A decimal number is called balanced if the sum of the digits before the decimal point is equal to the sum of the digits after the decimal point. For example, the number 25.133 is balanced. In each of the numbers 497 365.198 043 and 197 352.598 062, cross out some digits so that a) the largest balanced number is created, b) the balanced number with the most digits is created.

( $S$. Bednářová)
Let's take the number 497365.198043. The sum of the digits before the decimal point is 34, and the sum of the digits after it is 25. This means that before the decimal point, digits with a total value of 9 need to be crossed out. To keep as many digits as possible, we only cross out the 9 and get 47365.198043. If we want to get the largest possible number, we try to cross out as few digits as possible with the highest possible value before the decimal point. After the decimal point, we cross out digits so that the digits with higher values "move" forward. This way, we get the number 47365.19843.

Let's take the number 197352.598062. The sum of the digits before the decimal point is 27, and the sum of the digits after it is 30. If the number is to have as many digits as possible, we need to cross out one digit before the decimal point and one after it. The solutions are:

| 19732.59062 | crossed out 5,8 |
| :--- | :--- |
| 19752.59802 | crossed out 3,6 |
| 19735.98062 | crossed out $2,5$. |

If we want to get the largest possible number, we need to cross out one digit before the decimal point. After the decimal point, we cross out digits so that the digits with higher values "move" forward. This way, we get the number $19752.5982$.