Airline Excess Luggage Allowance

Let free allowance be $F$ kg and rate per kg be $V$. Raja excess charge: $(x - F)V = 1200$. Praja excess charge: $(60 - x - F)V = 2400$. Single person excess charge: $(60 - F)V = 5400$.

Dividing Praja by Raja: $\frac{60 - x - F}{x - F} = 2 \implies 60 - x - F = 2x - 2F \implies 3x - F = 60$. Dividing total by Raja: $\frac{60 - F}{x - F} = 4.5 \implies 60 - F = 4.5x - 4.5F \implies 4.5x - 3.5F = 60$. Solving $1.5 \times (3x - F = 60) \implies 4.5x - 1.5F = 90$. Subtracting: $2F = 30 \implies F = 15$ kg. $3x - 15 = 60 \implies 3x = 75 \implies x = 25$ kg (Raja's luggage). Praja's luggage $= 60 - 25 = 35$ kg.

From the system of equations derived above, $F = 15$ kg.