One gallon of fuel mixture contains antifreeze in the ratio of 5 parts fuel to one part antifreeze. to this is added half a gallon of mixture which is one third antifreeze and two thirds fuel. what is the ratio of fuel to antifreeze in the final mixture?

1 gal (initial) = (5/6) parts fuel & (1/6) parts antifreeze= total of 6 parts
0.5 gal (add'l) = (2/3) parts fuel & (1/3) parts antifreeze = total of 3 parts

adding the two
Fuel:          1gal*(5/6 parts) + 0.5gal*(2/3 parts) = 7/6 parts in 1.5 total gallons
Antifreeze:  1gal*(1/6 parts) + 0.5gal*(1/3 parts) = 2/6 parts in 1.5 total gallons

thus with a common denominator provides a ratio of 7:2 (fuel:antifreeze)