If $R$ is a commutative ring and $I$ and $J$ are ideals s.t. $I+J=R$ then show that $IJ=I\cap J$.

I've already shown that $IJ \subset I\cap J$, now I need to show the reverse inclusion. 

I'm a bit lost, so far i'm just figuring out what pieces I have to work with.

Such as:

$\forall r\in R$ $\exists i\in I ,j\in J$ s.t. $i+j=r$

$\forall ij\in IJ$, $ij=i\_1$ and $ij=j\_1$ for some $i\_1\in I$, $j\_1\in J$.

Also, if I let $x\in I\cap J$, then $x=i\_2=j\_2=i+j$ for some $i\_2\in I$, $j\_2\in J$

Anyone, having problem getting to the conclusion here, thanks in advance