# Thread: Help painting my bathroom!!

1. ## Help painting my bathroom!!

Here is the question:

I have three hoses and I wish to paint my bathroom. If I use my blue paint hose, I could paint it in 2 hours. If I use my green hose, I could do it in 3 hours. If I use my red hose, I could paint it in 4 hours. If I don't mind very much about what it looks like and use all three hoses at the same time, how long would it take to paint the room?

2. Originally Posted by bbbeagar

Here is the question:

I have three hoses and I wish to paint my bathroom. If I use my blue paint hose, I could paint it in 2 hours. If I use my green hose, I could do it in 3 hours. If I use my red hose, I could paint it in 4 hours. If I don't mind very much about what it looks like and use all three hoses at the same time, how long would it take to paint the room?
You're right.

Suppose you had a number of bedrooms all the same size.
You hose one with the blue paint and your friends hose with the other two.

In 12 hours, you hose 6 bedrooms.
In 12 hours, the friend with the green paint hoses 4 bedrooms.
In 12 hours, the third friend hoses 3 bedrooms.

In 12 hours, 13 bedrooms get hosed.

In 1 hour, how many get hosed ?
Or, how long does it take to hose 1 bedroom ?

Probably you are expected to work with fractions though (not sure).

3. Hello, bbbeagar!

Here is my baby-talk approach to "work problems".

I get 12/13 of an hour, but have no way of checking my answer:

Here is the question:

I have three hoses and I wish to paint my bathroom.
If I use my blue paint hose, I could paint it in 2 hours.
If I use my green hose, I could do it in 3 hours.
If I use my red hose, I could paint it in 4 hours.
If I use all three hoses at the same time,
how long would it take to paint the room?

Let $\displaystyle \,x$ = hours for all 3 hoses to paint the room.

The blue hose paints the room in 2 hours.
. . In one hour, it paints $\displaystyle \frac{1}{2}$ of the room.
. . In $\displaystyle \,x$ hours, it paints $\displaystyle \frac{x}{2}$ of the room.

The green hose paints the room in 3 hours.
. . In one hour, it paints $\displaystyle \frac{1}{3}$ of the room.
. . In $\displaystyle \,x$ hours, it paints $\displaystyle \frac{x}{3}$ of the room.

The red hose paints the room in 4 hours.
. . In one hour, it paints $\displaystyle \frac{1}{4}$ of the room.
. . In $\displaystyle \,x$ hours, it paints $\displaystyle \frac{x}{4}$ of the room.

Working together for $\displaystyle \,x$ hours,
. . the 3 hoses will paint: .$\displaystyle \frac{x}{2} + \frac{x}{3} + \frac{x}{4}$ of the room.

But in $\displaystyle \,x$ hours, the 3 hoses will paint the entire room (1 room).

There is our equation! .$\displaystyle \hdots\quad \dfrac{x}{2} + \dfrac{x}{3} + \dfrac{x}{4} \:=\:1$

Multiply by 12: .$\displaystyle 6x + 4x + 3x \:=\:12 \quad\Rightarrow\quad 13x \:=\:12$

Therefore: .$\displaystyle x \:=\:\dfrac{12}{13}\text{ hours} \;\approx\;\text{55 minutes, 23 seconds}$

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Checking the answer is a bit messy . . .

The blue hose paints $\displaystyle \frac{1}{2}$ the room in one hour.
. . In $\displaystyle \frac{12}{13}$ hour, it paints: .$\displaystyle \frac{12}{13}\cdot\frac{1}{2} \:=\:\frac{6}{13}$ of the room.

The green hose paints $\displaystyle \frac{1}{3}$ the room in one hour.
. . In $\displaystyle \frac{12}{13}$ hour, it paints: .$\displaystyle \frac{12}{13}\cdot\frac{1}{3} \:=\:\frac{4}{13}$ of the room.

The red hose paints $\displaystyle \frac{1}{4}$ the room in one hour.
. . In $\displaystyle \frac{12}{13}$ hour, it paints: .$\displaystyle \frac{12}{13}\cdot\frac{1}{4} \:=\:\frac{3}{13}$ of the room.

Working together for $\displaystyle \frac{12}{13}$ hour, the three hoses will paint:

. . . $\displaystyle \displaystyle\frac{6}{13} + \frac{4}{13} + \frac{3}{13} \;=\;\frac{13}{13} \;=\;1$ . . . the whole room!

4. Another way to think about this, and "work" problems in general, is that when several people, machines, "paint hoses", etc. work together, their rates of work add. With the red hose you can "paint it in 2 hours" or "2 hours per room", so, taking the reciprocal, its rate of work is "1/2 rooms per hour". With the blue hose, you can "do it in 3 hours" or "3 hours per room" so its rate of work is "1/3 room per hour". With the red hose, you can "paint it in 4 hours" or "4 hours per room", so its rate of work is "1/4 room per hour". Using all three to together (I don't even want to imagine the mess!) you can work at the rate of 1/2+ 1/3+ 1/4= 6/12+ 4/12+ 3/12= 13/12 "room per hour". The reciprocal of that is 12/13 "hours per room" so it would take, as you said, 12/13 hour to paint the room.

5. When you say you have no way of checking your answer,
you could consider this also.....

the full room is painted in $\displaystyle \displaystyle\frac{12}{13}$ hour (hopefully!).

To check this:

$\displaystyle \displaystyle\ 2=\frac{26}{13}\Rightarrow\frac{12}{26}$ of the room gets covered by blue in $\displaystyle \displaystyle\frac{12}{13}$ hour.

$\displaystyle \displaystyle\ 3=\frac{39}{13}\Rightarrow\frac{12}{39}$ of the room gets covered by green in $\displaystyle \displaystyle\frac{12}{13}$ hour.

$\displaystyle \displaystyle\ 4=\frac{52}{13}\Rightarrow\frac{12}{52}$ of the room gets covered by red in $\displaystyle \displaystyle\frac{12}{13}$ hour.

Add the fractions to find out if it is one room.

$\displaystyle \displaystyle\frac{12}{26}+\frac{12}{39}+\frac{12} {52}=\frac{6}{13}+\frac{4}{13}+\frac{3}{13}$