Problem
Overlaps equal tape count minus one
Ten strips of tape, each 9.5 cm long, are joined in a row so that every two neighboring strips overlap by 1.5 cm. Find the total length of the whole joined band.
Base-ten numbers
Your answer
How to solve
Strategy Draw a Diagram — A quick sketch of strips laid in a row shows that the overlapped (darker) sections sit between strips, so the number of overlaps is one less than the number of strips. From there the work splits into two simple subproblems: add up all the tape, then subtract the doubled-up overlap length. The arithmetic itself is decimal addition, multiplication, and subtraction.
1STEP 1
Count the overlaps from a diagram
Joints sit between strips, so 10 strips in a row make only 9 overlaps.
10 - 1 = 9 overlaps
Seeing the strips drawn out makes it obvious that joints sit between pieces, so there is always one fewer overlap than the number of pieces.
5.NBT.B.7Draw A DiagramTen strips glued in a row have 9 overlaps.
Why?
An overlap sits in each gap where two neighbouring strips meet, and a row of ten strips has nine such gaps.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The joined length is all the tape laid out less what the overlaps hide, because each overlap doubles up a stretch that the finished strip counts only once.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Total length if there were no overlaps
First pretend the strips do not overlap. Then the full length is just 10 strips, each 9.5 cm long, added together.
10 × 9.5 = 95 cm
Breaking the question into 'all the tape first' is the easy half to compute, since it is just one multiplication.
5.NBT.B.7Identify Subproblems3STEP 3
Total length lost to overlaps
Each overlap hides 1.5 cm, so 9 x 1.5 = 13.5 cm is swallowed.
9 × 1.5 = 13.5 cm
Every place two strips stick together, 1.5 cm of length is shared instead of added, so that length must be removed once for each overlap.
5.NBT.B.7Identify Subproblems4STEP 4
Subtract the overlap from the total
The real joined length is the full tape length minus the length swallowed by the overlaps.
95 - 13.5 = 81.5 cm
The same rule works for any number of strips: total = (strips x strip length) - ((strips - 1) x overlap), so the pattern generalizes cleanly.
5.NBT.B.7Look For A PatternAnswer
81.5 cm
The answer 81.5 cm is a little less than the 95 cm of tape used, which makes sense because overlapping always shortens the band. The units are centimeters, as asked, and the magnitude is sensible for 10 strips that are each about 8 cm of new length once joined.
Takeaway
When strips overlap, the number of overlaps is one less than the number of strips, so add all the tape and take away each shared piece once.
- Count the overlaps from a diagram
- Total length if there were no overlaps
- Total length lost to overlaps
- Subtract the overlap from the total
Where next?
Another one like thissuggested
Step back and solidifysuggested