6. A farmer presented 6 types of sour cream in barrels of $9,13,17,19,20,38$ liters at the market. On the first day, he sold sour cream from three barrels completely, and on the second day, from two more barrels completely. The volume of sour cream sold on the first day was twice the volume of sour cream sold on the second day. Which barrels were emptied on the first day? In your answer, indicate the largest possible sum of the volumes of sour cream sold on the first day.

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# Solution:

A total of 116 liters were delivered; $116=3 \cdot 38+2$; therefore, the unsold barrel, when divided by 3, gives a remainder of 2.

1) If it is 17 liters, then $116-17=99$, then on the second day, one third of 99 liters were sold, which is $33=13+20$, then on the first day, barrels with capacities of $9, 19, 38$ liters were sold, their total volume is 66 liters.

2) If it is 20 liters, then $116-20=96$; one third of 96 is $32=13+19$, then on the first day, barrels with capacities of $9, 17, 38$ liters were sold, their total volume is 2 liters less than the previous one;

3) If the 38-liter barrel is unsold, then $116-38=78$, two thirds of 78 is 52, 52 is less than 66.

Answer: 66.