Calculate resulting RGB from 2 colors, one is transparent

Question

I'm looking for a formula to convert them.

I know to convert a general transparency it is

alpha * new + ( 1 - alpha ) * old

I have:

Color A : RGB( 85, 113, 135 )
Color B : RGB( 43, 169, 225 )

Color A has 90% opacity and is laid on top of Color B, resulting in

Color C : RGB( 65, 119, 145 )

My question is, how does it get Color C? If I substitute Color B for another thing, how do I get Color C?

Here's another example, same base color:

Color A : RGB( 85, 113, 135 )
Color B : RGB( 45, 67, 82 )
--------
Color C : RGB( 65, 109, 131 )

Those are working examples done with images -- I'm trying to now calculate the remaining Color C so I can assign a background color.

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UPDATE, please see the accepted answer. The red in the above examples is strange -- the accepted answer has the correct formula for all the colors, I tested it in Photoshop.

Answer 1

It appears that your formula is precisely the formula used in your examples, calculated per component, and rounded up.

R_c := ceiling(R_a * alpha) + ceiling (R_b * (1 - alpha))

G_c := ceiling(G_a * alpha) + ceiling (G_b * (1 - alpha))
B_c := ceiling(B_a * alpha) + ceiling (B_b * (1 - alpha))

Strange, though, the R component doesn't appear to follow the rules. I'm inclined to wonder why.