Ok, no problem.
The error made when fueling Flight 143 was that the conversion factor used was in units of pounds/liter, when it should have been in kg/liter. The correct conversion factor would have been:
(1.77 pounds/L)(0.45359 kg/pound) = 0.803 kg/L
Assuming that the measurement of 7,682 liters was correct, the mass of fuel in the tanks at that time would have been:
(7682 L)(0.803 kg/L) = 6,168.646 kg
The amount that would need to be added to get the plane's fuel load up to 22,300 kg is then:
22,300 kg - 6,168.646 kg = 16,131.354 kg
The volume (in liters) that should have been added at that point is then:
(16,131.354 kg) / (0.803 kg/L) = 20,088.86 L
Regarding the two fractions, both 1.77 L / 1 kg and 1 L / 1.77 kg would have been the incorrect factor to use, because the 1.77 value is part of the conversion from pounds to liters, instead of from kg to liters. Ignoring the value and instead just focusing on the units of the two fractions, the fraction 1.77 L / 1 kg would mean that each kilogram of fuel occupied a volume of 1.77 liters. The fraction 1 L / 1.77 kg means that each liter has a mass of 1.77 kg.
To look at the fractions another way, 1.77 L / 1 kg is equivalent to 1.77 L/kg, while 1 L / 1.77 kg is equivalent to 0.565 L/kg. Clearly the two factors are very different values.
Both fractions could be used to convert from kg to liters, so it would just be a matter of which value was the correct factor. In the case of Flight 143 though, neither of those factors would have been correct.
I hope this helps.