Okay, let’s start with diameters, stick with diameters, and finish with diameters.
The area of a circle is A = (π) (D²) / 4. Where π = 3.14159 (approximately)
You could use A = (π) r² but since D = 2r then r = D/2 and r² = D²/4 so we can work with diameter instead of radius. Most drills I have used are specified by diameter, not radius.
We start with a 1/8-inch hole diameter, or D = 0.125 inches.
The area of that one hole is A = (π) (D²) / 4 = 0.01227 square inches (approximately). Then twenty of these holes would occupy an area of (20) (0.01227) = 0.2454 square inches (approximately). So, the area of a round hole with that area would have a squared diameter D² = (4) (0.2454) / (π) = 0.3125. Taking the square root of this, D = 0.5590.
If we work the problem backwards, from an assumed solution of 2 ½ inches diameter, then the area of that hole would be (π) (2.5)² / 4 = 4.9087 square inches. Divide this by 20 to obtain the area equivalent of twenty holes: 0.2454 square inches per hole. Solve for the squared diameter of each hole, D² = (4) (0.2454) / (π) = 0.3125. Taking the square root, obtain 0.5590 inches diameter.
Clearly, this does not equal 0.125 inches diameter as originally specified, so the assumed solution of 2 ½ inches diameter is wrong.