Assume Total Work

To make calculations easy, assume the total work is the Least Common Multiple (LCM) of the given days $10$, $15$, and $30$.

$\text{Total Work} = \text{LCM}(10, 15, 30) = 30 \text{ units}$

Calculate Efficiency of Each Person

Efficiency is the work done per day, calculated as $\text{Total Work} / \text{Time}$. Calculate this for A, B, and C.

$E_A = \frac{30}{10} = 3, \; E_B = \frac{30}{15} = 2, \; E_C = \frac{30}{30} = 1$

Calculate Combined Efficiency

Find the total work done by A, B, and C together in a single day by adding their individual efficiencies.

$E_{A+B+C} = 3 + 2 + 1 = 6 \text{ units/day}$

Calculate Final Time

Divide the total work by the combined daily efficiency to find the total days required to complete the work.

$\text{Total Time} = \frac{30}{6} = 5 \text{ days}$