Three workers, $P, Q,$ and $R$, can complete digging a drainage trench in $10\text{ days}, 15\text{ days},$ and $30\text{ days}$, respectively. If they all work together simultaneously, how many days will they take to complete the trench? ________ (answer in integer).
Answer Key:
5
Step-by-Step Solution & Explanation
**Method 1: Unitary Work Rate**
$\text{Combined rate} = \frac{1}{10} + \frac{1}{15} + \frac{1}{30} = \frac{3 + 2 + 1}{30} = \frac{6}{30} = \frac{1}{5}\text{ trench/day}$
$\text{Total time required} = \frac{1}{1/5} = 5\text{ days}$
**Method 2: LCM Total Work Units**
Let total work be $\text{LCM}(10, 15, 30) = 30\text{ units}$.
- Rate of $P = 30/10 = 3\text{ units/day}$
- Rate of $Q = 30/15 = 2\text{ units/day}$
- Rate of $R = 30/30 = 1\text{ unit/day}$
$\text{Combined rate} = 3 + 2 + 1 = 6\text{ units/day}$
$\text{Time} = \frac{30}{6} = 5\text{ days}$