Contribution margin and break-even

Contribution margin (CM) is what one unit leaves after the costs that come with it: CM = P(1 − f) − T − V, where P is the price, f the payment fee as a share of price, T the tutor payout and V video and AI per lesson. CM% = CM ÷ P is the same thing as a share of price.

Work the brackets first. A programming lesson at Klasna costs 800 UAH: 800 × 0.975 = 780 after a 2.5% fee, then 780 − 520 − 30 = 230 UAH. CM% = 230 ÷ 800 = 28.75%, so 28.75 kopecks of every hryvnia of price is left over.

Keep both forms. Hryvnias per lesson answer "how many lessons cover the fixed costs?"; the share answers "what does a discount do to the margin?". The fee multiplies the price because it is a share of it; subtracting 2.5 as if it were hryvnias is the classic slip.

Unit economics splits costs by one question: does this cost grow with each extra lesson? If yes, it is a variable cost and goes into the contribution margin: the payment fee, the tutor payout, video and AI. If no, it is a fixed cost F for the period: coordinators' salaries, the platform team, office rent. One more lesson doesn't change them.

The split matters because the two kinds of cost are covered in different places. Each lesson pays its own variable costs at once; what's left, the 230 UAH of a programming lesson, goes into a common pot that has to cover F. Subtract a salary inside CM and you count it once per lesson, thousands of times a month.

Some costs sit in between. A coordinator is fixed until the volume needs one more; then F jumps by a whole salary. That is a step cost, and it moves the break-even line rather than the margin.

Break-even is the volume whose total contribution covers the fixed costs: Q_BE = ⌈F ÷ c⌉, with c the contribution per unit. It is where the line profit = c × Q − F from The linear function crosses zero. Klasna's programming line carries F = 700,000 UAH a month and leaves c = 230 per lesson: 700,000 ÷ 230 ≈ 3,043.5, so 3,044 lessons.

The margin of safety says how far the actual volume Q sits above that line, as a share of Q: MoS = (Q − Q_BE) ÷ Q. With 3,500 lessons it is (3,500 − 3,044) ÷ 3,500 ≈ 13%: volume could fall by about 13% before the line makes a loss.

Below the line the same formula turns negative, and that is fine. In July, with 2,600 lessons, MoS = (2,600 − 3,044) ÷ 2,600 ≈ −17.08%: the month was 444 lessons short. The sign tells you which side of the line you're on; the size, how far.

F ÷ c rarely comes out whole, and a break-even volume is a count of lessons that must cover something. Counts like that round up, the rule from Rounding: at the end, and sometimes up.

Check it on the programming line: 700,000 ÷ 230 ≈ 3,043.5. Ordinary rounding could give 3,043, but 3,043 × 230 = 699,890 UAH, still 110 short of the fixed costs. Only 3,044 lessons (700,120) cover them. Rounding down would label a losing month as break-even.

The same holds for any F and c: with fixed costs of 520,000 and 184.50 per lesson, 520,000 ÷ 184.50 ≈ 2,818.4, so 2,819 lessons; 2,818 would leave 519,921, 79 UAH short.

The margin of safety is different. It is a share, not a count, so keep it as a decimal or a percent with sensible rounding at the end, like 13% or −17.08%.

A step cost is fixed over a range of volume and then jumps by a whole unit: one more coordinator, one more server, one more room. Between the steps it behaves like F; at a step, F rises by the full amount at once.

The contribution per lesson doesn't change, so the step shows up only in break-even. At Klasna, a coordinator at 38,000 UAH a month lifts the programming line's F from 700,000 to 738,000. Break-even moves from 3,044 to ⌈738,000 ÷ 230⌉ = ⌈3,208.7⌉ = 3,209 lessons: the new salary needs 38,000 ÷ 230 ≈ 165 extra lessons.

Two habits help. Before a hire, compute how many extra lessons pay for it and compare that with the volume you expect. And re-check the margin of safety after the step: with 3,500 lessons it falls from about 13% to (3,500 − 3,209) ÷ 3,500 ≈ 8.31%.

A discount comes almost entirely out of the margin, not out of the price as a whole. If the contribution margin is m (as CM%) and the discount takes d of the price out of it, each lesson now leaves m − d instead of m. To keep the same total contribution you need ΔV = d ÷ (m − d) more volume, or V × m ÷ (m − d) in all.

With a fee, the discount also shrinks the fee, so it takes d = discount × (1 − f) out of margin: 10% off at Klasna is 0.10 × 0.975 = 0.0975. Programming has m = 0.2875, so ΔV = 0.0975 ÷ 0.19 ≈ 51%: 3,500 lessons must become 3,500 × 0.2875 ÷ 0.19 ≈ 5,296. Check it directly: 720 × 0.975 − 550 = 152 per lesson, and 5,296 × 152 = 804,992 ≈ 3,500 × 230.

The denominator m − d is the trap. As d approaches m, it heads to zero and the volume explodes: 20% off needs about +211%.

When a fee is a share of the price, the amount that lands is price × (1 − f). To land a target X, solve for the price, as in Rearranging a formula: price = X ÷ (1 − f). That is the gross-up.

For Klasna to keep 970 UAH after a 3% payment fee: 970 ÷ 0.97 = 1,000. Check: 3% of 1,000 is 30, leaving 970. For a tutor to net 600 after a 20% marketplace commission: 600 ÷ 0.8 = 750, since 20% of 750 is 150.

The tempting shortcut is to multiply by 1 + f: 970 × 1.03 = 999.10, or 600 × 1.2 = 720. It always falls short, because the fee is taken from the bigger, grossed-up price, not from X: 720 × 0.8 is only 576. The gap is small at 3% and large at 20%, so always check by taking the fee off your answer.

ROI is a relative change, the idea from Percent vs percentage points: ROI = (B − C) ÷ C, the gain over the cost as a share of the cost. The question is what counts as the benefit B. Use the contribution the spending brought, not the revenue: revenue still has to pay the tutor, the fee and the video and AI.

A promotion costing 50,000 UAH that brings 80,000 of contribution has ROI = 30,000 ÷ 50,000 = 60%. One that brings 160,000 of programming revenue brings only 160,000 × 0.2875 = 46,000 of contribution: ROI = (46,000 − 50,000) ÷ 50,000 = −8%. On revenue it would look like +220%.

How it fits together: contribution per lesson is the building block: divided into fixed costs it gives break-even, as a share it prices a discount, grossed up it sets a price, and set against spending it gives ROI. The next lesson adds time: flows of customers and cash.