Paul can install a 300-square-foot hardwood floor in 18 hours. Matt can install the same floor in 22 hours. How long would it take Paul and Matt to install the floor working together?

Question

whpalmer4 · Answer

Find the unit rate of each worker.  Paul can do 300 sq ft/18 hours, and Matt can do 300 sq ft/22 hours.  Even simpler is to consider this in terms of 1 floor:

Paul's rate is 1 floor/18 hours
Matt's rate is 1 floor/22 hours

Together, to do one floor, they work at a combined rate of $$r =\frac{1 \text{ floor}}{18 \text{ hours}} + \frac{1 \text{ floor}}{22\text{ hours}}$$

To do 1 floor, plug that rate into the rate equation and solve for the time.

$$x = r t$$$$t = \frac{x}{r} = \frac{1 \text{ floor}}{\frac{1 \text{ floor}}{18 \text{ hours}} + \frac{1 \text{ floor}}{22\text{ hours}}} =  \frac{1 \cancel{ \text{ floor}}}{\frac{1 \cancel{\text{ floor}}}{18 \text{ hours}} + \frac{1 \cancel{\text{ floor}}}{22\text{ hours}}} = \frac{1}{\frac{1}{18}+\frac{1}{22}}\text{ hours}$$

anonymous · Answer

Thank you!

whpalmer4 · Answer

Got an answer?  I'll check it for you...

anonymous · Answer

I got 13.2

whpalmer4 · Answer

well, let's check:

in 13.2 hours, Paul does 13.2/18 of a floor, or 0.7333
in 13.2 hours, Matt does 13.2/22 of a floor, or 0.6

If you add those together, the result will be 1 if you've done the problem correctly.  Have you? :-)

whpalmer4 · Answer

you could "cheat" and do this with decimals, but I think you should be sufficiently proficient using fractions to do it with fractions, too.

whpalmer4 · Answer

$$\frac{1}{18} + \frac{1}{22} = \frac{1}{18}*\frac{22}{22} + \frac{1}{22}*\frac{18}{18} = \frac{22}{22*18} + \frac{18}{22*18} = \frac{40}{396}$$$$\frac{1}{\frac{40}{396}} = 1*\frac{396}{40} = \frac{396}{40} = \frac{99}{10} = 9.9$$

After 9.9 hours, we have 9.9/18 + 9.9/22 = 0.55 + 0.45 = 1.0 jobs completed

anonymous · Answer

Aww, I got it wrong :( Thank you for explaining it though! I understand it now :)