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:
$$\boxed{ \frac{M_{1} \ast D_{1} \ast T_{1}}{W_{1}} = \frac{M_{2} \ast D_{2} \ast T_{2}}{W_{2}}}$$
Here, $\frac{(15 \text{H} + 5 \text {R}) 30 \text{D}}{\text{W}} = \frac{(5 \text{H} + 15\text{R}) 60\text{D}}{\text{W}}$
$\Rightarrow (15 \text{H} + 5 \text {R}) 30 = (5 \text{H} + 15\text {R}) 60$
$\Rightarrow 15 \text{H} + 5 \text {R} = (5 \text{H} + 15\text{R}) 2$
$\Rightarrow 15 \text{H} + 5 \text {R} = 10 \text{H} + 30 \text{R} $
$\Rightarrow 15 \text{H} – 10 \text {H} = 30 \text{R} – 5 \text{R} $
$\Rightarrow 5 \text{H} = 25 \text{R}$
$\Rightarrow \text{H} = 5 \text{R} \quad \longrightarrow (1)$
$\Rightarrow \boxed { \frac { \text{H}} { \text{R}} = \frac{5}{1} }$
$\textbf{First Method:}$
The total work done by human and robot is:
- Total work $=(15 \text {H} + 5 \text {R}) 30 $
- Total work $=(15 (5\text{R})+ 5 \text{R}) 30 $
- Total work $=(75\text{R} + 5\text{R})30$
- Total work $= 80 \text{R}\times 30$
- Total work $= 2400\text{R}$ units.
$\therefore$ Number of days in which $15$ humans finish the work $ = \frac{2400\text{R}}{15 \text {H}} =\frac{2400\text{R}}{15 ( 5\text{R})} = \frac{2400}{75} = 32\; \text {days}.$
$\textbf{Second Method:}$ Let fifteen humans working together takes $x$ days to finish the job.
The, $\frac{(15\text{H}) \ast x}{\text{W}} = \frac{(15\text{H} + 5\text{R}) \ast 30}{\text{W}}$
$\Rightarrow (15\text{H}) \ast x = (15\text{H} + \text{H}) \ast 30 \quad [\because \text{From equation (1)}]$
$\Rightarrow (15\text{H}) \ast x = (16\text{H}) \ast 30$
$\Rightarrow x = 32\;\text{days}.$
Correct Answer $: \text{B}$