# Contribution margin and break-even > The unit-economics formulas as pure math: CM = P(1 − f) − T − V, CM%, break-even ⌈F ÷ c⌉, margin of safety, step costs, the volume a discount needs, gross-up and ROI. - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: Unit economics, the bridge - Lesson 31/36, ~29 min - Source: https://productovi.com/courses/math-basics/contribution-and-break-even ## What one lesson leaves **Alex Tutor:** Back on Klasna's product team for a two-lesson bridge: we put the math of this course together into unit economics. The finance course reads these numbers as business; here we make sure the formulas hold no surprises. **Alex Tutor:** Take a programming lesson: price 800 UAH, a 2.5% payment fee, 520 to the tutor, 30 for video and AI. **Contribution margin** is what's left after the costs that come with each lesson. $$ CM = P \cdot (1 - f) - T - V \qquad CM\% = \frac{CM}{P} $$ *P: price; f: the fee as a share of price; T: tutor payout; V: video and AI per lesson.* **Sofia:** Brackets first: 800 × 0.975 = 780. Then 780 − 520 − 30 = 230 UAH. And CM% = 230 ÷ 800 = 28.75%. ![A programming lesson priced at 800 UAH loses 20 to the fee, 520 to the tutor and 30 to video and AI, leaving 230 of contribution margin, 28.75% of the price.](https://productovi.com/courses/math-basics/diagrams/m8-l1-cm-bar.png) *The price splits into the costs each lesson brings and what is left.* **Sofia:** So CM% is the share of each hryvnia of price that's left over to pay for everything fixed? **Alex Tutor:** Exactly: 28.75% of every hryvnia. Keep both numbers: hryvnias per lesson for break-even, the share for discounts. > **Exercise:** An English lesson costs 700 UAH with a 2.5% fee, 450 to the tutor and 30 for video and AI. What is its contribution margin, in UAH? **Alex Tutor:** Which costs belong in that subtraction, and which don't? > **Exercise:** Which of these Klasna costs is NOT subtracted when you compute the contribution margin of one programming lesson? ## Break-even and margin of safety **Alex Tutor:** The programming line carries 700,000 UAH of fixed costs a month. **Break-even** is the number of lessons whose contribution covers them: 700,000 ÷ 230 ≈ 3,043.5, rounded **up** to 3,044. $$ Q_{BE} = \left\lceil \frac{F}{c} \right\rceil \qquad MoS = \frac{Q - Q_{BE}}{Q} $$ *F: fixed costs; c: contribution per unit; ⌈ ⌉: round up; Q: actual volume; MoS: margin of safety.* **Sofia:** With 3,500 lessons the margin of safety is (3,500 − 3,044) ÷ 3,500 ≈ 13%. And in a month below break-even it would be negative? **Alex Tutor:** Yes, and that's fine mathematically: a negative MoS says how far you'd have to grow to reach the line. A **step cost** moves the line: one more coordinator at 38,000 makes F = 738,000 and break-even 3,209. > **Exercise:** Break-even is 3,044 lessons. In July the programming line delivered 2,600. What is the margin of safety, in percent? Two decimals; use a minus sign if it is negative. **Sofia:** Why always round up? 3,043 is closer to 3,043.5 than 3,044 is. **Alex Tutor:** Because 3,043 lessons leave 3,043 × 230 = 699,890: still 110 short. Rounding down would call a losing month break-even. **Alex Tutor:** Same rule as in the first module: counts that must cover something, like lessons or coordinators, round up. **Sofia:** And the margin of safety I can keep as a decimal, since it's a share, not a count. > **Exercise:** Fixed costs are 520,000 UAH a month and each lesson leaves 184.50 UAH. How many lessons a month are needed to break even? **Alex Tutor:** Now the step cost from before, as a number. > **Exercise:** The programming line delivers 3,500 lessons a month, leaves 230 UAH per lesson and has fixed costs of 700,000 UAH. Klasna hires one more coordinator at 38,000 UAH a month. What is the margin of safety after the hire, in percent? Two decimals. ## What a discount costs in volume **Alex Tutor:** Now a discount d, as a share of price. With a fee, each hryvnia of price only leaves (1 − f) of it, so the margin loses d × (1 − f) of the price. For 10% off: 0.0975. $$ \Delta V = \frac{d}{m - d} \qquad V_{new} = V \cdot \frac{m}{m - d} $$ *m: CM% before the discount; d: the share of price the discount takes out of margin, d × (1 − f) when there is a fee; ΔV: the extra volume needed to keep total contribution.* **Sofia:** 0.0975 ÷ (0.2875 − 0.0975) = 0.0975 ÷ 0.19 ≈ 51%. Half as many lessons again, for 10% off. **Alex Tutor:** Check it directly: 720 × 0.975 − 550 = 152 per lesson, and 230 ÷ 152 ≈ 1.513. The denominator m − d shrinks fast, so 20% off needs about +211%. **Sofia:** So the thinner the margin to start with, the faster the required volume runs away. ![With a 28.75% contribution margin, a 5% discount needs 20% more lessons to keep the same contribution, 10% needs 51%, 15% needs 104% and 20% needs 211%.](https://productovi.com/courses/math-basics/diagrams/m8-l1-discount-volume.png) *As the discount approaches the margin, the volume needed explodes.* ![A cat stands in the middle of a level seesaw, with stacks of coins on one end and a heavy weight on the other.](https://productovi.com/courses/math-basics/inline/m8-l1-i1.webp) *Break-even is the point where the two sides balance.* > **Exercise:** The programming line sells 3,500 lessons at CM% m = 0.2875. A 10% discount takes d = 0.0975 of price out of margin. How many lessons are needed to keep the same total contribution? Round to a whole lesson. ## Gross-up and ROI **Alex Tutor:** Two more pieces. When a fee is taken from the price, you **gross up** by dividing: for a tutor on a marketplace to net 600 after a 20% commission, the price is 600 ÷ 0.8 = 750, not 600 × 1.2 = 720. **Sofia:** Because 20% is taken from 750, not from 600: 720 × 0.8 is only 576. **Alex Tutor:** And **ROI** is a relative change: (benefit − cost) ÷ cost. A promotion costing 50,000 that brings 80,000 of contribution has ROI = 30,000 ÷ 50,000 = 60%. Use contribution, not revenue, as the benefit. > **Exercise:** A payment provider takes 3% of the price. Klasna wants 970 UAH to land after the fee. What price should it charge? **Alex Tutor:** And ROI, with the benefit measured properly. > **Exercise:** A promotion on programming lessons cost 50,000 UAH and brought 160,000 UAH of extra revenue. Programming's contribution margin is 28.75% of price. What is the promotion's ROI? ## Your turn: 12% off programming **Alex Tutor:** Leo wants 12% off programming lessons in September: "It's only 12%, so we need maybe 12% more lessons to make up for it." **Sofia:** The formula says otherwise, and by a lot. **Alex Tutor:** Then show him the number and why it's so much bigger than 12%. > **Exercise:** Reply to Leo, Klasna's CEO, in 3–4 sentences: how many more programming lessons does a 12% discount need to keep the same total contribution (m = 0.2875, 2.5% fee), and why is it not 12%? ## Where you will meet this > **In short** > > - **CM = P(1 − f) − T − V**; **CM% = CM ÷ P**. > - **Break-even = ⌈F ÷ c⌉**, always rounded up. **Margin of safety = (Q − Q_BE) ÷ Q**; negative below the line. A step cost raises F and moves the line. > - A discount needs **ΔV = d ÷ (m − d)** more volume (d = discount × (1 − f) with a fee); new volume **V × m ÷ (m − d)**. > - **Gross-up:** to keep X after a share f is taken, charge X ÷ (1 − f). **ROI = (B − C) ÷ C**, with contribution as B. > **These lessons use it** > > - [Contribution margin: what one lesson leaves](/courses/finance-for-product/contribution-margin): CM and CM% for each package. > - [Fixed, variable and step costs](/courses/finance-for-product/fixed-variable-and-step-costs): which costs go into F. > - [Break-even and margin of safety](/courses/finance-for-product/break-even): the line for each month and what moves it. > - [Price vs volume: how many customers can a discount lose?](/courses/finance-for-product/price-vs-volume): the business reading of d ÷ (m − d). > - [Scenarios and sensitivity](/courses/finance-for-product/scenarios-and-sensitivity): a break-even price, grossed up by ÷ 0.975. > - [CAC, payback and ROI](/courses/product-analytics/cac-payback-roi): ROI on gross profit at a stated month. **Alex Tutor:** Next time, the last bridge: flows of customers and cash over time. - Previous lesson: [Other curves and power](https://productovi.com/courses/math-basics/other-curves-and-power.md) - Next lesson: [Flows: customers and cash](https://productovi.com/courses/math-basics/flows-customers-and-cash.md) --- Productovi Learning. 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