Stamp Collector and the Secret Microprint: Magnification by a Convex Lens

Problem Statement

A stamp collector uses a magnifying glass (convex lens) to examine very small "microprint" on a rare 19th-century stamp.

Let the focal length of the magnifying glass be $f$f. The stamp surface is placed perpendicular to the optical axis of the lens. When the distance between the stamp and the lens is $a$a, the microprint observed through the lens appears as an upright virtual image larger than the actual object.

Let the actual length of the microprint be $L$L. Find the length $L'$L′ of the virtual image formed by the lens.

Assume the lens is thin and only paraxial rays are considered. The size of the virtual image $L'$L′ is defined as $L' = m \times L$L′=m×L, where $m$m is the magnification.

Constraints

• Focal length of magnifying glass: $f = 120$f=120 $\text{mm}$mm
• Distance from stamp to lens: $a = 90$a=90 $\text{mm}$mm
• Actual length of microprint: $L = 7$L=7 $\text{mm}$mm

Input Format

Find the length $L'$L′ of the virtual image in $\text{mm}$mm and give the answer as a positive integer.