I have some trouble figuring how to do this:

If Sally can paint a house in 4 hours, and John can paint the same house in 6 hour, how long will it take for both of them to paint the house together?

A. 2 hours and 24 minutes
B. 3 hours and 12 minutes
C. 3 hours and 44 minutes
D. 4 hours and 10 minutes
E. 4 hours and 33 minutes

I have no clue how they got those answers...

Here's my take on this, boi:-
In 1 hour, Sally can paint a quarter of the house, and John can paint one sixth. So, between them, they can paint five-twelfths of the house in one hour (ie. a quarter plus a sixth).

Therefore, between them, they can paint ONE twelfth of the house in one fifth of an hour, which is 12 minutes.

So, to paint the WHOLE house (ie. TWELVE twelfths of it), they will take 12 x 12 minutes. This is 144 minutes - 2 hours & 24 minutes.

To the original O.P.

if you've done quite a number of these types of problems, you will notice that

if Sally takes x hours to do a certain job and

John takes y hours to do the same job,

then the same job done by the two together will be done in \frac{xy}{x+y} hours.

To the original O.P.,

if you've done quite a number of these types of problems, you will notice that

if Sally takes x hours to do a certain job and

John takes y hours to do the same job,

then the same job done by the two together will be done in \frac{xy}{x+y} hours.

Another way I've seen double rates done is

\frac{1}{r_1}+\frac{1}{r_2}=\frac{1}{r_n}

if you do a bit of algebra work you get it into the same format as chris-infj posted. If you do it this way, don't forget to inverse your answer!

here: {r_1} is the first rate, {r_2} is the second rate, and {r_n} is the combined rate

And additionally, if there are more than one 'person' concerned, then, we go:

\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + ... = \frac{1}{x_n}

Where n is the number of 'people' concerned here.