The requested source-image relation is the thin lens equation, using positive real-object distance

u

v

\frac{1}{f} = \frac{1}{u} + \frac{1}{v} .

Here

f > 0

for a converging lens, and

v < 0

denotes a virtual image. This is usually distinguished from Lensmaker's equation,

1/ f = (n - 1) (1/ R_{1} - 1/ R_{2})

for a thin lens in air with the usual signed surface radii. The latter relates focal length to the lens material and curvature, rather than to object and image positions.^[ref]

For a magnifying glass held close to the eye, compare the angle

h / u

with the unaided near-point angle

h / D_{n}

, where

D_{n} = 25 cm

. Maximum usable angular magnification occurs with the virtual image at the near point, so

v = - D_{n}

. Thus

u = \frac{f D _{n}}{D _{n} + f} = \frac{100}{29} cm ≃ 3.45 cm, M_{m a x} = \frac{D _{n}}{u} = 1 + \frac{D _{n}}{f} = 7.25.

The object lies inside the front focal length. The emerging light rays diverge, while their backward extensions meet at an upright enlarged virtual image on the object's side. With the image at infinity instead, the relaxed-eye angular magnification would be