. By how many $\text{cm}$ should each overlap be?

Your answer

Show solution

1 · Understandwhat's really being asked

7 tape strips, each 12 cm long, are joined into one long row by overlapping each neighboring pair by the same amount. The finished tape is 60 cm (60 cm) long. We must find how many cm each overlap is.

Givens

7 strips, each 12 cm long
Strips overlap by the same amount between neighbors
Final joined length is 60 cm = 60 cm

Unknowns

The length of each overlap in cm

Constraints

All overlaps are equal
With 7 strips in a row there are 7 - 1 = 6 overlap regions

2 · Planchoose the strategy

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

The figure shows that joining strips makes them overlap, and the key pattern is that the number of overlaps is one fewer than the number of strips. The total length lost equals the combined overlaps, which we then split equally.

3 · Execute4 carry out the plan

1Total length of the separate strips

#7 Identify Subproblems 3.OA.A.3

Add up all 7 strips before any overlapping: 7 strips of 12 cm.

7 \times 12 = 84 \text{ cm}

If they were laid end to end with no overlap, the lengths simply add.

2Find what was lost

#1 Draw a Diagram 3.MD.D.8

The finished tape is 60 cm = 60 cm. Overlapping makes the tape shorter, so the amount lost is the separate total minus the final length.

84 - 60 = 24 \text{ cm}

Every overlap hides some tape; the missing 24 cm is all the hidden overlap together.

3Count the overlaps

#5 Look for a Pattern 3.OA.D.9

The overlap regions sit between neighboring strips. With 7 strips there are 6 such overlaps (one fewer than the number of strips).

7 - 1 = 6

Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.

4Find each overlap

#7 Identify Subproblems 3.OA.A.3

The 24 cm of hidden tape is shared equally among the 6 overlaps.

24 \div 6 = 4 \text{ cm}

Equal overlaps means equal sharing, so divide the total overlap by how many there are.

Answer: 4 cm

4 · Reviewdoes it hold up?

With 4 cm overlaps: total 84 cm minus 6 overlaps times 4 cm = 84 - 24 = 60 cm = 60 cm, matching the target. An overlap of 4 cm is smaller than a 12 cm strip, which is sensible.

Another way: Build up by addition: first strip is the full length, then each added strip contributes the strip length minus the overlap; setting 12 + 6 times (12 - overlap) = 60 also gives overlap = 4 cm.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 7 by 12 and dividing the lost length by the number of overlaps
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Reasoning about combined lengths and the length lost to overlapping
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the number of overlaps is one fewer than the number of strips

💡Takeaway. When you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!

Variant 3 easy

You want to join $4$ strips of tape, each $20 \text{ cm}$ long, by overlapping them by the same amount as shown in the figure, to make one long tape that is $65 \text{ cm}$ long. By how many $\text{cm}$ should each overlap be?

Your answer

Show solution

1 · Understandwhat's really being asked

4 tape strips, each 20 cm long, are joined into one long row by overlapping each neighboring pair by the same amount. The finished tape is 65 cm (65 cm) long. We must find how many cm each overlap is.

Givens

4 strips, each 20 cm long
Strips overlap by the same amount between neighbors
Final joined length is 65 cm = 65 cm

Unknowns

The length of each overlap in cm

Constraints

All overlaps are equal
With 4 strips in a row there are 4 - 1 = 3 overlap regions

2 · Planchoose the strategy

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

The figure shows that joining strips makes them overlap, and the key pattern is that the number of overlaps is one fewer than the number of strips. The total length lost equals the combined overlaps, which we then split equally.

3 · Execute4 carry out the plan

1Total length of the separate strips

#7 Identify Subproblems 3.OA.A.3

Add up all 4 strips before any overlapping: 4 strips of 20 cm.

4 \times 20 = 80 \text{ cm}

If they were laid end to end with no overlap, the lengths simply add.

2Find what was lost

#1 Draw a Diagram 3.MD.D.8

The finished tape is 65 cm = 65 cm. Overlapping makes the tape shorter, so the amount lost is the separate total minus the final length.

80 - 65 = 15 \text{ cm}

Every overlap hides some tape; the missing 15 cm is all the hidden overlap together.

3Count the overlaps

#5 Look for a Pattern 3.OA.D.9

The overlap regions sit between neighboring strips. With 4 strips there are 3 such overlaps (one fewer than the number of strips).

4 - 1 = 3

Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.

4Find each overlap

#7 Identify Subproblems 3.OA.A.3

The 15 cm of hidden tape is shared equally among the 3 overlaps.

15 \div 3 = 5 \text{ cm}

Equal overlaps means equal sharing, so divide the total overlap by how many there are.

Answer: 5 cm

4 · Reviewdoes it hold up?

With 5 cm overlaps: total 80 cm minus 3 overlaps times 5 cm = 80 - 15 = 65 cm = 65 cm, matching the target. An overlap of 5 cm is smaller than a 20 cm strip, which is sensible.

Another way: Build up by addition: first strip is the full length, then each added strip contributes the strip length minus the overlap; setting 20 + 3 times (20 - overlap) = 65 also gives overlap = 5 cm.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 4 by 20 and dividing the lost length by the number of overlaps
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Reasoning about combined lengths and the length lost to overlapping
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the number of overlaps is one fewer than the number of strips

💡Takeaway. When you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!

Variant 4 medium

You want to join $6$ strips of tape, each $15 \text{ cm}$ long, by overlapping them by the same amount as shown in the figure, to make one long tape that is $65 \text{ cm}$ long. By how many $\text{cm}$ should each overlap be?

Your answer

Show solution

1 · Understandwhat's really being asked

6 tape strips, each 15 cm long, are joined into one long row by overlapping each neighboring pair by the same amount. The finished tape is 65 cm (65 cm) long. We must find how many cm each overlap is.

Givens

6 strips, each 15 cm long
Strips overlap by the same amount between neighbors
Final joined length is 65 cm = 65 cm

Unknowns

The length of each overlap in cm

Constraints

All overlaps are equal
With 6 strips in a row there are 6 - 1 = 5 overlap regions

2 · Planchoose the strategy

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

The figure shows that joining strips makes them overlap, and the key pattern is that the number of overlaps is one fewer than the number of strips. The total length lost equals the combined overlaps, which we then split equally.

3 · Execute4 carry out the plan

1Total length of the separate strips

#7 Identify Subproblems 3.OA.A.3

Add up all 6 strips before any overlapping: 6 strips of 15 cm.

6 \times 15 = 90 \text{ cm}

If they were laid end to end with no overlap, the lengths simply add.

2Find what was lost

#1 Draw a Diagram 3.MD.D.8

The finished tape is 65 cm = 65 cm. Overlapping makes the tape shorter, so the amount lost is the separate total minus the final length.

90 - 65 = 25 \text{ cm}

Every overlap hides some tape; the missing 25 cm is all the hidden overlap together.

3Count the overlaps

#5 Look for a Pattern 3.OA.D.9

The overlap regions sit between neighboring strips. With 6 strips there are 5 such overlaps (one fewer than the number of strips).

6 - 1 = 5

Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.

4Find each overlap

#7 Identify Subproblems 3.OA.A.3

The 25 cm of hidden tape is shared equally among the 5 overlaps.

25 \div 5 = 5 \text{ cm}

Equal overlaps means equal sharing, so divide the total overlap by how many there are.

Answer: 5 cm

4 · Reviewdoes it hold up?

With 5 cm overlaps: total 90 cm minus 5 overlaps times 5 cm = 90 - 25 = 65 cm = 65 cm, matching the target. An overlap of 5 cm is smaller than a 15 cm strip, which is sensible.

Another way: Build up by addition: first strip is the full length, then each added strip contributes the strip length minus the overlap; setting 15 + 5 times (15 - overlap) = 65 also gives overlap = 5 cm.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 6 by 15 and dividing the lost length by the number of overlaps
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Reasoning about combined lengths and the length lost to overlapping
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the number of overlaps is one fewer than the number of strips

💡Takeaway. When you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!

Variant 5 medium

You want to join $3$ strips of tape, each $40 \text{ cm}$ long, by overlapping them by the same amount as shown in the figure, to make one long tape that is $1 \text{ m } 10 \text{ cm}$ long. By how many $\text{cm}$ should each overlap be?

Your answer

Show solution

1 · Understandwhat's really being asked

3 tape strips, each 40 cm long, are joined into one long row by overlapping each neighboring pair by the same amount. The finished tape is 1 m 10 cm (110 cm) long. We must find how many cm each overlap is.

Givens

3 strips, each 40 cm long
Strips overlap by the same amount between neighbors
Final joined length is 1 m 10 cm = 110 cm

Unknowns

The length of each overlap in cm

Constraints

All overlaps are equal
With 3 strips in a row there are 3 - 1 = 2 overlap regions

2 · Planchoose the strategy

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

The figure shows that joining strips makes them overlap, and the key pattern is that the number of overlaps is one fewer than the number of strips. The total length lost equals the combined overlaps, which we then split equally.

3 · Execute4 carry out the plan

1Total length of the separate strips

#7 Identify Subproblems 3.OA.A.3

Add up all 3 strips before any overlapping: 3 strips of 40 cm.

3 \times 40 = 120 \text{ cm}

If they were laid end to end with no overlap, the lengths simply add.

2Find what was lost

#1 Draw a Diagram 3.MD.D.8

The finished tape is 1 m 10 cm = 110 cm. Overlapping makes the tape shorter, so the amount lost is the separate total minus the final length.

120 - 110 = 10 \text{ cm}

Every overlap hides some tape; the missing 10 cm is all the hidden overlap together.

3Count the overlaps

#5 Look for a Pattern 3.OA.D.9

The overlap regions sit between neighboring strips. With 3 strips there are 2 such overlaps (one fewer than the number of strips).

3 - 1 = 2

Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.

4Find each overlap

#7 Identify Subproblems 3.OA.A.3

The 10 cm of hidden tape is shared equally among the 2 overlaps.

10 \div 2 = 5 \text{ cm}

Equal overlaps means equal sharing, so divide the total overlap by how many there are.

Answer: 5 cm

4 · Reviewdoes it hold up?

With 5 cm overlaps: total 120 cm minus 2 overlaps times 5 cm = 120 - 10 = 110 cm = 1 m 10 cm, matching the target. An overlap of 5 cm is smaller than a 40 cm strip, which is sensible.

Another way: Build up by addition: first strip is the full length, then each added strip contributes the strip length minus the overlap; setting 40 + 2 times (40 - overlap) = 110 also gives overlap = 5 cm.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 3 by 40 and dividing the lost length by the number of overlaps
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Reasoning about combined lengths and the length lost to overlapping
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the number of overlaps is one fewer than the number of strips

💡Takeaway. When you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!

Variant 6 medium

You want to join $5$ strips of tape, each $18 \text{ cm}$ long, by overlapping them by the same amount as shown in the figure, to make one long tape that is $74 \text{ cm}$ long. By how many $\text{cm}$ should each overlap be?

Your answer

Show solution

1 · Understandwhat's really being asked

5 tape strips, each 18 cm long, are joined into one long row by overlapping each neighboring pair by the same amount. The finished tape is 74 cm (74 cm) long. We must find how many cm each overlap is.

Givens

5 strips, each 18 cm long
Strips overlap by the same amount between neighbors
Final joined length is 74 cm = 74 cm

Unknowns

The length of each overlap in cm

Constraints

All overlaps are equal
With 5 strips in a row there are 5 - 1 = 4 overlap regions

2 · Planchoose the strategy

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

The figure shows that joining strips makes them overlap, and the key pattern is that the number of overlaps is one fewer than the number of strips. The total length lost equals the combined overlaps, which we then split equally.

3 · Execute4 carry out the plan

1Total length of the separate strips

#7 Identify Subproblems 3.OA.A.3

Add up all 5 strips before any overlapping: 5 strips of 18 cm.

5 \times 18 = 90 \text{ cm}

If they were laid end to end with no overlap, the lengths simply add.

2Find what was lost

#1 Draw a Diagram 3.MD.D.8

The finished tape is 74 cm = 74 cm. Overlapping makes the tape shorter, so the amount lost is the separate total minus the final length.

90 - 74 = 16 \text{ cm}

Every overlap hides some tape; the missing 16 cm is all the hidden overlap together.

3Count the overlaps

#5 Look for a Pattern 3.OA.D.9

The overlap regions sit between neighboring strips. With 5 strips there are 4 such overlaps (one fewer than the number of strips).

5 - 1 = 4

Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.

4Find each overlap

#7 Identify Subproblems 3.OA.A.3

The 16 cm of hidden tape is shared equally among the 4 overlaps.

16 \div 4 = 4 \text{ cm}

Equal overlaps means equal sharing, so divide the total overlap by how many there are.

Answer: 4 cm

4 · Reviewdoes it hold up?

With 4 cm overlaps: total 90 cm minus 4 overlaps times 4 cm = 90 - 16 = 74 cm = 74 cm, matching the target. An overlap of 4 cm is smaller than a 18 cm strip, which is sensible.

Another way: Build up by addition: first strip is the full length, then each added strip contributes the strip length minus the overlap; setting 18 + 4 times (18 - overlap) = 74 also gives overlap = 4 cm.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 5 by 18 and dividing the lost length by the number of overlaps
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Reasoning about combined lengths and the length lost to overlapping
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the number of overlaps is one fewer than the number of strips

💡Takeaway. When you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!

Variant 7 hard

You want to join $3$ strips of tape, each $30 \text{ cm}$ long, by overlapping them by the same amount as shown in the figure, to make one long tape that is $80 \text{ cm}$ long. By how many $\text{cm}$ should each overlap be?

Your answer

Show solution

1 · Understandwhat's really being asked

3 tape strips, each 30 cm long, are joined into one long row by overlapping each neighboring pair by the same amount. The finished tape is 80 cm (80 cm) long. We must find how many cm each overlap is.

Givens

3 strips, each 30 cm long
Strips overlap by the same amount between neighbors
Final joined length is 80 cm = 80 cm

Unknowns

The length of each overlap in cm

Constraints

All overlaps are equal
With 3 strips in a row there are 3 - 1 = 2 overlap regions

2 · Planchoose the strategy

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

The figure shows that joining strips makes them overlap, and the key pattern is that the number of overlaps is one fewer than the number of strips. The total length lost equals the combined overlaps, which we then split equally.

3 · Execute4 carry out the plan

1Total length of the separate strips

#7 Identify Subproblems 3.OA.A.3

Add up all 3 strips before any overlapping: 3 strips of 30 cm.

3 \times 30 = 90 \text{ cm}

If they were laid end to end with no overlap, the lengths simply add.

2Find what was lost

#1 Draw a Diagram 3.MD.D.8

The finished tape is 80 cm = 80 cm. Overlapping makes the tape shorter, so the amount lost is the separate total minus the final length.

90 - 80 = 10 \text{ cm}

Every overlap hides some tape; the missing 10 cm is all the hidden overlap together.

3Count the overlaps

#5 Look for a Pattern 3.OA.D.9

The overlap regions sit between neighboring strips. With 3 strips there are 2 such overlaps (one fewer than the number of strips).

3 - 1 = 2

Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.

4Find each overlap

#7 Identify Subproblems 3.OA.A.3

The 10 cm of hidden tape is shared equally among the 2 overlaps.

10 \div 2 = 5 \text{ cm}

Equal overlaps means equal sharing, so divide the total overlap by how many there are.

Answer: 5 cm

4 · Reviewdoes it hold up?

With 5 cm overlaps: total 90 cm minus 2 overlaps times 5 cm = 90 - 10 = 80 cm = 80 cm, matching the target. An overlap of 5 cm is smaller than a 30 cm strip, which is sensible.

Another way: Build up by addition: first strip is the full length, then each added strip contributes the strip length minus the overlap; setting 30 + 2 times (30 - overlap) = 80 also gives overlap = 5 cm.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 3 by 30 and dividing the lost length by the number of overlaps
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Reasoning about combined lengths and the length lost to overlapping
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the number of overlaps is one fewer than the number of strips

💡Takeaway. When you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!

Variant 8 hard

You want to join $4$ strips of tape, each $25 \text{ cm}$ long, by overlapping them by the same amount as shown in the figure, to make one long tape that is $88 \text{ cm}$ long. By how many $\text{cm}$ should each overlap be?

Your answer

Show solution

1 · Understandwhat's really being asked

4 tape strips, each 25 cm long, are joined into one long row by overlapping each neighboring pair by the same amount. The finished tape is 88 cm (88 cm) long. We must find how many cm each overlap is.

Givens

4 strips, each 25 cm long
Strips overlap by the same amount between neighbors
Final joined length is 88 cm = 88 cm

Unknowns

The length of each overlap in cm

Constraints

All overlaps are equal
With 4 strips in a row there are 4 - 1 = 3 overlap regions

2 · Planchoose the strategy

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

The figure shows that joining strips makes them overlap, and the key pattern is that the number of overlaps is one fewer than the number of strips. The total length lost equals the combined overlaps, which we then split equally.

3 · Execute4 carry out the plan

1Total length of the separate strips

#7 Identify Subproblems 3.OA.A.3

Add up all 4 strips before any overlapping: 4 strips of 25 cm.

4 \times 25 = 100 \text{ cm}

If they were laid end to end with no overlap, the lengths simply add.

2Find what was lost

#1 Draw a Diagram 3.MD.D.8

The finished tape is 88 cm = 88 cm. Overlapping makes the tape shorter, so the amount lost is the separate total minus the final length.

100 - 88 = 12 \text{ cm}

Every overlap hides some tape; the missing 12 cm is all the hidden overlap together.

3Count the overlaps

#5 Look for a Pattern 3.OA.D.9

The overlap regions sit between neighboring strips. With 4 strips there are 3 such overlaps (one fewer than the number of strips).

4 - 1 = 3

Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.

4Find each overlap

#7 Identify Subproblems 3.OA.A.3

The 12 cm of hidden tape is shared equally among the 3 overlaps.

12 \div 3 = 4 \text{ cm}

Equal overlaps means equal sharing, so divide the total overlap by how many there are.

Answer: 4 cm

4 · Reviewdoes it hold up?

With 4 cm overlaps: total 100 cm minus 3 overlaps times 4 cm = 100 - 12 = 88 cm = 88 cm, matching the target. An overlap of 4 cm is smaller than a 25 cm strip, which is sensible.

Another way: Build up by addition: first strip is the full length, then each added strip contributes the strip length minus the overlap; setting 25 + 3 times (25 - overlap) = 88 also gives overlap = 4 cm.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 4 by 25 and dividing the lost length by the number of overlaps
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Reasoning about combined lengths and the length lost to overlapping
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the number of overlaps is one fewer than the number of strips

💡Takeaway. When you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!

Variant 9 hard

You want to join $5$ strips of tape, each $22 \text{ cm}$ long, by overlapping them by the same amount as shown in the figure, to make one long tape that is $98 \text{ cm}$ long. By how many $\text{cm}$ should each overlap be?

Your answer

Show solution

1 · Understandwhat's really being asked

5 tape strips, each 22 cm long, are joined into one long row by overlapping each neighboring pair by the same amount. The finished tape is 98 cm (98 cm) long. We must find how many cm each overlap is.

Givens

5 strips, each 22 cm long
Strips overlap by the same amount between neighbors
Final joined length is 98 cm = 98 cm

Unknowns

The length of each overlap in cm

Constraints

All overlaps are equal
With 5 strips in a row there are 5 - 1 = 4 overlap regions

2 · Planchoose the strategy

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

The figure shows that joining strips makes them overlap, and the key pattern is that the number of overlaps is one fewer than the number of strips. The total length lost equals the combined overlaps, which we then split equally.

3 · Execute4 carry out the plan

1Total length of the separate strips

#7 Identify Subproblems 3.OA.A.3

Add up all 5 strips before any overlapping: 5 strips of 22 cm.

5 \times 22 = 110 \text{ cm}

If they were laid end to end with no overlap, the lengths simply add.

2Find what was lost

#1 Draw a Diagram 3.MD.D.8

The finished tape is 98 cm = 98 cm. Overlapping makes the tape shorter, so the amount lost is the separate total minus the final length.

110 - 98 = 12 \text{ cm}

Every overlap hides some tape; the missing 12 cm is all the hidden overlap together.

3Count the overlaps

#5 Look for a Pattern 3.OA.D.9

The overlap regions sit between neighboring strips. With 5 strips there are 4 such overlaps (one fewer than the number of strips).

5 - 1 = 4

Each new strip after the first adds exactly one overlap, so overlaps are one fewer than strips.

4Find each overlap

#7 Identify Subproblems 3.OA.A.3

The 12 cm of hidden tape is shared equally among the 4 overlaps.

12 \div 4 = 3 \text{ cm}

Equal overlaps means equal sharing, so divide the total overlap by how many there are.

Answer: 3 cm

4 · Reviewdoes it hold up?

With 3 cm overlaps: total 110 cm minus 4 overlaps times 3 cm = 110 - 12 = 98 cm = 98 cm, matching the target. An overlap of 3 cm is smaller than a 22 cm strip, which is sensible.

Another way: Build up by addition: first strip is the full length, then each added strip contributes the strip length minus the overlap; setting 22 + 4 times (22 - overlap) = 98 also gives overlap = 3 cm.

Standardsmin grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Multiplying 5 by 22 and dividing the lost length by the number of overlaps
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Reasoning about combined lengths and the length lost to overlapping
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the number of overlaps is one fewer than the number of strips

💡Takeaway. When you overlap strips in a row, the overlaps are always one fewer than the strips, so the hidden length just splits evenly among them!

Overlaps are one fewer than the strips · 9 practice problems

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