# Rearranging a formula: solve for what you need > Do the same to both sides and undo in reverse order: break-even Q = F/c, runway, the regression intercept, sample size, a fraction inside a denominator, and why a gross-up divides by (1 − f). - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: The language of formulas - Lesson 7/36, ~25 min - Source: https://productovi.com/courses/math-basics/rearranging-a-formula ## Same formula, different question **Alex Tutor:** Back on Klasna's product team, where a formula from the finance course has raised a fair question. **Sofia:** The finance course writes break-even as Q = F/c, but I first learned it as profit = c·Q − F. Are those two different formulas? **Alex Tutor:** One formula, asked two questions. The second one answers "how much profit at this volume?", the first "what volume gives zero profit?". Turning one into the other is **rearranging**. **Alex Tutor:** One rule does all the work: an equation is a balance, so whatever you do to one side, do to the other. Undo the operations around the unknown in reverse order, outermost first. ![A cat sits behind a level balance scale with an orange block in each paw, ready to add one to each pan.](https://productovi.com/courses/math-basics/inline/m2-l3-i1.webp) *Whatever goes on one side goes on the other.* **Alex Tutor:** First, the very first move. Which one keeps the balance? > **Exercise:** You want to solve 174·Q − 870,000 = 0 for Q. Which first step is correct? **Alex Tutor:** Klasna's fixed costs are F = 870,000 UAH a month and each lesson leaves c = 174 UAH. Set profit to zero: 174·Q − 870,000 = 0. Add 870,000 to both sides, then divide both by 174: Q = 870,000 ÷ 174 = **5,000 lessons**. ![Three steps: c·Q − F = 0; add F to both sides to get c·Q = F; divide both sides by c to get Q = F/c, here 870,000 ÷ 174 = 5,000 lessons.](https://productovi.com/courses/math-basics/diagrams/m2-l3-solve-steps.png) *Each step does the same thing to both sides.* > **Exercise:** A new subject has fixed costs of 612,000 UAH a month and leaves 180 UAH per lesson. Solve 180·Q − 612,000 = 0 for Q: how many lessons a month break even? ## Runway, intercepts and sample sizes **Alex Tutor:** The bootstrapped course gives personal runway as R = (S − Z)/(E − I): savings S minus a cushion Z you never touch, divided by monthly expenses E minus income I. Suppose you want the opposite: how much to save for 12 months. **Alex Tutor:** Multiply both sides by (E − I), then add Z: S = Z + R·(E − I). With E = 45,000, I = 15,000 and Z = 60,000: S = 60,000 + 12 × 30,000 = **420,000 UAH**. **Sofia:** Same trick for regression? The line ŷ = b₀ + b₁x passes through the point of means, so ȳ = b₀ + b₁x̄... **Alex Tutor:** ...and subtracting b₁x̄ from both sides gives **b₀ = ȳ − b₁x̄**. Square roots work the same way: SE = √(p(1 − p)/n), so square both sides and swap SE and n: **n = p(1 − p)/SE²**. That's how a test's sample size is found. > **Exercise:** You want 9 months of runway: R = 9. Monthly expenses are E = 52,000 UAH, income I = 20,000 and the untouchable cushion Z = 40,000. Using S = Z + R·(E − I), how much do you need to have saved, in UAH? **Alex Tutor:** Now the regression intercept from a moment ago, with numbers. > **Exercise:** A regression of monthly lessons on ad spend has slope b₁ = 25 lessons per thousand UAH. Average spend is x̄ = 60 thousand UAH and average lessons ȳ = 9,000. The line passes through (x̄, ȳ). What is the intercept b₀ = ȳ − b₁x̄? > **Exercise:** A conversion rate is about p = 0.10, and you want its standard error to be SE = 0.006. Using n = p(1 − p)/SE², how many users do you need? ## A fraction in the denominator **Alex Tutor:** Dividing by a share is where intuition slips. An investor pays X = 3 million UAH for a share s = 15% of a company. The whole company is then worth post-money = X/s = 3 ÷ 0.15 = **20 million**. **Sofia:** Dividing by 0.15 made the number bigger. **Alex Tutor:** Dividing by a fraction is multiplying by its flip: ÷ 0.15 is × 6.67. Same with a loan where you pay only interest: 3,000 UAH a month at r = 2% a month means a balance of 3,000 ÷ 0.02 = 150,000. > **Exercise:** An investor pays 4 million UAH for 16% of a company. What is the post-money valuation, X/s? **Alex Tutor:** And the interest-only loan from before, the other way round. > **Exercise:** On an interest-only loan you pay 4,500 UAH a month at 1.5% a month. How large is the balance? ## The gross-up trap **Alex Tutor:** A partner platform keeps 20% of each lesson price. A tutor wants 400 UAH in hand. What price should the lesson have? **Sofia:** 400 plus 20%, so 480? **Alex Tutor:** Check it: the platform takes 20% of 480, which is 96, and the tutor gets 384. The fee is a share of the **price**, not of 400. Write it as an equation: P·(1 − 0.2) = 400, so P = 400 ÷ 0.8 = **500**. ![A price of 500 splits into 400 for the tutor and 100 for the 20% fee. A price of 480 (400 × 1.2) splits into 384 and 96, leaving the tutor 16 short.](https://productovi.com/courses/math-basics/diagrams/m2-l3-gross-up.png) *A fee on the price needs ÷ (1 − f), not × (1 + f).* **Alex Tutor:** So a gross-up is P = cost/(1 − f). Multiplying by (1 + f) is right only when the fee is charged on top of the cost, like a markup. Always ask: a share of what? > **Exercise:** A platform keeps 25% of the lesson price. A tutor wants 600 UAH in hand. What should the price be? ## Your turn: price for a net amount **Alex Tutor:** Now a slightly different case: the platform keeps 15%, and the tutor wants 510 UAH per lesson in hand. **Sofia:** I'll check the wrong version too, so the difference is visible. > **Exercise:** A partner platform keeps 15% of the lesson price. A tutor wants 510 UAH per lesson in hand, and a colleague suggests a price of 510 × 1.15. Find the right price, show how much the tutor would really get at the colleague's price, and explain the mistake in one sentence. ## Where you will meet this > **In short** > > - An equation is a balance: do the same to both sides and undo the operations around the unknown in reverse order. > - Break-even: c·Q − F = 0 gives **Q = F/c**. Runway R = (S − Z)/(E − I) gives **S = Z + R·(E − I)**. > - **b₀ = ȳ − b₁x̄**; from SE = √(p(1 − p)/n), **n = p(1 − p)/SE²**. > - Dividing by a share makes a number bigger: post-money = **X/s**, interest-only balance = **interest/r**. > - A fee taken from the price: **P = cost/(1 − f)**, not cost × (1 + f). > **These lessons use it** > > - [Break-even and margin of safety](/courses/finance-for-product/break-even): Q_BE = ⌈F/c⌉. > - [Personal runway](/courses/bootstrapped-product/personal-runway): R = (S − Z)/(E − I). > - [Simple linear regression](/courses/statistics-for-pm-and-marketers/linear-regression): the intercept b₀ = ȳ − b₁x̄. > - [Power analysis and sample size](/courses/statistics-for-pm-and-marketers/power-and-sample-size) and [How big and how long](/courses/product-analytics/test-size): a formula solved for n. > - [Price vs volume](/courses/finance-for-product/price-vs-volume): the volume a discount needs, solved from the margins. > - [Debt vs equity](/courses/finance-for-product/debt-vs-equity): post-money = investment ÷ share. > - [Building it without pain: pay yourself first](/courses/personal-finance/pay-yourself-first): a monthly amount solved from a goal and a date. **Alex Tutor:** Next time: inequalities, intervals and formulas with cases, the "if this, then that" of math. - Previous lesson: [The Σ sign: sums, limits and running totals](https://productovi.com/courses/math-basics/the-sum-sign.md) - Next lesson: [Inequalities, intervals and formulas with cases](https://productovi.com/courses/math-basics/inequalities-and-intervals.md) --- Productovi Learning. 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