7.3. There are 30 logs, the lengths of which are 3 or 4 meters, and their total length is one hundred meters. How many cuts can be made to saw all these logs into pieces 1 meter long? (Each cut saws exactly one log).
Answer: 70.

Solution. First method The total length of the logs is 100 meters. If it were one log, 99 cuts would be needed. Since there are 30 logs, 29 cuts have already been made. Therefore, $99-29=70$ cuts remain to be made.

Second method Let's find the number of logs of each type. If all were 3-meter logs, their total length would be 90 meters. Since the total length is 100 meters, there are 10 logs that are 4 meters long and 20 logs that are 3 meters long. For each 4-meter log, three cuts are required, and for each 3-meter log, two cuts are required. In total: $10 \cdot 3 + 20 \cdot 2 = 70$ cuts.

Grading criteria.

“+” A complete and well-reasoned solution is provided

“±” A generally correct reasoning is provided, but a computational error is made

“±” Only the correct answer is provided or the correct answer is obtained from a specific example

“-” The problem is not solved or is solved incorrectly