Technique

Take total work as the LCM of the times

Arithmetic • Time and Work

Statement

If workers finish a job alone in

t_{1}, t_{2}, \dots

days, set the total work to

LCM (t_{1}, t_{2}, \dots)

units. Each worker's rate is then

\frac{total}{t _{i}}

units per day, a whole number.

How to use it

1Add rates when workers act together; subtract when one opposes (an outlet pipe).
2Time $= \frac{total work}{combined rate}$ .
3Any total works; the LCM is chosen only to keep the arithmetic in integers.

A

finishes a job in

12

days and

B

6

days. How long together?

Step-by-step

1
Total work $= LCM (12, 6) = 12$ units
Why: Choose the total so both rates are integers.
2
Rates: $A = 1$ , $B = 2$ units/day, combined $= 3$
3
Time $= \frac{12}{3} = 4$ days

Answer

4

days

Watch out

Rates add, times do NOT. Averaging the two times is the standard wrong answer.

Builds on

\times

Also known as

LCM method for work, unitary work method, pipes and cisterns rates