Question

A convex mirror is needed to produce an image that is nine-tenths the size of an object and located 21 cm behind the mirror. What focal length should be specified?

Answer 1

First of all the lens-maker's equation is correctly written as:

1/f = 1/u + 1/v,

And the appropriate sign convention is worked out for each specific case in question.

cases:
convex or concave lens, image distance is negative only for virtual images
(projected images) that is images formed on the same side of the lens.
plane or spherical mirror, image distance: v is negative, since image is virtual

Now, 1/u + 1/v => combine fractions by using the common denominator: uv

1/u + 1/v = (v + u)/uv = 1/f

f = uv/(u+v)

Note: always write the general lens equation with all positive terms,
the appropriate sign will come out in the wash.