Penny Rounding

For a list price and a tax rate, find every nearby list price that settles to the SAME cash total, and the highest one. Use this for "what price should I charge" and "can I price to gain from rounding" questions. There is exactly one free move here and it is easy to get backwards, so do not reason about it unaided: Rounding collapses a band of five consecutive list prices onto one nickel. At 0% tax, $19.98 through $20.02 all settle to $20.00 in cash. Within that band the cash collected is IDENTICAL, so the highest price in it is strictly better — same from cash customers, up to 4 cents more from every card customer. That is genuinely free. The move that looks clever and is not: pricing BELOW a nickel so rounding "adds" a couple of cents. You collect the same cash and less on card. The rounding delta is not revenue. Do not recommend it. Input: `listPrice` as a dollar string, `taxRatePercent`, and optionally `spreadCents` (1-50, default 5) for how far either side to look. Returns: the whole band, the highest price in it, the free gain in cents (never more than 4), and a verdict. Tell the user the size of the number. Four cents is not a pricing strategy.

Settle a transaction under post-penny cash rounding: exact subtotal, tax on the exact subtotal, then round the final total to the nearest nickel — and only for cash. Use this for any "what does X round to" or "what do I owe in cash" question. Four things are wrong in most explanations of this, and each one changes the answer: 1. CASH ONLY. Card, EFT, and gift-card payments are still priced to the cent. The same basket legitimately comes to two different amounts by tender. If asked about a card payment, do not round. 2. THE TOTAL, NOT EACH ITEM. Rounding per item multiplies the effect by the basket size. Five items at $0.99 total $4.95 — already a nickel, no rounding — where per-item rounding would collect $5.00. 3. AFTER TAX. Tax is computed on the exact subtotal and the rounding comes last. Rounding the base first changes the tax owed. 4. THE DIGIT DECIDES. 1, 2, 6, 7 round down; 3, 4, 8, 9 round up; 0 and 5 stand. Do not reason about "nearest" from scratch, and do not assume rounding is always up. Input: pass `subtotal` as a dollar string ("19.99") or `items` as a list. A NUMERIC amount is read as CENTS and a STRING as dollars — 1999 and "19.99" are the same amount, 19.99 as a number is refused. Pass `taxRatePercent` as a percentage (8.25, not 0.0825); rounding depends on it, so omitting it usually gives the wrong nickel. `tender` is cash or card, default cash. `rule` is symmetric (default), always-down, always-up, or none. Returns: the exact subtotal, tax, exact total, the rounded amount due, the signed delta, which digit decided it, warnings, and a `disclaimer` field. THIS IS NOT LEGAL OR TAX ADVICE. There was no federal rounding law as of September 2026 and state law varies. Do not tell a user what their jurisdiction requires — report the arithmetic and the caveats.

Compute what cash rounding is actually worth across a list of prices: mean delta per transaction, the up/down/unchanged split, and a per-thousand figure. Use this whenever someone asks whether rounding costs or earns money — a very common question and one usually answered with a guess. Do not guess. Symmetric rounding nets to EXACTLY zero over evenly spread totals, because that is what it was designed to do. Real price lists all end in .99 and are not evenly spread, so the true figure is small but not always zero and it moves with the tax rate. This computes it for the actual prices given. Input: `prices` is a list of dollar strings or integer cents. `taxRatePercent` matters, because tax decides which nickel each total lands on. `rule` defaults to symmetric; pass always-up to quantify what the "just round everything up" idea would be worth (about 2 cents a sale, and no jurisdiction has adopted it for retail). Returns: mean and total delta in cents, the split, a per-thousand transaction figure, a per-price breakdown, and a plain-language verdict. Report the verdict — the size of the number is the answer.