Case Math Under Pressure

Learning objectives

By the end you can:

1. Convert cleanly between "percentage of," percentage-point change, and percentage (relative) change, and never again confuse a point move with a relative move, out loud or on paper.
2. Combine successive percentage changes with the exact correction-term trick (not naive addition), including the equal-and-opposite trap (why "up 20%, down 20%" is a net loss) and the reciprocal recovery trap (why undoing a decline always takes a bigger percentage gain than the decline itself), and quantify precisely how large an error the naive shortcut makes.
3. Estimate a compound annual growth rate (CAGR) fast, using the Rule of 72 (and its 70 / 69.3 cousins) plus the one always-true direction fact about naive linear guesses, landing within a defensible point or two without a calculator.
4. Build a correctly weighted blended average from two or more segments, explain why an unweighted average of ratios is a live error, and predict, and quantify, the pure mix-shift effect when segment weights move but underlying rates do not.
5. Compute breakeven volume, breakeven revenue, and margin of safety from fixed cost, price, and variable cost; and predict (with a number, not just a direction) how a small price or cost change moves the breakeven point, and why that move is often disproportionately large.
6. Use a ratio as a live consistency check: reverse-engineer one disclosed figure from another to test whether a third claimed figure could possibly be true, and catch an order-of-magnitude mismatch on sight rather than after the meeting.
7. Apply a small, memorized toolkit of friendly-number decomposition (round-and-correct multiplication, the ×5/×15/×25/÷5/÷25 factor tricks, and the core fraction-to-percent table) to multiply, divide, and estimate fast, and know, case by case, how small the resulting error actually is.
8. Carry a rounding discipline that keeps cumulative error inside roughly 2% across a multi-step chain, state your roundings and assumptions out loud as you compute, and never let doing the math in your head quietly become doing the math in silence.
9. (AI-augmented objective.) Use an AI tool to generate a bottomless, leveled stream of case-math drills and to pressure-test your worked method between practice sessions, while keeping every ounce of the live arithmetic in your own head, because no case room, client meeting, or steering committee lets you open a chat window mid-sentence.

Objective 9 is a standing habit, not a separate gate: the mastery check and the mapped drill test objectives 1–8 only, and nobody grades how you practiced. But the discipline inside objective 9 (draft with AI, verify by hand, never depend on it live) is exactly what keeps the other eight fast and trustworthy over time.

Prerequisites & connections

Builds on. Nothing, formally. CN2.01 is the first node of the Casing & Quantitative Fluency branch, and (per this galaxy's resolved progression rule) every branch's opening module carries no hard prerequisite from anywhere else in the galaxy; you can start here cold with nothing but ordinary school arithmetic (fractions, decimals, percentages). Two things make it click faster, both advisory and neither required. First, CN0.01's placement diagnostic already samples six items from exactly this territory, so a strong score there is a signal (not a requirement) that this module will move quickly for you. Second, CN1.01 and CN1.02 (issue trees and hypothesis-driven problem solving) build the structuring habit this arithmetic will eventually sit inside; if you have already met them, you will recognize immediately where each of these five operations gets deployed inside a real tree.

Leans on the Finance galaxy, never re-teaches it (R1). If you have been through the Finance galaxy, you have already done slow, careful, on-paper versions of every operation this module speeds up. M2.01 (Ratio Toolkit I. Profitability & DuPont) built the margin stack and the ROE/DuPont algebra that a blended-margin question (objective 4) is a fast-forward version of; M2.02 and M2.03 (Liquidity/Solvency/Efficiency; Returns on Capital) built the ratio and turnover fluency that objective 6's consistency checks lean on; M4.02 (Unit Economics) built contribution margin and breakeven from the ground up as a business-model diagnostic, objective 5 is the same arithmetic, stripped of the diagnostic apparatus and sped up for live use; and M3.01 (Time Value of Money) built compounding and discounting with full rigor, objective 3's CAGR work is that same compounding, mentally approximated under a clock. This module does not rebuild any of that judgment. It extracts the raw arithmetic those modules lean on and drills it for speed, no-calculator reliability, and out-loud narration, because a client meeting will not pause while you reconstruct a DuPont tree from first principles. If you have never touched the Finance galaxy, nothing above is required, every technique here is taught from scratch, with its own worked examples.