What is the balancing (constant-difference) trick for subtraction, and why is it exact?

Add the same amount to both numbers to make the one you're subtracting a round number — the difference never changes.

Because (a + k) − (b + k) = a − b, shifting both ends by the same k leaves the gap identical:

4.32 − 1.95 → add 0.05 to both → 4.37 − 2.00 = 2.37.

Check: 4.32 − 1.95 = 2.37. ✓ Pick k to round the subtrahend (the number being taken away) to something clean, and the subtraction becomes trivial.