# Batman can clean up all of the crime in gotham city in 8hr working alone. robin can do it alone in 12 hr. If robin starts crime fighting at 8am and batman joins at 10am , then at what time will they have all of the crime cleaned up?

Jun 4, 2016

finish time is 2pm

#### Explanation:

$\textcolor{red}{\text{Time spans are sometimes easier to work out on 24 hour clock}}$

Let the total amount of work to be done be $W$

Let Batman's rate of work be ${b}_{\text{w}}$

Let Robin's rate of work be ${r}_{\text{w}}$

Thus " "8b_("w")=W" and "12r_("w")=W

$\implies \text{hourly rate "-> b_("w")=W/8" and "r_("w")=W/12" } \ldots \ldots \ldots \ldots \left(1\right)$

Let Robin's work time be $t$

Let Batman's work time be $t - 2$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Then [b_("w")xx(t-2)]+[r_("w")xxt]=W" "..........(2)

Substitute in (2) using (1)

$\left[\frac{W}{8} \times \left(t - 2\right)\right] + \left[\frac{W}{12} \times t\right] = W \text{ } \ldots \ldots \ldots . \left({2}_{a}\right)$

$W \left(\left[\frac{1}{8} \times \left(t - 2\right)\right] + \left[\frac{1}{12} \times t\right]\right) = W$

Divide both sides by $W$

$\frac{t - 2}{8} + \frac{t}{12} = 1$

$\frac{12 \left(t - 2\right) + 8 t}{8 \times 12} = 1$

Multiply both sides by $\left(8 \times 12\right) \to 96$

$12 t - 24 + 8 t = 96$

$20 t - 24 = 96$

$20 t = 120$

Divide both sides by 20

$t = 6 \text{ hours "larr" time for Robbin}$
'~~~~~~~~~~~~~~~~~~~~~~~
Robbin starts at 8am so finish time is using 24 hour clock

$0800 \leftarrow \text{ Robbin's start time}$
$\underline{0600} \leftarrow \text{ add hours worked}$
$1400 \text{ hours "larr" 24 hour clock}$

$1400 \text{ hours "-> 2" pm}$