Time and Work Aptitude

Cheatsheet Content

### Basics of Time and Work - **Work (W):** The amount of task to be completed. Usually assumed as 1 unit if not specified. - **Time (T):** The duration taken to complete the work. - **Efficiency (E):** The amount of work done per unit of time. - **Fundamental Relation:** Work = Efficiency × Time (W = E × T) - **Inverse Proportionality:** Efficiency is inversely proportional to Time for a constant amount of work. - If E increases, T decreases (and vice-versa). - $E \propto \frac{1}{T}$ or $E_1 T_1 = E_2 T_2$ (for same work) - **Units:** Ensure consistency in units for time (hours, days, minutes) and work (units, items, fraction of total work). ### Individual Work - If a person can complete a work in 'D' days, then the work done by that person in 1 day is $\frac{1}{D}$ of the total work. - If a person does $\frac{1}{D}$ of the work in 1 day, then they will complete the total work in 'D' days. - **Example:** If A can do a piece of work in 10 days, A's 1-day work is $\frac{1}{10}$. ### Combined Work - **Adding Efficiencies:** If multiple people work together, their individual efficiencies (work per unit time) add up. - If A can do a work in $D_A$ days and B can do the same work in $D_B$ days: - A's 1-day work = $\frac{1}{D_A}$ - B's 1-day work = $\frac{1}{D_B}$ - (A+B)'s 1-day work = $\frac{1}{D_A} + \frac{1}{D_B} = \frac{D_A + D_B}{D_A D_B}$ - Time taken by A and B together to complete the work = $\frac{D_A D_B}{D_A + D_B}$ days. - For three people A, B, C working together: - (A+B+C)'s 1-day work = $\frac{1}{D_A} + \frac{1}{D_B} + \frac{1}{D_C}$ - Time taken by A, B, C together = $\frac{1}{\frac{1}{D_A} + \frac{1}{D_B} + \frac{1}{D_C}}$ days. ### Ratio Method for Efficiency & Time - If the ratio of efficiencies of A and B is $E_A : E_B = x : y$, then the ratio of time taken by them to complete the same work is $T_A : T_B = y : x$. - **Example:** If A is twice as efficient as B ($E_A : E_B = 2 : 1$), then A will take half the time B takes ($T_A : T_B = 1 : 2$). ### LCM Method (Unitary Method) - This is a highly effective method for solving Time and Work problems, especially when multiple people or varying times are involved. - **Steps:** 1. Find the LCM of the days/hours given for each person/entity to complete the work. This LCM represents the 'Total Work Units'. 2. Calculate the 'Efficiency' (work units per day/hour) for each person/entity by dividing the Total Work Units by their respective days/hours. 3. Use these efficiencies to solve the problem (e.g., find combined efficiency, calculate time for a fraction of work, etc.). - **Example:** A takes 10 days, B takes 15 days. - LCM(10, 15) = 30 units (Total Work). - A's efficiency = $\frac{30}{10} = 3$ units/day. - B's efficiency = $\frac{30}{15} = 2$ units/day. - Combined efficiency (A+B) = $3+2=5$ units/day. - Time for A+B to complete work = $\frac{30}{5} = 6$ days. ### Men, Days, and Hours Formula (Chain Rule) - If $M_1$ men can do $W_1$ work in $D_1$ days, working $H_1$ hours per day, and $M_2$ men can do $W_2$ work in $D_2$ days, working $H_2$ hours per day, then: - $$\frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2}$$ - **Important:** If any parameter is constant or not given, exclude it from the formula. - If work is the same ($W_1 = W_2$), then $M_1 D_1 H_1 = M_2 D_2 H_2$. - If hours are constant, then $\frac{M_1 D_1}{W_1} = \frac{M_2 D_2}{W_2}$. - If work and hours are constant, then $M_1 D_1 = M_2 D_2$. (Inverse proportionality between Men and Days) ### Alternating Work - When people work on alternate days/hours. - **Steps:** 1. Calculate the work done by each person per day/hour. 2. Calculate the total work done in a cycle (e.g., 2 days if two people alternate). 3. Find how many full cycles are needed to complete most of the work. 4. Calculate the remaining work and the time taken by the person who starts the next turn. - **Example:** A does work in 10 days, B in 15 days. They work alternately, A starting. - A's 1-day work = $\frac{1}{10}$, B's 1-day work = $\frac{1}{15}$. - Work in 2 days (1 cycle) = $\frac{1}{10} + \frac{1}{15} = \frac{3+2}{30} = \frac{5}{30} = \frac{1}{6}$ of total work. - Total work = 1. To complete 1 unit of work, it takes $6 \times 2 = 12$ days. (This is for a simple case where work completes exactly on a cycle end). - If not exact, say 5 cycles complete $\frac{5}{6}$ work in 10 days. Remaining work = $\frac{1}{6}$. A does it in $\frac{1/6}{1/10} = \frac{10}{6} = 1\frac{2}{3}$ days. Total time = $10 + 1\frac{2}{3} = 11\frac{2}{3}$ days. ### Pipes and Cisterns - This is a direct application of Time and Work principles. - **Inlet Pipe:** Fills the tank (positive work). Its efficiency is positive. - **Outlet Pipe/Leak:** Empties the tank (negative work). Its efficiency is negative. - If an inlet pipe fills a tank in 'x' hours, it fills $\frac{1}{x}$ of the tank in 1 hour. - If an outlet pipe empties a tank in 'y' hours, it empties $\frac{1}{y}$ of the tank in 1 hour. - If both are open: - Net work done in 1 hour = $\frac{1}{x} - \frac{1}{y}$ (assuming $x ### Negative Work (Leaks, Destroyers) - When some entity works against the completion of the task (e.g., a leak in a tank, a person destroying work). - Treat efficiency of negative work as negative. - **Total Work Rate = (Sum of Positive Work Rates) - (Sum of Negative Work Rates)** - **Example:** A builds a wall in 20 days. B destroys the same wall in 30 days. - A's 1-day work = $\frac{1}{20}$. - B's 1-day work = $-\frac{1}{30}$ (negative as it's destroying). - Combined 1-day work (if both work) = $\frac{1}{20} - \frac{1}{30} = \frac{3-2}{60} = \frac{1}{60}$. - Time to build wall = 60 days. ### Shared Wages - Wages are always distributed in proportion to the work done by each person, or in proportion to their efficiencies if they work for the same amount of time. - If A and B complete a work together for 'Total Wage': - Ratio of work done by A : B = $\frac{1}{D_A} : \frac{1}{D_B}$ (if they work for full duration). - A's share = Total Wage $\times \frac{\text{Work done by A}}{\text{Total Work Done}}$ - A's share = Total Wage $\times \frac{\text{A's Efficiency}}{\text{Total Efficiency}}$ (if time is same) - **Example:** A and B complete a work for Rs. 1000. A does $\frac{2}{5}$ of the work, B does the rest. - B does $1 - \frac{2}{5} = \frac{3}{5}$ of the work. - Ratio of work A:B = $\frac{2}{5} : \frac{3}{5} = 2:3$. - A's share = $\frac{2}{5} \times 1000 = Rs. 400$. - B's share = $\frac{3}{5} \times 1000 = Rs. 600$. ### Group Efficiency - When a group of people (e.g., men, women, children) have different individual efficiencies. - **Steps:** 1. Establish a relationship between the efficiencies of different groups (e.g., 1 Man = 2 Women). 2. Convert all workers into a single type (e.g., all men or all women). 3. Apply the Men-Days-Hours formula or LCM method. - **Example:** 6 Men or 8 Women can complete a work in 10 days. - This means 6 Men have the same efficiency as 8 Women. - $6M = 8W \Rightarrow 3M = 4W \Rightarrow 1M = \frac{4}{3}W$. - To find time taken by 3 Men and 4 Women: - Convert to Women: 3 Men + 4 Women = $3(\frac{4}{3}W) + 4W = 4W + 4W = 8W$. - Since 8 Women complete work in 10 days, 3 Men and 4 Women will also complete it in 10 days. - Alternatively, convert to Men: 3 Men + 4 Women = $3M + 4(\frac{3}{4}M) = 3M + 3M = 6M$. - Since 6 Men complete work in 10 days, 3 Men and 4 Women will also complete it in 10 days. ### Problem Solving Tips - **Understand the Question:** Clearly identify what is given (time, number of workers, total work, conditions) and what needs to be found. - **Choose a Method:** - **LCM Method:** Best for most problems involving multiple workers and different times. - **Fraction Method:** Good for simple combined work problems or when work is given as a fraction. - **M-D-H Formula:** Ideal for problems involving changes in men, days, hours, or work quantity. - **Standardize Units:** Ensure all time units (days, hours, minutes) are consistent. - **Break Down Complex Problems:** For scenarios like "work starts with A, B joins after some days, C leaves", break the problem into stages. - **Assume Total Work:** If total work is not given, assume it to be 1 unit or use the LCM of times as total work units. - **Negative Work:** Always account for negative work (e.g., leaks, destroyers) by subtracting their efficiency. - **Wages:** Remember wages are proportional to work done. - **Practice:** Consistent practice with varied problem types is key to mastering Time and Work.