Correct Answer: 77. (3)

Solution: 77. (3) Original number of people associated with plan $=x$

$x$ people would have done the work in $12 x$ days.

$\mathrm{M} _{1} \mathrm{D} _{1}=\mathrm{M} _{2} \mathrm{D} _{2}$

$\Rightarrow 12 x=x \times 1+(x-4) \times 1+(x$ $-8) \times 1+(x-12) \times 1 \ldots \ldots$ to 18 terms

$\Rightarrow 12 x=18 x-(4+8+12+\ldots$. to 17 terms)

$\Rightarrow 12 x$ $=18 x-\left[\frac{17}{2}[2 \times 4+(17-1) \times 4]\right]$

For arithmetic series, $\mathrm{S}_{n}=\frac{n}{2}$ $[2 a+(n-1) d]$

$\Rightarrow 12 x=18 x-\frac{17}{2}(8+64)$

$\Rightarrow 12 x=18 x-\frac{17}{2} \times 72$

$\Rightarrow 12 x=18 x-612$

$\Rightarrow 18 x-12 x=612$

$\Rightarrow 6 x=612$

$\Rightarrow x=\frac{612}{6}=102$