## 20. Math Puzzle $1 / 67$

At the Office of Invention, two improvement suggestions are submitted to reduce the production time of a part. However, these suggestions are mutually exclusive.

The first suggestion requires 20 hours to adjust the production equipment and results in a time savings of $30\%$ per part. The second suggestion requires only 10 hours of preparation time but yields only a $10\%$ time savings.

For which quantities is the first suggestion more economical, and for which quantities is the second suggestion more economical, if the current method requires 10 working hours for one part?
The savings amount to $3x-20$ and $x-10$. The time spent is equal when: $3x-20=x-10 \rightarrow x=5$. Therefore, the first proposal is more cost-effective if $x>5$.

At 5 units, although the production time is the same for both methods, it is still $5 \text{ h}$ longer than the old method. In fact, the first proposal is only time-efficient from 7 units onwards, and the second from 11 units onwards. Therefore, in any case, the first proposal will be preferred from 7 units onwards, and for quantities below 7, production will continue using the old method.