3. When manufacturing a metal wedge, the waste is $12.5\%$ of the material used. In this way, from one piece of metal, exactly 100,000 such wedges are made. All the waste obtained is melted back into one piece of metal, and from it, more such wedges are made in the same way. This process is repeated until it is possible to make at least one such wedge from the waste.

Determine the total number of wedges obtained, including the initial 100,000 wedges.
Solution. The waste amounts to $\frac{12.5}{100}=\frac{1}{8}$ of the used material.

According to this, from the waste generated by the production of the first 100,000 wedges, $100000 \div 8 = 12500$ wedges can be made with $\frac{1}{8}$ waste. From the new waste, we get $12500 \div 8 = 1562$ wedges, with waste amounting to $\frac{4}{8}$ of one wedge. Now we have $\frac{1562 + 4}{8} = \frac{1566}{8}$ leftovers, from which 195 wedges can be made with a remainder of $\frac{6}{8}$. Further, we have 201:8-25 (remainder is 1); 26:8=3 (remainder is 2). Finally, there is not enough material to continue production. Therefore, a total of $100000 + 12500 + 1562 + 195 + 25 + 3 = 114285$ wedges have been produced with a remainder of $\frac{5}{8}$ material for making a new wedge.