Work Problems

1/5.5 + 1/7.5 =52/165, therefore the job will be done in:

165 /52 =~3 hours and about 10 minutes.

Aris can paint a room in 5 ½ hours.
His brother can do the same job in 7 ½ hours.
How long will it take them to paint the room together?

Let t = time required when working together

$\begin{array}{|rcll|} \hline t_1 &=& 5\frac12 \\ t_2 &=& 7\frac12 \\\\ \frac{1}{t} &=& \frac{1}{t_1} + \frac{1}{t_2} \\ \frac{1}{t} &=& \frac{t_1+t_2}{t_1\cdot t_2} \\ \mathbf{t} &\mathbf{=}& \mathbf{ \frac{t_1\cdot t_2}{t_1+t_2} }\\\\ t & = & \frac{t_1\cdot t_2}{t_1+t_2} \qquad & \qquad t_1 = 5\frac12 = \frac{11}{2} \qquad t_2 = 7\frac12 = \frac{15}{2}\\ t & = & \frac{\frac{11}{2}\cdot \frac{15}{2}}{\frac{11}{2}+\frac{15}{2}} \\ t & = & \frac{ \frac{11\cdot 15}{4} } { \frac{11+15}{2} } \\ t & = & \frac{ \frac{165}{4} }{ \frac{26}{2} } \\ t & = & \frac{165}{4} \cdot \frac{2}{26} \\ t & = & \frac{165}{2\cdot26} \\ t & = & \frac{165}{52} \\ t & = & 3.17307692308~ \text{hours}\\ \hline \end{array}$

It will take them 3.173 hours to paint the room together.