Mom is sewing towels from fabric that is $120 \mathrm{~cm}$ wide. The finished towel has dimensions of $60 \mathrm{~cm} \times 38 \mathrm{~cm}$. When cutting the fabric, $2 \mathrm{~cm}$ must be added to each edge for finishing. How many centimeters of fabric does Mom need to buy at a minimum to sew 10 towels?
First, we need to realize what piece of fabric mom needs to cut out for one dishcloth. It will be a rectangle with dimensions $42 \mathrm{~cm}$ and $64 \mathrm{~cm}$. This means that we cannot fit even three dishcloths "in width" or two dishcloths "in height" on the fabric (let alone if we rotate one of them).

Let's sketch the individual placements (not entirely):
a)

![](https://cdn.mathpix.com/cropped/2024_04_17_516c8fdd311781851c06g-2.jpg?height=342&width=1014&top_left_y=777&top_left_x=541)
b)

![](https://cdn.mathpix.com/cropped/2024_04_17_516c8fdd311781851c06g-2.jpg?height=343&width=1014&top_left_y=1122&top_left_x=541)
c)

![](https://cdn.mathpix.com/cropped/2024_04_17_516c8fdd311781851c06g-2.jpg?height=354&width=1014&top_left_y=1465&top_left_x=544)
d)

![](https://cdn.mathpix.com/cropped/2024_04_17_516c8fdd311781851c06g-2.jpg?height=345&width=1014&top_left_y=1815&top_left_x=544)

In layout a), $10 \cdot 42=420 \mathrm{~cm}$ of fabric will be needed.

In layout b), $5 \cdot 64=320 \mathrm{~cm}$ of fabric will be needed.

In layout c), we need to determine which of the two lengths is greater. From the sketch, we cannot precisely see whether the greater length will be taken up by 4 dishcloths in the top row or 6 dishcloths in the bottom row: $4 \cdot 64=256$ and $6 \cdot 42=252$. In layout c), $256 \mathrm{~cm}$ of fabric will be needed.

There is one more option for placing five and five dishcloths, which we see in diagram d). In this case, we save an additional $2 \mathrm{~cm}$ of fabric, so mom will need to buy $254 \mathrm{~cm}$ of fabric.