Sam and Jane paint a room

Question

Suppose it takes Sam 5 hours to paint a room and it takes Jane $x$ hours to paint the same room. If it takes Sam and Jane $\frac{2}{3}x$ hours to paint the room together, what equation can be used to determine the value of $x$?

My mathematical intuition tells me that the equation is given by $$\frac{5x}{x+5} = \frac{2}{3}x.$$ However, I need to try and explain this to my younger high school cousin and seem to be having a hard time finding a good explanation . Can anyone please give me an explanation that I might be able to make her understand why that would be the equation to solve the problem?

Answer 1

My professor solved kind of the same problem in my class. Here is how it goes:
Let $A$ be the finished painting process and $t$ the time to finish it.
$S$-Sam and $J$-Jane
$$ A = tS \Longrightarrow A = 5S \Longrightarrow S = \frac{A}{5}\\ A = tJ \Longrightarrow A = xJ \Longrightarrow J = \frac{A}{x} $$ And when they work together we have:$$ A = \frac{2}{3}x(S+J)\\ A = \frac{2}{3}x \left( \frac{A}{5} + \frac{A}{x} \right)\\ 1 = \frac{2}{3} x \left( \frac{1}{5} + \frac{1}{x} \right) $$ So the equation for $t$ is $$\frac{2}{3} x \left( \frac{1}{5} +\frac{1}{x} \right) = 1.$$

Answer 2

I would put it in this way:
..then in $1$ h Sam paints $1/5$ of the room and Jane $1/x$ .
in $2/3x$ hours they will paint $2/3x(1/5+1/x)$ of a room and this shall $1$. Thus ..