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Humans and robots can both perform a job but at different efficiencies. Fifteen humans and five robots working together take thirty days to finish the job, whereas five humans and fifteen robots working together take sixty days to finish it. How many days will fifteen humans working together (without any robot) take to finish it?

1. $36$
2. $32$
3. $45$
4. $40$

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 $: \text{B}$

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