\displaystyle \textbf{Rules for solving problems based on time and work}

\displaystyle \textbf{1. Work done in one day}

\displaystyle \text{If a person can complete a piece of work in }n\text{ days, then the work}
\displaystyle \text{done by the person in }1\text{ day is }\frac{1}{n}\text{ of the total work}.

\displaystyle \text{One day's work} =\frac{1}{\text{Number of days required to complete the work}}.

\displaystyle \text{For example, if A can complete a work in }10\text{ days, then A's one day's work}
\displaystyle =\frac{1}{10}\text{ of the total work}.
\displaystyle \\

\displaystyle \textbf{2. Time required from one day's work}

\displaystyle \text{If a person completes }\frac{1}{n}\text{ of a work in }1\text{ day, then the person}
\displaystyle \text{will require }n\text{ days to complete the whole work}.

\displaystyle \text{Number of days required to complete the work} =\frac{1}{\text{One day's work}}.

\displaystyle \text{For example, if A completes }\frac{1}{8}\text{ of a work in }1\text{ day, then A}
\displaystyle \text{will require }8\text{ days to complete the whole work}.
\displaystyle \\

\displaystyle \textbf{3. Time required to complete the remaining work}

\displaystyle \text{If a part of the work has already been completed and the remaining work is}
\displaystyle \text{to be completed by A, then the time required is given by:}

\displaystyle \text{Number of days required} =\frac{\text{Remaining part of the work}}{\text{A's one day's work}}.

\displaystyle \text{For example, if }\frac{2}{5}\text{ of a work remains and A's one day's work is } \frac{1}{10},
\displaystyle \text{then the number of days required} =\frac{2/5}{1/10}=4\text{ days}.
\displaystyle \\

\displaystyle \textbf{4. Relationship between efficiency and time}

\displaystyle \text{If A is }m\text{ times as efficient as B, then in the same amount of time,}
\displaystyle \text{A will do }m\text{ times as much work as B}.

\displaystyle \text{Therefore, the ratio of the efficiencies of A and B is}
\displaystyle A:B=m:1.

\displaystyle \text{For the same amount of work, time taken is inversely proportional to efficiency}.

\displaystyle \therefore \text{Ratio of work done by A and B in the same time}=m:1.

\displaystyle \therefore \text{Ratio of time taken by A and B for the same work}=1:m.
\displaystyle \\

\displaystyle \textbf{5. Work done by two or more persons together}

\displaystyle \text{If A's one day's work is }\frac{1}{a}\text{ and B's one day's work is }\frac{1}{b},
\displaystyle \text{then their combined one day's work is }\frac{1}{a}+\frac{1}{b}.

\displaystyle \text{Therefore, time taken by A and B working together}
\displaystyle =\frac{1}{\frac{1}{a}+\frac{1}{b}} =\frac{ab}{a+b}\text{ days}.

\displaystyle \text{Similarly, if several persons work together, their individual rates of work}
\displaystyle \text{are added to obtain their combined rate of work}.
\displaystyle \\

\displaystyle \textbf{6. Division of wages}

\displaystyle \text{When two or more persons work together, the total wages are divided among}
\displaystyle \text{them in the ratio of the amount of work actually done by each person}.

\displaystyle \text{Work done}=\text{Rate of work}\times\text{Time worked}.

\displaystyle \text{If A and B work for the same amount of time, their wages are divided in the}
\displaystyle \text{ratio of their efficiencies or their one-day work rates}.

\displaystyle \text{If they work for different lengths of time, the ratio of wages is}
\displaystyle (\text{A's efficiency}\times\text{A's time}): (\text{B's efficiency}\times\text{B's time}).
\displaystyle \\

\displaystyle \textbf{Important relationship}

\displaystyle \text{For a fixed amount of work, efficiency and time are inversely proportional}.

\displaystyle \text{Efficiency}\propto\frac{1}{\text{Time}}.

\displaystyle \therefore \frac{\text{Efficiency of A}}{\text{Efficiency of B}} =\frac{\text{Time taken by B}}{\text{Time taken by A}}.
\displaystyle \\

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