Sumvia

Business Math Calculators

Sumvia is a small publication about the arithmetic of running a business — margins, markup, break-even points, returns, growth rates, tax, commission, and pay. Each topic gets one focused calculator page: the formula, an interactive tool, and worked examples that are generated at build time by the same tested code that powers the tool, so the explanation can never drift from the calculation. Everything runs in your browser; nothing you type is sent to a server.

Pricing and margins

What to charge, what you keep, and what a change in price really does.

  • Profit Margin — gross margin from cost and price, with the equivalent markup alongside
  • Markup — set a selling price from cost and a target markup
  • Profit — net profit and margin from revenue and total costs
  • Price Change — the true percentage behind a rise, a drop, or a rollback
  • Discount — sale prices, stacked discounts, and what the seller gives up
  • Unit Price — compare pack sizes on cost per unit instead of shelf price
  • VAT — add or strip value-added tax at any rate, both directions

Returns and growth

Whether a spend paid off, and how fast something actually grew.

  • ROI — return on investment, with the annualized rate when you give it a duration
  • CAGR — compound annual growth rate — the honest average over multiple years
  • Break-Even — the units or revenue where a product stops losing money
  • SaaS Metrics — LTV, LTV:CAC, payback months, and honest annual churn from four inputs
  • Simple Interest — I = P·r·t for loans and deposits that do not compound

Pay and sales

What a sale pays, what a job pays, and what the hours add up to.

  • Commission — flat and tiered commission on a sale or a period of sales
  • Salary — convert between hourly, weekly, monthly, and annual pay
  • Overtime — regular and overtime earnings at 1.5×, 2×, or a custom multiplier
  • Time Card — weekly hours from clock-in and clock-out times, breaks deducted

The confusion these pages exist to prevent

Most business-math mistakes are not arithmetic slips — they are the right formula applied to the wrong base. The two most expensive examples, with the numbers computed by our engine:

Margin is not markup. Buy at 60 and sell at 100 and you have made 40 of profit — but that one sale is a 66.67% markup (profit over cost) and only a 40% margin (profit over price). Quote a supplier the wrong one and you misprice by a wide margin, literally. The profit margin calculator always shows both percentages side by side so the two cannot be confused.

Average growth is not compound growth. A value that doubles in five years grew 100% in total, which feels like 20% a year — but the honest compound rate is 14.87% per year, because each year grows on top of the last. Dividing total growth by the number of years overstates every multi-year trend you will ever quote. The CAGR calculator computes the compound rate and shows the gap.

Start with the guides

If you want the reasoning before the tool, the guides hub explains each corner of the arithmetic in plain language, with every figure computed the same way the calculators compute theirs. The one that underpins half of this cluster is margin vs. markup arithmetic — it works through why a single profit figure reads as two different percentages and gives the exact identity that converts one to the other. The full set:

Inside the pricing and margins tools

This cluster turns on a single distinction the engine never lets you lose: margin is measured on revenue, markup on cost. The same 40 profit on a 60 cost sold at 100 is a 66.67% markup but a 40% margin, because the first divides the profit by the cost and the second by the larger selling price. The profit margin page starts from a known cost and price and prints both; the markup page runs it the other way — give it a cost and a target markup and it sets the price. A fictional 80 cost marked up 25% sets the price at 100, a profit of 20, which is a 20% margin — proof that a markup percentage is always the larger of the two for the same profitable sale.

The profit tool steps up from one sale to a whole line: revenue minus total costs, with per-unit figures when you supply a quantity. On a fictional 5,000 of revenue against 3,200 of costs over 400 units, net profit is 1,800 — a 36% margin — and each unit carries 12.5 of revenue, 8 of cost, and 4.5 of profit. The price change tool reports a move in both directions, because they are not mirror images: raising a fictional 100 to 120 is a 20% increase, yet the reverse trip back down is a 16.67% decrease — the two use different baselines.

The discount tool multiplies stacked reductions rather than adding them: a fictional 50 price cut 20% and then a further 10% is not off 30% but 28%, a sale price of 36 and 14 saved, because the second cut lands on the already-reduced price. Unit price puts two pack sizes on the same cost-per-unit footing so the cheaper shelf price and the cheaper unit price can part ways. And VAT runs in both directions — rates differ by country and category, so you supply yours and nothing is assumed. Adding a fictional 15% to a net 200 gives a gross of 230 with 30 of tax; backing the same rate out of a gross 230 returns a net of 200, by dividing rather than subtracting the percentage.

Inside the returns and growth tools

Here the recurring trap is treating a multi-year total as if it were an annual rate. The ROI tool reports the total return — a fictional 15,000 returned on a 10,000 outlay is a net 5,000 and a 50% return — and, when you give it a duration, the annualized rate that compounds to the same place: spread over three years, that 50% total is only 14.47% a year. The CAGR tool is that annualized idea on its own: a value that doubles over five years grew 100% in total but compounds at 14.87% a year, not the 20% a naive division would suggest. Both take a root of a growth ratio, which is exactly why they disagree with the arithmetic most people reach for first.

Break-even asks a different question — how many units cover the fixed costs before a product stops losing money. Each sale contributes its price minus its variable cost; on fictional fixed costs of 12,000 with a price of 50 and a variable cost of 20, the contribution margin is 30 a unit (a 60% contribution-margin ratio), so break-even sits at 400 units, or 20,000 of revenue. When the price does not exceed the variable cost, the tool reports that break-even is never reached rather than returning an infinity. The SaaS metrics page applies the same contribution thinking to subscriptions, turning four inputs into lifetime value, the LTV:CAC ratio, and payback months. Simple interest stays deliberately finance-lite — I = P·r·t, where interest never earns interest: a fictional 2,000 at 5% for three years earns 300 of interest for a total of 2,300. Most real accounts compound, and the page says so plainly rather than quietly switching formulas.

Inside the pay and sales tools

Every amount in this cluster is gross — pre-tax — and the pages state it rather than implying a take-home figure. The commission tool takes a sale amount and a rate, and optionally splits the result between an agent and a broker. A fictional 8,000 sale at 5% is a 400 commission, leaving 7,600 to the seller; a 60% agent share of that commission is 240 to the agent and 160 to the broker — an effective 3% and 2% of the sale respectively, since the share is of the commission, not of the sale.

The salary tool moves one gross figure across every pay period on a standard 52-week year. A fictional 25 an hour at 40 hours a week is 1,000 a week, 4,333.33 a month, and 52,000 a year — all the same annual sum, re-sliced. Overtime pays regular hours at the base rate and overtime hours at a multiplier that defaults to 1.5× — a convention, not a legal rule, and always editable, with an optional second tier for arrangements like double time. A fictional 40 regular hours at 20 an hour is 800; six overtime hours at the 30 overtime rate add 180, for 980 across 46 hours. The time card tool works the hours out in the first place, totalling a week from clock-in and clock-out times with breaks deducted.

How every page here is built

Each calculation engine is a small, typed function with automated tests over standard inputs, decimals, negatives, zeros, and edge cases. The worked examples in the prose and FAQs are computed by that same engine when the site is built — a wrong figure fails the build rather than getting published. The full discipline is documented on the methodology page, and about Sumvia explains who is behind it. Results here are aids, not advice: verify anything consequential before you act on it.

Frequently asked questions

Does anything I type get sent to a server?

No. Every calculator runs entirely in your browser; the values you enter are processed on your device and never transmitted. There is no account, no login, and nothing stored — the static page you downloaded is the whole product.

Where do the worked examples and FAQ numbers come from?

Every figure on these pages is computed at build time by the same tested engine that powers the interactive tool, using clearly labelled fictional inputs. If an engine change ever made a published number wrong, the site would fail to build rather than ship copy that disagreed with the code.

Do the calculators work in any currency?

Yes — the engines are unit-agnostic. They compute on the numbers you enter whatever the currency, and they apply no currency conversion and no jurisdiction-specific rounding of money. A result is a cleanly rounded presentation of a more precise internal value; the details are on the methodology page.

How are results rounded and displayed?

Internally the math keeps full double-precision; before display each value is normalized to strip binary artifacts, then shown at a fixed decimal cap. A tiny nonzero result that would otherwise round to "0" falls back to significant digits so it stays visible. The exact precision and display figures live in methodology's rounding policy, and the rounding guide walks the 0.1 + 0.2 and one-third cases through in full.

Why does removing VAT divide instead of subtracting the rate?

Because the tax was charged on the net price, not the gross. Backing a fictional 15% out of a gross 230 returns a net of 200 with 30 of tax — reached by dividing by one plus the rate, not by taking 15% off the gross, which lands on a different and wrong figure. Adding and removing VAT are covered in the VAT guide.

Is annualized ROI the same as CAGR?

They use the same per-year-growth arithmetic — both take a root of a ratio rather than dividing by the number of years. ROI also reports the total return (50% over three years in the example, which annualizes to 14.47%), and only annualizes when you give it a duration, whereas CAGR is always a yearly rate. The ROI and CAGR guide sets them side by side.

What happens if I enter an impossible value?

The engine reports it rather than guessing. A percentage of a zero base, a price at or below the variable cost on the break-even tool, or a churn of zero on the SaaS tool returns a clear, human-readable message instead of an infinity or an error code — the calculator says what is undefined and why.

Do these pages give financial, tax, or pricing advice?

No. They compute the stated formula on the numbers you supply and stop there — no "healthy margin", "good ROI", or "price it at" verdicts. The arithmetic is an aid; choosing the right formula for a real decision, and verifying anything consequential, stays with you.

Beyond the business cluster

Sumvia is also the studio behind a small network of single-purpose sites, each doing one job on its own domain: