Problem 4.8. A merchant has 6 bags weighing 13, 15, 16, 17, 21, and 24 kg. One bag is filled with turnips, and each of the remaining bags is either onions or carrots. The merchant knows that the total weight of the carrots is twice the total weight of the onions. In which bag can the turnips be? List all possible options.
Answer: 13 or 16.

Solution. The total weight of onions is a whole number of kilograms, and carrots weigh twice as much. Therefore, the total weight of onions and carrots is a whole number of kilograms divisible by 3, and they are in all bags except one. The total weight of all bags is $13+15+$ $16+17+21+24=106$ kg, so the total weight of onions and carrots is one of the following numbers:

- $106-13=93$ kg;
- $106-15=91$ kg;
- $106-16=90$ kg;
- $106-17=89$ kg;
- $106-21=85$ kg;
- $106-24=82$ kg.

Only 93 and 90 are suitable.

Rape could have been in the bag weighing 13 kg if the onions were in the bags weighing 15 and 16 kg, and the carrots were in the bags weighing 17, 21, and 24 kg. Indeed, $17+21+24=62=2 \cdot(15+16)$.

Rape could also have been in the bag weighing 16 kg if the onions were in the bags weighing 13 and 17 kg, and the carrots were in the bags weighing 15, 21, and 24 kg. Indeed, $15+21+24=60=$ $2 \cdot(13+17)$

## 5th grade