Placement Preparation

• [Fundamental model] Work = Rate × Time. When the entire job is treated as 1, rate is measured as job per day or job per hour.
• [Reciprocal rate] If A alone completes a job in a days, A's one-day work (rate) is 1/a and one-hour work is 1/(total working hours).
• [Work completed] Work completed in t days by a worker who needs a days = t/a.
• [Remaining work] Remaining work after t days = 1 - t/a, provided the worker's efficiency stays constant.
• [Time from work and rate] Required time = Required work ÷ Effective rate.
• [Combined positive rates] If workers cooperate independently, their rates add: R_total = R1 + R2 + ... + Rn.
• [Two-person combined time] If A and B take a and b days separately, together they take ab/(a+b) days.
• [Three-person combined time] If A, B and C take a, b and c days separately, together they take abc/(ab+bc+ca) days.
• [Unknown partner's rate] If A+B finish in t days and A alone in a days, B's rate = 1/t - 1/a.
• [Unknown partner's time] If A+B finish in t days and A alone in a days (a>t), B alone takes at/(a-t) days.
• [Negative or destructive worker] If a person/machine undoes work, subtract its rate: R_net = sum of productive rates - sum of destructive rates.
• [Efficiency and time] For equal work, efficiency is inversely proportional to time: E1/E2 = T2/T1.
• [Efficiency ratio to time ratio] If efficiencies A:B = m:n, their times for the same job are n:m.
• [Time ratio to efficiency ratio] If times A:B = m:n, their efficiencies are n:m.
• [Percentage more efficient] If A is p% more efficient than B, EA:EB = (100+p):100 and TA:TB = 100:(100+p).
• [Percentage less efficient] If A is p% less efficient than B, EA:EB = (100-p):100 and TA:TB = 100:(100-p).
• [Time after efficiency increase] If efficiency increases by p%, new time = old time × 100/(100+p).
• [Time after efficiency decrease] If efficiency decreases by p%, new time = old time × 100/(100-p).
• [Combined time from efficiency ratio] If efficiencies A:B = m:n and together they take t days, A alone takes t(m+n)/m and B alone takes t(m+n)/n.
• [Rate from partial work] If A completes fraction f of a job in t days, A's rate = f/t and full-job time = t/f.
• [Equivalent work units] Choose total work as the LCM of individual times. A worker taking a days then performs LCM/a units per day.
• [Worker-day model] For identical workers and fixed hours/efficiency, total work ? number of workers × days.
• [Worker-hour model] For fixed efficiency, total work ? workers × days × hours per day.
• [General work proportionality] W1/W2 = (M1×D1×H1×E1)/(M2×D2×H2×E2), where M=workers, D=days, H=hours/day and E=relative efficiency.
• [Same-work cross multiplication] For the same work: M1D1H1E1 = M2D2H2E2.
• [Join/leave phase equation] For multiple phases, total work = ?(rate active in phase × duration of phase). Set this sum equal to 1.
• [Leaves before completion] If A leaves k days before completion time T, A works T-k days while workers who remain work T days.
• [Joins after a delay] If B joins after k days and total time is T, the first worker works T days while B works T-k days.
• [Alternate-day cycle] For A and B working on alternate days, two-day cycle work = RA+RB. Count full cycles, then handle the final partial day.
• [Rotating-team cycle] For a repeating schedule of teams, cycle work = sum of each scheduled day's effective rate.
• [Periodic absence] For absences every mth, nth, etc. day, use an LCM-length calendar cycle and add only the rates of those present each day.
• [Work-rest schedule] Calendar duration = required working days + rest days that occur before completion. Do not add a rest after the job is finished.
• [Equivalent workforce] If x men = y women for the same time and work, one man = y/x women and efficiency M:W = y:x.
• [Mixed workforce rate] Convert all worker types into one equivalent unit, then use equivalent workers × days × hours.
• [Wage division] For equal wage rate per unit of work, wages are divided in the ratio of actual work contributions: Ei×Ti.
• [Deadline workforce] Required workers = remaining work units ÷ (remaining days × work per worker per day).
• [Minimum whole workers] When a fractional number of workers is obtained, round upward because a partial worker cannot meet the deadline.
• [Actual-rate correction] If progress differs from plan, calculate actual team rate from completed work/elapsed time before estimating remaining manpower.
• [Changing workforce] Total work in worker-days = ?(workers active in each phase × phase days × relative efficiency).
• [Arithmetic-progression workforce] If workforce changes by d each day, total worker-days in n days = n/2[2a+(n-1)d].
• [Arithmetic-progression productivity] If daily output forms an AP with first output a and increase d, total output after n days = n/2[2a+(n-1)d].
• [Geometric-progression productivity] If daily output forms a GP with first output a and ratio r, total after n days = a(r^n-1)/(r-1), r?1.
• [Hourly-rate conversion] If a worker completes in d days at h hours/day, hourly rate = 1/(dh).
• [Overlapping shifts] Daily work = sum of each worker's hourly rate × actual hours worked that day, including overlap without double-counting time.
• [Machine productivity] Output = machines × days × hours/day × output per machine-hour × efficiency factor.
• [Defect/rejection adjustment] Effective productive rate = nominal rate × accepted fraction. If rejection is q%, accepted fraction is 1-q/100.
• [Rework after rejection] Net completed work = gross completed work × accepted fraction; rejected work must be added back to remaining work.
• [Efficiency changes by phase] For phase i, time_i = work_fraction_i ÷ adjusted_rate_i. Total time is the sum of all phase times.
• [Project-scope change] If project size changes, express completed and remaining work in units of the original project before computing the new balance.
• [Equal-finish work allocation] To make teams with rates R1:R2 finish simultaneously, divide work in the same ratio R1:R2.
• [Pairwise combined rates] If AB, BC and CA rates are p, q and r, then A+B+C rate = (p+q+r)/2.
• [Individual rate from pairwise rates] A=(AB+AC-BC)/2, B=(AB+BC-AC)/2, C=(AC+BC-AB)/2.
• [All-three and pair rate] If ABC rate and AB rate are known, C rate = ABC-AB; similarly for any omitted worker.
• [Rate chain] Build chained efficiencies on a common base. Example: A=2B and B=1.5C gives A:B:C=6:3:2.
• [Time-difference equation] If A and B together take t days and their individual times differ by k, solve 1/x + 1/(x+k) = 1/t.
• [Multiple projects] Keep a separate work ledger for each project. Worker transfers change future rates, not work already completed.
• [Work share from equal time] If teams work for the same duration, their completed-work ratio equals their effective-rate ratio.
• [Percentage work in percentage time] If a worker completes x% of work in y% of the usual time, effective efficiency relative to normal = x/y.
• [Completion within a day] Final fraction of a day = remaining work ÷ that day's active rate.

• [Normalize to one job] Treat the whole job as 1 when equations are simple. This prevents unnecessary large numbers.
• [Use LCM units] When times are integers, take total work as their LCM; daily outputs become whole numbers and hard fractions disappear.
• [Invert time ratios] Immediately reverse a time ratio to get the efficiency ratio.
• [Convert percentages to small ratios] 25% more ? 5:4, 20% less ? 4:5, 50% more ? 3:2, 33?% more ? 4:3.
• [Two-person direct formula] For separate times a and b, use ab/(a+b) instead of repeatedly adding fractions.
• [Unknown partner direct formula] If together time is t and A alone time is a, B time = at/(a-t).
• [Pairwise half-sum trick] With AB, BC and CA times, convert to rates and halve their sum to get the all-three rate.
• [Pairwise cancellation trick] A's rate = ½[(AB rate)+(AC rate)-(BC rate)]; use analogous expressions for B and C.
• [Cycle before calendar] For alternate or rotating schedules, find work per complete cycle first, then inspect only the leftover days.
• [Do not average times] Average rates, not completion times. The arithmetic mean of individual times is generally wrong.
• [Check combined-time sanity] A productive combination must take less time than the fastest individual. Otherwise, recheck signs or data.
• [Partial-day finish] Never force a full final day. Divide the remaining work by the active rate on that exact day.
• [Starting worker matters] In alternate-day problems, identify who starts; it can change the answer when completion occurs mid-cycle.
• [Worker-day ledger] Write total required worker-days, subtract completed worker-days, then divide the balance by the new workforce.
• [Update from actual progress] If the team finishes only a stated fraction by a checkpoint, ignore the original assumed rate and use actual progress.
• [Deadline quick ratio] For identical efficiency and unchanged work, workers are inversely proportional to available days.
• [Equivalent-worker conversion] Convert women, children, machines, or shifts into one base unit before any multiplication.
• [Wages follow contribution] Use efficiency × actual time, not merely attendance days, to divide payment.
• [Negative work sign] Use a minus sign for leakage, damage, rework or undoing. A net rate ?0 means the job will never finish.
• [Rest-day counting] Count only rest days that occur before the final working period; there is no rest after completion.
• [Hourly conversion first] When daily hours differ, convert every worker to hourly rate before combining.
• [AP workforce shortcut] For workers increasing/decreasing regularly, use the AP sum instead of writing every day's workforce.
• [GP output shortcut] For doubling/halving productivity, use the geometric-series sum.
• [Scope-change ledger] Keep completed work in original-job units, revise total scope, and subtract; do not reset progress.
• [Rejection shortcut] If q% of output is rejected continuously, multiply that worker's rate by (100-q)/100.
• [Ceiling rule] For required workers or machines, always take the next integer unless the result is already whole.
• [Option testing] In MCQs, test answer choices using rate × time; this can avoid solving a quadratic.
• [Fraction discipline] Keep exact fractions until the final step. Early decimal rounding often changes the final partial day.
• [Phase table] For join/leave questions, create columns: phase, active workers, rate, time, work done.
• [Periodic schedule LCM] For 'every 3rd/4th/5th day' conditions, build one LCM-length attendance cycle.
• [Equal-finish split] Split work in the ratio of team rates so both teams take the same time.
• [Rate-chain base] For chained comparisons, choose a common base that removes decimals before combining.
• [Time-difference quadratic] Let faster time be x and slower time x+k; use product/sum after clearing denominators.
• [Project-transfer rule] Finish calculations for work already done before transferring workers; only future workforce changes.
• [Same-work cancellation] In M×D×H×E comparisons, cancel common factors before multiplying large numbers.
• [Fraction-of-work shortcut] If f of a job takes t days, full-job time is t/f instantly.
• [Percentage-time shortcut] Completing x% work in y% time means efficiency multiplier x/y.
• [Alternating destructive work] Include the destructive rate on every day it operates, even when different productive workers alternate.
• [Repeating pattern marker] Write day numbers under one cycle and mark exactly who works; this prevents off-by-one errors.
• [Feasibility check] Individual rates derived from pairwise data must all be positive; zero or negative values indicate impossible or redundant data.