Silent speed is how candidates burn the case. The sheet rewards an audible setup that keeps about four chunks in play; the eight moves below are how you stay inside that budget while the interviewer still hears the logic.
Thesis: mental math in cases is not a race to a precise answer. It is recoding ugly numbers into familiar chunks so you can talk, compute, and sanity-check at once. After this page you will own eight named moves, one worked example each, and a five-minute drill path on CoachNed's Quick Math. Scoring habits and formula sheets live elsewhere; this piece is pure technique.
Sibling posts cover the practice program and the reference sheet. Here the only job is the move set: factoring, percent-of-percent, the 1/x table, splitting, compensation rounding, powers of ten, growth approximations, and fraction–decimal swaps.
Why chunking beats speed
George Miller's classic review showed that short-term capacity is measured in chunks, not raw digits, and that intelligent grouping expands what you can hold (Miller, 1956). Nelson Cowan's later synthesis tightened the typical focus-of-attention limit to roughly three to five chunks when rehearsal and long-term knowledge are controlled (Cowan, 2001). Dual-task work on mental arithmetic finds that executive working memory is taxed hardest by counting and decomposition strategies, less so by retrieval of known facts (Cragg et al., 2017).
In the room that means: if you try to multiply 47 × 83 as eight separate digit operations while also tracking units and a story, you overflow. If you recode 47 × 83 as "(50 − 3) × 83," you hold two chunks and retrieve a product. Firms ask you to keep communicating through the arithmetic—BCG's public case prep says you will "perform quick calculations" while structuring and talking (BCG case interview preparation); McKinsey's older Problem Solving Test coaching guide told candidates to approximate and use shortcuts rather than grind exact arithmetic under time (McKinsey PST coaching guide). Chunking is what makes that possible.
Ned's rule. If you cannot say the move in six words before you compute—"factor 25 out," "one over twelve is eight percent"—do not start the multiply. You are about to invent digits.
Move 1: Factoring
Pull a common factor so the remaining multiply is on friendly integers. The point is not elegance; it is reducing how many novel digits you must hold while speaking.
Worked. Harborlight sells 2,400 units at $18.75. Revenue = 2,400 × 18.75. Factor: 2,400 × (20 − 1.25) = 48,000 − 3,000 = $45,000. Or 2,400 × (75/4) = (2,400 / 4) × 75 = 600 × 75 = 45,000. Same answer, two doors; pick the one you can narrate.
Drill. Compute 1,600 × 37.5 and 3,600 × 12.5 without a calculator. Time yourself; restart if you go silent for more than two seconds. Then invent a third: 2,800 × 15. Say the factor before you multiply.
Move 2: Percent-of-percent
Never multiply "twelve times eight" when both are percents. Convert one to a decimal once, then take that percent of the other base. Most zero-drop disasters I see are percent-of-percent mistakes dressed as arithmetic.
Worked. Gross margin 28%, COGS fall 15% of COGS. Cost saving as share of revenue ≈ 0.28 × 0.15 = 0.042, or 4.2 points of revenue—not 43%, not 4.2% of COGS alone without stating the base. If COGS is 72% of revenue, a 15% COGS cut is 0.72 × 0.15 = 10.8 points of revenue. Different story. Name the base every time.
Drill. Take 22% of 35% of $80m. Say the bases out loud: "twenty-two percent of thirty-five percent of eighty million." Then flip it: 35% of 22% of $80m should match.
Move 3: The 1/x table
Memorize a short reciprocal table so division becomes multiplication.
| x | 1/x (approx) | Say it as |
|---|---|---|
| 3 | 0.333 | one-third |
| 4 | 0.25 | a quarter |
| 5 | 0.20 | one-fifth |
| 6 | 0.167 | one-sixth ≈ 17% |
| 7 | 0.143 | one-seventh ≈ 14% |
| 8 | 0.125 | one-eighth |
| 9 | 0.111 | one-ninth ≈ 11% |
| 11 | 0.091 | one-eleventh ≈ 9% |
| 12 | 0.083 | one-twelfth ≈ 8% |
| 16 | 0.0625 | one-sixteenth |
Worked. Payback on a $48m plant earning $6m/year cash: 48 / 6 = 8 years, or 48 × (1/6) ≈ 48 × 0.167 ≈ 8 years. Margin of 1/12 of revenue: about 8%. Share of 1/7: about 14%. Once those four are automatic, division stops being a silent panic.
Drill. Convert 1/6, 1/7, 1/9, and 1/12 to percent in under ten seconds each. Then invert: what fraction is ~9%, ~14%, ~8%? Write the table once from memory; check against the table above.
Move 4: Splitting
Break a product or quotient into a sum of easy pieces. Splitting is the move you use when factoring does not bite and the 1/x table does not apply.
Worked. 47 × 26. Split 26 = 20 + 6: 47 × 20 = 940; 47 × 6 = 282; total 1,222. Or 47 × 26 = 47 × (25 + 1) = 1,175 + 47 = 1,222. For division, 910 / 26: 26 × 30 = 780; remainder 130; 26 × 5 = 130; so 35.
Drill. Do 63 × 28 and 81 × 19 by splitting only. Announce the split before any multiply. Add 1,040 / 32 as a quotient split.
Move 5: Compensation rounding
Round one factor to a friendly number, then compensate the other direction. This is the move McKinsey's old test guide was pointing at when it told candidates to approximate rather than grind.
Worked. 498 × 36. Round 498 → 500 (+2). 500 × 36 = 18,000. Compensation: 2 × 36 = 72. Answer 17,928. Say: "I rounded up by two, so I subtract seventy-two."
Compensation beats rounding every factor the same way. Rounding 498 → 500 and 36 → 35 without compensating stacks error you cannot explain. In a case, announce the round; the interviewer can follow. Silent rounding looks like guessing.
Drill. 1,995 × 42 and 3,010 × 19 with one announced round and one compensation each. For each, state the error direction before the final number.
Move 6: Powers of ten
Park the zeros, do the core multiply, then replace the zeros with words. Dropped zeros are the most common first-round math fail I score; this move is the fix.
Worked. 45,000 × 0.08. Core: 45 × 8 = 360. Zeros: 45,000 is 45 × 10³; 0.08 is 8 × 10⁻²; product scale 10¹ → 3,600. Say "forty-five thousand times eight percent is thirty-six hundred," not a muttered 3600. Sanity: 8% of 45k must be a few thousand, not a few hundred and not forty thousand.
Drill. 2.4m × 0.015 and 75k × 1.2. Speak every answer in words once before digits. If the spoken magnitude feels wrong, you dropped a power of ten—stop and repark the zeros.
Move 7: Growth approximations
For small rates, (1 + g)^n ≈ 1 + ng. For two rates, (1 + a)(1 + b) ≈ 1 + a + b when both are small. Keep the cross term when it matters. This is not a finance identity lesson; it is how you avoid saying "+3%" when the product is +2.8%.
Worked. Volume up 4%/year for three years: 1.04³ ≈ 1 + 3 × 0.04 = 1.12 (true 1.04³ ≈ 1.125; you are ~0.5 points low—say so). Price up 6% and volume down 3%: net ≈ 1.06 × 0.97 ≈ 1 + 0.06 − 0.03 − 0.0018 ≈ +2.8%, not +3%. For longer horizons or double-digit rates, do not lean on the linear shortcut—switch to compounding one year at a time or ask for a calculator if the firm allows it.
Drill. Approximate 1.05^4 and 1.08 × 0.95. State the cross term when you drop it. Then redo 1.12 × 0.90 with the cross term kept.
Move 8: Fraction–decimal swaps
Flip between fraction and decimal when one form is easier to multiply. Candidates often freeze on 0.375 × 64 when 3/8 of 64 is obvious, or the reverse.
Worked. Take 3/8 of $64m. As decimal: 0.375 × 64. Split: 0.4 × 64 = 25.6; subtract 0.025 × 64 = 1.6; $24m. As fraction: (3 × 64) / 8 = 192 / 8 = 24. Same chunk, different door. For 5/12 of a market, either 0.417 ≈ 42% or keep the fraction until the last step.
Drill. Compute 5/12 of 84 and 7/8 of 960 both ways. Prefer the door that leaves an integer mid-step. If both doors leave messes, compensate-round the base first.
One audible pass that uses three moves
Harborlight's Midland DC ships 18,400 cases/week at $12.50/case. A 6% price rise and a 4% volume drop are on the table. What happens to weekly revenue?
- Powers of ten + factoring. Base revenue: 18,400 × 12.50. Factor 12.50 = 50/4: (18,400 / 4) × 50 = 4,600 × 50 = $230,000.
- Growth approximation with cross term. New factor ≈ (1.06)(0.96) = 1 + 0.06 − 0.04 − 0.0024 = 1.0176.
- Percent-of-percent / compensation. 230,000 × 1.0176 ≈ 230,000 + 230,000 × 0.018 − 230,000 × 0.0004 ≈ 230,000 + 4,140 − 92 ≈ $234,000.
So-what in one line: weekly revenue rises about $4k (~1.8%) despite the volume loss—the price move more than offsets. That is what the sheet hears: identity, move, number, implication.
Practice this today
Pick one move. Run ten problems of that shape only. Then mix three moves on a single revenue or margin line and narrate every step. CoachNed's Quick Math is built for that loop—timed prompts, no calculator, free forever. When you want scored case turns around the math, start is open for seven days without a card; after that a recruiting season is $120 or $49 a month.
Frequently asked questions
How fast should case mental math be?
Fast enough that the interviewer does not wait, slow enough that they hear the setup. In first rounds I treat a clean 20–40 second calculation with units as a pass; a whispered exact number with a wrong base as a fail. Speed without chunks is noise.
Do I need to memorize a long 1/x table?
No. The ten reciprocals in the table above cover almost every case division you will see. Add 1/15 (~6.7%) and 1/20 (5%) if you want a twelfth and a twentieth for payback and share-of-wallet lines.
Is a calculator ever allowed in case interviews?
Live interviewer cases are almost always oral and paper. Online assessments vary by firm and year; treat any calculator permission as an exception the recruiter states, not a default. The eight moves still help when you must estimate under a clock.
Should I round every number in a case?
Round one factor on purpose, announce it, and compensate when needed. Rounding every input the same direction is how candidates invent a market that is 30% off and still sound confident.
How is this different from case math practice programs?
This page is the technique catalog. Practice volume, scoring rubrics, and formula sheets are separate jobs. Learn the eight moves here; then drill them on Quick Math until retrieval is automatic.
Sources
- The Magical Number Seven, Plus or Minus Two (Miller, 1956 PDF) — chunking expands short-term capacity beyond raw digits. Checked 2026-09-24.
- The magical number 4 in short-term memory (Cowan, 2001) — typical focus-of-attention limit of about three to five chunks. Checked 2026-09-24.
- When is working memory important for arithmetic? (Cragg et al., 2017) — executive load differs by arithmetic strategy (counting, decomposition, retrieval). Checked 2026-09-24.
- BCG case interview preparation — public description that cases include quick calculations alongside structure and communication. Checked 2026-09-24.
- McKinsey Problem Solving Test coaching guide — firm guidance to approximate and use shortcuts under time. Checked 2026-09-24.
CoachNed is not affiliated with McKinsey, BCG, Bain, Deloitte, or any firm named above.