Work Rate Problem: Calculating C's Task Time

This solution outlines the steps to find how long C alone takes to complete a task, given the work rates of A and B and their combined effort.

Work Rates: A and B Individual Task Time

• A's daily rate: $\frac{1}{30}$ of the task.
• B's daily rate: $\frac{1}{40}$ of the task.

Combined Work Rate: A and B Together

The combined daily rate for A and B is:

$ \text{Rate}_{A+B} = \frac{1}{30} + \frac{1}{40} = \frac{4+3}{120} = \frac{7}{120} \text{ task/day} $

Work done by A and B in $10$ days:

$ \text{Work}_{A+B, 10 \text{ days}} = 10 \times \frac{7}{120} = \frac{7}{12} \text{ task} $

Remaining Work Calculation

The work remaining after A and B worked for 10 days is:

$ \text{Remaining Work} = 1 - \frac{7}{12} = \frac{5}{12} \text{ task} $

C completes this $\frac{5}{12}$ of the task in $5$ days.

C's Work Rate Determination

C's daily work rate is calculated as:

$ \text{Rate}_C = \frac{\text{Remaining Work}}{\text{Time}} = \frac{5/12 \text{ task}}{5 \text{ days}} = \frac{1}{12} \text{ task/day} $

C's Total Task Time Calculation

The total time C requires alone to complete the entire task (1 task) is:

$ \text{Time}_C = \frac{1}{\text{Rate}_C} = \frac{1}{1/12} = 12 \text{ days} $

Based on the calculations, C alone takes $12\text{ days}$ to complete the task.