# Problem Solving

• Jun 19th 2008, 05:47 PM
not happy jan
Problem Solving
Harry, Annie and Mitch decided to paint the living room. Harry could paint the room by himself in 3 hours. Annie could do it in 4 hours and Mitch would take 6 hours on his own. If they all work together and dont get in each others way, how long will the job take?
• Jun 19th 2008, 05:54 PM
TheEmptySet
Quote:

Originally Posted by not happy jan
Harry, Annie and Mitch decided to paint the living room. Harry could paint the room by himself in 3 hours. Annie could do it in 4 hours and Mitch would take 6 hours on his own. If they all work together and dont get in each others way, how long will the job take?

Since Harry can paint the room in 3 hours he can paint the $\frac{1}{3}$ of the room in 1 hour.

Using the same argument for the other's we get

Annie $\frac{1}{4}$ of the room in 1 hour

Mitch $\frac{1}{6}$ of the room in 1 hour

Now we can add the rates that each of them work

$\underbrace{\left(\frac{1}{3}+\frac{1}{4} +\frac{1}{6}\right)}_{\mbox{how fast they are together}}\cdot \underbrace{t}_{time}=\underbrace{1}_{\mbox{one room}}$

Simplifying we get

$\frac{3}{4}t=1 \iff t=\frac{4}{3}$ so it will take them One and one thirds hours or 1 hour and 20 min.

I hope this helps.
• Jun 19th 2008, 05:56 PM
Jhevon
Why are you not happy, Jan?
Quote:

Originally Posted by not happy jan
Harry, Annie and Mitch decided to paint the living room. Harry could paint the room by himself in 3 hours. Annie could do it in 4 hours and Mitch would take 6 hours on his own. If they all work together and dont get in each others way, how long will the job take?

just add their rates. let painting the living room be the job

for Harry, he does $\frac {1 \mbox{ Job}}{3 \mbox{ Hours}}$

Annie does $\frac {1 \mbox{ Job}}{4 \mbox{ Hours}}$

Mitch does $\frac {1 \mbox{ Job}}{6 \mbox{ Hours}}$

together they can do: $\frac 13 + \frac 14 + \frac 16 = \frac 34 = \frac 1{4/3}$ that is, $\frac {1 \mbox{ Job}}{{\color{red}4/3 \mbox{ Hours}}}$

EDiT: Thanks a lot, TES :p you look different with long hair