Arithmetic

Time and Work formulas

At a glance

job = LCM of days     rate = job ÷ days     rates add, days do not
two people: ab / (a + b)
pairs: (A+B) + (B+C) + (C+A) = 2(A+B+C)
efficiency ratio = inverse of time ratio
M₁D₁H₁ / W₁ = M₂D₂H₂ / W₂
together: abc/(ab + bc + ca)     A alone from A+B (x) and B (y): xy/(y − x)
emptying pipe or leak = negative rate;  fill a, empty b → ab/(b − a)

Work is a rate: the LCM method

Take the job as the LCM of the days; rates add, days do not

job = LCM of the days     rate = job ÷ days
together: days = job ÷ (sum of rates) = ab / (a + b) for two people

Groups and pairs

(A+B) + (B+C) + (C+A) = 2 (A + B + C)
C alone = (A + B + C) − (A + B)
A, B, C alone in a, b, c days  →  together abc / (ab + bc + ca) days
A + B together in x days, B alone in y  →  A alone xy / (y − x) days

Shortcut: If halving the pair sum gives fractions, take the job as twice the LCM. Every rate then comes out whole.

Efficiency ratios

efficiency A : B = time B : time A

Man-days: work as an area

Work = men × days: same area, different shapes

M men, D days, H hours a day, W amount of work.

M₁ × D₁ × H₁ / W₁ = M₂ × D₂ × H₂ / W₂

Pipes and cisterns

net rate = (sum of filling rates) − (sum of emptying rates)
fills in a hours, empties in b hours (b > a)  →  full in ab / (b − a) hours

Formulas stick when you use them

The Time and Work lesson has worked problems and quick checks for every formula here. Free to start.

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