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A contractor agreed to construct a $6 \; \text{km}$ road in $200 \; \text{days}.$ He employed $140$ persons for the work. After $60 \; \text{days},$ he realized that only $1.5 \; \text{km}$ road has been completed. How many additional people would he need to employ in order to finish the work exactly on time $?$
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Let the additional people need be $x.$

If $M_{1}$ person can do $W_{1}$ work in $D_{1}$ days working $T_{1}$ hours in a day and $M_{2}$ person can do $W_{2}$ work in $D_{2}$ days working $T_{2}$ hours in a day then the relationship between them is :
$$ \boxed {\frac{M_{1} \ast D_{1} \ast T_{1}}{W_{1}} = \frac{M_{2} \ast D_{2} \ast T_{2}}{W_{2}}} $$

Now, $ \dfrac{140 \times 60}{1.5} = \dfrac{(x+140) \times 140}{4.5} $

$ \Rightarrow 180 = x + 140 $

$ \Rightarrow x = 180 – 140 $

$ \Rightarrow \boxed{x = 40} $

$\therefore$ The number of additional people needed  is $40.$

Correct Answer$: 40$
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