Guides

Sumvia’s guides are the companion reading to its calculators. Where a calculator takes your numbers and returns an answer, a guide explains the arithmetic behind that answer: what the engine is actually computing, why it uses the conventions it does, where it deliberately stops, and the misreadings that turn a correct calculation into the wrong one. They are the engine’s definitions and conventions made legible — the same small, tested functions that power the tools, described in words rather than run on your inputs.

Two things hold across every guide. First, each one is anchored to specific calculators and quotes the formula those calculators use, so the prose and the code cannot drift apart. Second, every figure printed in a guide is computed at build time by that same engine, on fictional inputs labelled as examples — not a single number is typed in by hand. If an example ever disagreed with the code, the site would fail to build. How that engine is implemented, tested, and rounded is set out in full on the methodology page.

Margin vs. markup arithmetic

Margin and markup divide the same profit by different bases — margin by the selling price, markup by the cost — so a single sale reports two different percentages. This guide works through both formulas, shows the exact algebraic identity that converts one into the other in each direction, and names the base errors that quietly cost money, such as quoting a markup as though it were a margin. It draws on the profit margin and markup calculators.

Break-even quantity and contribution margin

Break-even is a single division — fixed costs over the contribution margin — yet nearly every mistake lives in that denominator rather than in the division. The guide explains what the contribution margin is, keeps it distinct from a profit margin, traces how sharply the break-even point moves when the price changes, and marks where the formula stops matching a business with more than one product or stepwise fixed costs. It is built on the break-even calculator.

ROI and CAGR explained

Return on investment and compound annual growth rate measure the same growth from two different starting points: money out against money in for ROI, a value seen at two dates for CAGR. This guide shows why a plain total return and a per-year rate differ once compounding is accounted for, and why annualising an ROI turns out to be the identical arithmetic as a CAGR. It draws on the ROI and CAGR calculators.

VAT added and removed

Adding VAT to a net price is one multiplication; recovering the net hidden inside a gross total is a division — and treating that division as a subtraction of the rate is the mistake this guide heads off. It shows the three figures the engine returns, why the subtraction shortcut lands short by an amount that widens as the rate grows, and why adding then removing at one rate returns the original price. Every rate used is a labelled fictional example, never a country’s, and the tool behind it is the VAT calculator.

Commission and discount arithmetic

A commission is one rate on one base, but an agent’s share is a percentage of the commission rather than of the sale, and a tiered schedule is composed slice by slice rather than entered as a single blended rate. The guide covers both, explains why two stacked discounts multiply instead of adding, and shows how the engine labels the direction of a price change — and why a cut and an equal-looking rise do not cancel. It draws on the commission, discount, and price change calculators.

Salary conversions and overtime pay

The salary converter never jumps straight from one pay period to another; it annualises the amount you enter through a single 52-week pivot and divides that annual total back into each period, which is why a month is not four weeks and bi-weekly is not semi-monthly. The overtime engine treats its multiplier as an editable convention, scaling only the overtime hours and leaving regular pay untouched. Both deal strictly in gross, pre-tax amounts, exactly as the engine states, and the tools are the salary and overtime calculators.

How Sumvia rounds and displays numbers

One rounding policy stands behind every calculator, and this guide demonstrates it working: full IEEE-754 precision internally, normalisation to twelve significant digits to strip binary floating-point artifacts, and at most six decimal places on screen — with a fallback to six significant digits so a genuinely tiny result is never shown as zero. It also covers the supported input range and why a figure copied off the screen into further hand calculations carries the rounded value, not the exact one the engine kept.

The calculators the guides refer to

Every guide points back to the tools it describes. The sixteen calculators the series draws on:

Where to go next

Start from a calculator if you already have numbers in front of you, or from a guide above if you want the arithmetic first; each links back to the other. The methodology page documents how every engine is built, tested, and rounded, and the about page explains what Sumvia is and why it narrowed to business math. If a figure ever looks wrong, the methodology page sets out how to report it.

Every figure in these guides is computed at build time by the same tested engine that powers the calculators, on fictional example inputs, and the tools themselves run entirely in your browser. Each guide implements the general mathematical definition of its topic; your jurisdiction’s or contract’s conventions may differ, and nothing here is legal, tax, accounting, or investment advice. Implementation and rounding details are on the methodology page.