## 4. Cutting the Twine

A twine of length $60 \mathrm{~m}$ is cut into equal pieces of length $3 \mathrm{~m}$. From the resulting pieces, six are cut into segments of length $10 \mathrm{~cm}$, six into segments of length $5 \mathrm{~cm}$, and the remaining into segments of length $6 \mathrm{~cm}$. How many cuts were made in the twine?

Result: $\quad 939$
## Solution.

If the parts are 3 meters long, then there are $60: 3=20$ parts, meaning the rope is cut at 19 points, or in other words, it is cut 19 times for this part.

Six parts of 3 meters in length are cut into pieces of 10 cm in length, which means that from each part of 3 meters, or $300 \text{ cm}$, 300 : $10=30$ pieces are obtained, meaning each such part is cut at 29 points. In total, it is cut $6 \cdot 29=\underline{174}$ times.

Six parts of 3 meters in length are cut into pieces of 5 cm in length, which means that from each part of 3 meters, or $300 \text{ cm}$, $300: 5=60$ pieces are obtained, meaning each such part is cut at 59 points. In total, it is cut $6 \cdot 59=\underline{354}$ times.

The remainder, or 20-6-6 = 8 parts of 3 meters in length, are cut into pieces of 6 cm in length, which means that from each part of 3 meters, or $300 \text{ cm}$, $300: 6=50$ pieces are obtained, meaning each such part is cut at 49 points. In total, it is cut $8 \cdot 49=\underline{392}$ times.

The total number of cuts is $19+174+354+392=939$.