Scale chain calculator

Each step converts its percentage into an exact factor, and the factors multiply:

k_total = k₁ × k₂ × … × kₙ

Because every step is kept as a fraction, a chain of ten resizes has exactly the same precision as one. This is the concrete benefit of exact arithmetic: rounding each step to four decimals and multiplying would drift, and the drift grows with every copy.

A three-step chain, worked end to end

A 1:100 plan is reduced to 80% for a smaller sheet, enlarged to 125% by the next office, then put through a 90% fit-to-page print. Three plausible steps, and the paperwork says the two middle ones cancel:

k_total = 0.80 × 1.25 × 0.90 = 0.90 = 90%
step scales: 1:100 → 1:125 → 1:100 → 1:111.11

80% and 125% are exact reciprocals, so they do cancel — the residue is entirely the 90% step. The drawing is now at 1:111.11, a scale no rule reads, and a wall measured with a 1:100 scale rule reads 11% short.

The correction, and the mistake that undoes it

The correction is 1 ÷ k_total = 1 ÷ 0.9 = 1.111… = 111.111…%, and it applies to the current copy:

current copy × 111.111…% → 0.90 × 1.1111… = 1.000 ✓ (1:100 restored)
original × 111.111…% → 1.00 × 1.1111… = 1.111 ✗ (1:90 — further away)

Note also that the correction is not 110%: subtracting the 10% you appear to have lost leaves you at 0.9 × 1.10 = 0.99, still 1% out and still off-standard. Reversing a chain is a reciprocal, never a subtraction — which is why the correction is reported here as an exact fraction (10/9) as well as a decimal.