A vendor has six jugs of punch with capacities of 13, 15, 16, 21, 21, and 22 gallons. One jug, which he keeps for himself, is full of blue p

A vendor has six jugs of punch with capacities of 13, 15, 16, 21, 21, and 22 gallons. One jug, which he keeps for himself, is full of blue punch. The other five jugs are full of red punch. Without opening any of the jugs, he sells 1/3 of the red punch to Randy and the remaining 2/3 of the red punch to Lisa. How many gallons of the blue punch did he keep for himself?

 

13, 15, 16, 21, 21, and 22 gallons jugs

five full with red and one full with blue.

 

$${\mathtt{13}}{\mathtt{\,\small\textbf+\,}}{\mathtt{15}}{\mathtt{\,\small\textbf+\,}}{\mathtt{16}}{\mathtt{\,\small\textbf+\,}}{\mathtt{21}}{\mathtt{\,\small\textbf+\,}}{\mathtt{21}}{\mathtt{\,\small\textbf+\,}}{\mathtt{22}} = {\mathtt{108}}$$

 

Let x be the gallons of blue

 

$${\frac{\left({\mathtt{108}}{\mathtt{\,-\,}}{\mathtt{X}}\right)}{{\mathtt{3}}}}$$   = one or more of those jugs add together.  

 

  $$\\\frac{108-X}{3}=\frac{108}{3}-\frac{X}{3}=36-\frac{X}{3}$$

 

So X (Blue) must be divisable by 3   This means the blue possibilities are  15, and 21

 

$$\\36-\frac{X}{3}\\\\$$

If the blue is 15 gallons then the sum of all the others is 108-15=93

So 1 third is 31. There is no way to make 31 gallons from those bottles without opening them.

 

If the blue is 21 gallons then the sum of all the others is  108-21=87

So 1 third is 29 gallons. 13 gallons + 16 gallons = 29 gallons.

 

So the vendor keeps 21 gallons of blue.

Randy gets 13gallons + 16 gallons = 29 gallons of red   and

Lisa gets   21 gallons +   15 gallons +  22 gallons  = 58 gallons of red

 

check.   Lisa gets twice as much as Randy so that works out well :)