Rearranging a formula: solve for what you need

Rearranging a formula means asking the same formula a different question. Profit = c·Q − F answers "how much profit at this volume?"; Q = F/c answers "what volume gives zero profit?". They are one formula, rearranged.

One rule does all the work: an equation is a balance, so whatever you do to one side, you do to the other. Then undo the operations around the unknown in reverse order, the outermost first.

In 174·Q − 870,000 = 0, Q is first multiplied by 174, then 870,000 is subtracted. Undo in reverse: add 870,000 to both sides (174·Q = 870,000), then divide both sides by 174 (Q = 5,000).

Legal moves: add or subtract the same number on both sides; multiply or divide both sides by the same non-zero number; square both sides. Not legal: changing one side only, or dividing some terms of a side and not others.

Break-even is the classic rearrangement. Profit is c·Q − F: each lesson leaves c, and fixed costs F come off the top. Break-even is the volume at which profit is zero, so set c·Q − F = 0 and solve for Q:

1. add F to both sides: c·Q = F;
2. divide both sides by c: Q = F/c.

With Klasna's F = 870,000 UAH a month and c = 174 UAH a lesson, Q = 870,000 ÷ 174 = 5,000 lessons. A subject with F = 540,000 and c = 150 needs 540,000 ÷ 150 = 3,600.

When the division isn't exact, round up, as in module 1: a fraction of a lesson short still leaves a loss.

Check by putting the answer back: 150 × 3,600 − 540,000 = 0. If the check fails, a step was done to one side only.

When the unknown sits inside a fraction, undo the division first, then whatever is left.

Personal runway in the bootstrapped course is R = (S − Z)/(E − I): savings S minus a cushion Z you never touch, divided by monthly expenses E minus income I. To find how much to save for a target runway, solve for S:

1. multiply both sides by (E − I): R·(E − I) = S − Z;
2. add Z to both sides: S = Z + R·(E − I).

With E = 45,000, I = 15,000, Z = 60,000 and a target of 12 months: S = 60,000 + 12 × 30,000 = 420,000 UAH.

Two slips are common: multiplying by E alone instead of the whole bracket (E − I), and forgetting to add the cushion back. Check by putting the answer into the original: (420,000 − 60,000) ÷ 30,000 = 12 months.

A regression line ŷ = b₀ + b₁x has a slope b₁ and an intercept b₀, the predicted value when x is zero. Statistics computes the slope first; the intercept then comes from one fact and one rearrangement.

The fact: a least-squares line always passes through the point of means, (x̄, ȳ). So ȳ = b₀ + b₁x̄.

The rearrangement: subtract b₁x̄ from both sides, and b₀ = ȳ − b₁x̄.

Say a line of monthly lessons against ad spend has b₁ = 30 lessons per thousand UAH, with average spend x̄ = 50 thousand and average lessons ȳ = 8,000. Then b₀ = 8,000 − 30 × 50 = 6,500 lessons.

Read the result for what it is: the line's starting height, not a promise of what zero spending would bring, especially when no month in the data had spending near zero.

A square root is undone by squaring both sides. The standard error of a conversion rate is SE = √(p(1 − p)/n), and a test plan needs the reverse: how many users n give a chosen SE?

1. square both sides: SE² = p(1 − p)/n;
2. multiply both sides by n: n·SE² = p(1 − p);
3. divide both sides by SE²: n = p(1 − p)/SE².

For p = 0.2 and a target SE = 0.01: n = 0.2 × 0.8 ÷ 0.0001 = 1,600 users.

The shape of the result tells a story: SE is squared in the denominator, so halving the error needs four times the users. That's why tests chasing small effects get long.

Slips: forgetting to square SE (dividing by 0.01 instead of 0.0001 gives 16 users), and squaring only part of a side. Check by putting n back: √(0.16 ÷ 1,600) = √0.0001 = 0.01.

Dividing by a number below 1 makes the result bigger, and intuition often resists that. Dividing by a fraction is multiplying by its flip: ÷ 0.5 is × 2, ÷ 0.15 is × 6.67.

That's how a post-money valuation works. An investor pays X for a share s of a company, so X = s × value, and rearranging gives value = X/s. Paying 3 million UAH for 15% means the company is worth 3 ÷ 0.15 = 20 million after the deal. The pre-money value, before the new money arrives, is post-money minus the investment: 20 − 3 = 17 million.

Sanity check: s × value should give back X. 15% of 20 million is 3 million, so it holds.

The slip to avoid is multiplying instead: 3 × 0.15 = 0.45 million is less than the money paid, which can't be what the whole company is worth.

The same move turns an interest payment into the loan behind it. On a loan where you pay only interest, each month's payment is interest = balance × r, with r the monthly rate. Rearranged: balance = interest ÷ r.

3,000 UAH a month at r = 2% a month means a balance of 3,000 ÷ 0.02 = 150,000 UAH. Again, dividing by a small share makes the number much bigger, and again that is correct: a small rate on a big balance gives a modest payment.

Two habits keep this right. Write the rate as a decimal before dividing: 2% is 0.02, not 2, and dividing by 2 gives 1,500, a hundred times too small. And check every rearranged answer by putting it back into the original formula: 2% of 150,000 is 3,000. The check takes seconds and catches most slips.

A partner platform keeps 20% of each lesson price, and a tutor wants 400 UAH in hand. "400 plus 20%" gives 480, but the platform takes 20% of 480, which is 96, and the tutor gets 384. The fee is a share of the price, not of the 400.

Write it as an equation: P·(1 − f) = net. Solve for P: P = net/(1 − f) = 400 ÷ 0.8 = 500. Check: 20% of 500 is 100, and 400 is left.

Multiplying by (1 + f) is right only when the fee is charged on top of the cost, like a markup. So always ask: a share of what?

How it fits together: keep the balance, undo the operations in reverse order, square to undo a root, remember that dividing by a share makes a number bigger, and check by substituting back. Next lesson: inequalities, intervals and formulas with cases.