🛠️ Work & Time Basics

Instead of calling the job $1$, call the total work the LCM of all the given times. Then every rate becomes a whole number, so you just add and divide.

• Step 1. A takes 12 days, B takes 6 days. Total work $= \text{LCM}(12,6) = 12$ units.

• Step 2. Rate $= \dfrac{\text{total work}}{\text{time}}$. So A does $\dfrac{12}{12}=1$ unit/day, B does $\dfrac{12}{6}=2$ units/day.

• Step 3. Working together they do $1+2=3$ units/day.

• Step 4. Time together $= \dfrac{\text{total work}}{\text{combined rate}} = \dfrac{12}{3}=4$ days.

Two-worker shortcut: if A takes $a$ days and B takes $b$ days, together they take $\dfrac{a\times b}{a+b}$ days (product over sum). Here $\dfrac{12\times 6}{12+6}=\dfrac{72}{18}=4$ days.

Fraction of work done in $t$ days $= \dfrac{t}{\text{time for full job}}$. If A finishes in 12 days, in 3 days A has done $\dfrac{3}{12}=\dfrac14$ of it, and $\dfrac34$ is left.