Time and Work
Table of Content:
Time and Work Formulas
Basic Concepts
The basic concept of Time and Work problems is that work is generally considered to be 1 unit and the efficiency of a person is the amount of work done in a unit of time.
Basic Formulas
Work
\[ \text{Work} = \text{Rate} \times \text{Time} \]Rate
\[ \text{Rate} = \frac{\text{Work}}{\text{Time}} \]Time
\[ \text{Time} = \frac{\text{Work}}{\text{Rate}} \]Work Done by Multiple Entities
Combined Work Rate
\[ \text{Combined Work Rate} = \text{Rate}_1 + \text{Rate}_2 + \cdots + \text{Rate}_n \]Total Time
If A can complete a work in \( T_A \) days and B can complete the same work in \( T_B \) days, then:
\[ \text{Total Time} = \frac{T_A \times T_B}{T_A + T_B} \]Work Efficiency
Efficiency
\[ \text{Efficiency} = \frac{1}{\text{Time Taken to Complete Work}} \]Work Sharing
Share of Work
If A and B together can complete a work in \( T \) days and A alone can complete the work in \( T_A \) days, then B alone can complete the work in \( T_B \) days, where:
\[ \frac{1}{T} = \frac{1}{T_A} + \frac{1}{T_B} \]Work Involving Multiple Workers
Work Done in Given Days
If A, B, and C together can complete a work in \( T \) days, then the work done by A, B, and C together in \( D \) days is:
\[ \text{Work Done} = \frac{D}{T} \]Example Problems
Example 1: Finding Time
If A can complete a work in 10 days and B can complete the same work in 15 days, how long will it take for both to complete the work together?
\[ \text{Total Time} = \frac{10 \times 15}{10 + 15} = \frac{150}{25} = 6 \text{ days} \]Example 2: Finding Efficiency
If A can complete a work in 8 days, what is A's efficiency?
\[ \text{Efficiency} = \frac{1}{8} = 0.125 \text{ units/day} \]