Work done by $300$ men for $50$ days $ = 300M \times 50D = 15000MD$
Lets say total work is $'W'\,\,\text{unit}.$
$\implies \dfrac{1}{2} W = 15000 MD$
$\implies W = 30000 MD$
Remaining work $ = 30000MD - 15000MD = 15000 MD$
Work done by $200$ men for $40$ days $ = 200M \times 40D = 8000 MD$
Remaining work $ = 15000 MD - 8000MD = 7000 MD$
Remaining time $ = 155 - (50 + 40 + 15) = 155 - 105 = 50$ days
Lets say remaining work is done by $'x'$ men in $50$ days.
$\implies x M \times 50D = 7000 MD$
$\implies x = 140$ men is needed.
So, the correct answer is $(A).$