#1**+1 **

I think this question is one where some people might want to pull out their hair in frustration. I suggest gathering all your information into an organized table.

Number of Rooms Painted | Rate | Time | ||

Paulo | ||||

Irina | ||||

Paulo + Irina | ||||

We may not be able to fill in all the information, but that is OK! Do not forget what the end goal is! The end goal is to figure out how much time it will take if Paulo and Irina work together. This is the information we know:

- Irina painted 1 room
- Irina painted that room in 9 hours
- Paulo painted the same room
- Paulo painted that room in 8 hours
- Together, Paulo and Irina will paint one room

Now that we know all this information, let's fill in the table.

Number of Rooms Painted | Rate | Time | ||

Paulo | 1 | 8 hours | ||

Irina | 1 | 9 hours | ||

Paulo + Irina | 1 | |||

Now, what is the painting speed of Paulo? I think you would agree that Paulo can paint at 1 room per 8 hours. This is the rate for Paulo! We can use the same logic for Irina. Irina paints at a speed of 1 room per 9 hours. Let's fill that information in!

Number of Rooms Painted | Rate | Time | ||

Paulo | 1 | \(\frac{1\text{ room}}{8\text{ hours}}\) | 8 hours | |

Irina | 1 | \(\frac{1\text{ room}}{9\text{ hours}}\) | 9 hours | |

Paulo + Irina | 1 | |||

In order to fill in the rest of the table, we will have to introduce variables! I will use "x" to represent the number of hours it takes for Paulo and Irina collectively to paint this room. This means that the rate of them together is \(\frac{1\text{ room}}{x \text{ hours}}\). Let's fill that in, as well!

Number of Rooms Painted | Rate | Time | ||

Paulo | 1 | \(\frac{1\text{ room}}{8\text{ hours}}\) | 8 hours | |

Irina | 1 | \(\frac{1\text{ room}}{9\text{ hours}}\) | 9 hours | |

Paulo + Irina | 1 | \(\frac{1\text{ room}}{x\text{ hours}}\) | x hours | |

We can assume that, if Paulo and Irina work together at their individual rates, then the rate will be the sum.

Therefore, \(\frac{1}{8}+\frac{1}{9}=\frac{1}{x}\). Let's solve this!

\(\frac{1}{8}+\frac{1}{9}=\frac{1}{x}\) | The best way, I believe, to solve this equation is to multiply by the LCM of the denominators. in this case, that number would be \(72x\). |

\(9x+8x=72\) | Add the like terms. |

\(17x=72\) | Divide by 17 on both sides. |

\(x=\frac{72}{17}\approx\text{4.24 hours}\) | |

Therefore, the correct answer choice is the last one (which is really just a manipulation of the equation I created).

TheXSquaredFactor
Mar 5, 2018

#2**+1 **

Thanks, X^{2}

Here's another way to do this

Irma paints 1/9 of a room in one hour

Paulo paints 1/8 of a room in one hour

So.....Rate * Time = Total Job Finished

(1/9)x + (1/8)x = 1

(9 + 8) x

________ = 1 cross-multiply

72

17x = 72 divide both sides by 17

x = 72/17 ≈ 4.24 hrs

Notice something else, NSS.... if we add 1/8 and 1/9 and then take the reciprocal of that answer, we will get the total time !!!!

CPhill
Mar 5, 2018