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Posted on : 06-09-2018 Posted by : Admin

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Let us try to understand the basic concept of work and time with the example of construction of a building. To complete the construction in stipulated time few points are to be noted,

- The more the number of workers, the faster the work is completed.
- The efficiency is different for different workers.
- The more the number of working hours, the more the work completed.

In the exam, we need to calculate the following in work and time questions

- Time taken to complete the given work.
- Time needed by single person to complete a given work
- Amount of work actually completed in given time by ‘n’ number of persons

One of the methods is to use the days as denominator while solving the questions. But, the best trick is to find the efficiency of the workers in percentage. When we say X can complete a given work in 2 days, it means that C complete 50% of the work in one day. The following table can be used to calculate the efficiency in percentage.

Days to complete work |
Work completed in one day |
Efficiency in percentage |
---|---|---|

x | $\frac{1}{\mathrm{x}}$ | $\frac{100}{\mathrm{x}}$% |

1 | $\frac{1}{1}$ | 100% |

2 | $\frac{1}{2}$ | 50% |

3 | $\frac{1}{3}$ | 33.33% |

4 | $\frac{1}{4}$ | 25% |

5 | $\frac{1}{5}$ | 20% |

6 | $\frac{1}{6}$ | 16.6% |

7 | $\frac{1}{7}$ | 14.2% |

8 | $\frac{1}{8}$ | 12.5% |

9 | $\frac{1}{9}$ | 11.1% |

10 | $\frac{1}{10}$ | 10% |

11 | $\frac{1}{11}$ | 9.09% |

- The work and persons are directly proportional to each other. In other words more work, more men or less work, less men
- The time and persons are inversely proportional to each other. In other words more men, less time or less men, more time
- The work and time are directly proportional to each other. In other words more work, more time or less work, less time

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The following are few formulae and short tricks for solving questions relating to work and time:

**Formula 1: **

If M_{1} persons can do W_{1} work in D_{1} days working T_{1} hours a day and M_{2} persons can do W_{2} work in D_{2} days working T_{2} hours a day then M_{1}W_{2}D_{1}T_{1}= M_{2}W_{1}D_{2}T_{2}.

*For more clarity on this formula click here... Proportionality concept*

**Example:** 5 people can finish 10 projects in 6 days working 6 hours a day. In how many days can 12 men complete 16 projects working 8 hours a day?

**Solution: **According to the above given formula, If M_{1} persons can do W_{1} work in D_{1} days working T_{1} hours a day and M_{2} persons can do W_{2} work in D_{2} days working T_{2} hours a day then,

M_{1}W_{2}D_{1}T_{1 }= M_{2}W_{1}D_{2}T_{2}

$5\times 16\times 6\times 6=12\times {\mathrm{D}}_{2}\times 8\times 10\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\mathrm{D}}_{2}=\frac{5\times 16\times 6\times 6}{12\times 8\times 10}=3\mathrm{days}$

Hence, 16 projects can be completed in 3 days by 12 men working 8 hours a day

**Formula 2: **

If person P can do a work in x days and person Q can do the same work in y days then, one day work of P and Q together is $\frac{1}{\mathrm{x}}{+}\frac{1}{\mathrm{y}}{=}\frac{\mathrm{x}+\mathrm{y}}{\mathrm{xy}}$

**Example:** If P can do a work in 5 days and Q can do a work in 7 days, then in how many days will P and Q working together will do the same work?

**Solution: **According to the above given formula, If person P can do a work in x days and person Q can do the same work in y days then, one day work of P and Q together is $\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}=\frac{\mathrm{x}+\mathrm{y}}{\mathrm{xy}}$

so, $\frac{{5}{+}{5}}{5\times 5}=\frac{10}{25}=0.4\mathrm{days}$

Hence, is both of them work together, work will be completed in 0.4 days

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**Formula 3:**

P and Q can do a work in x days, Q and R can do the same work in y days, P and R can do it in z days. If they all work together P, Q, R can do that work in $\frac{\mathrm{xyz}}{\mathrm{xy}+\mathrm{yz}+\mathrm{zx}}$

**Example: **If P, Q, R can do a piece of work in 6 days, 12 days and 5 days respectively. If all of them work together, how long will they take to complete the work?

**Solution: **According to the above given formula, P and Q can do a work in x days, Q and R can do the same work in y days, P and R can do it in z days. If they all work together P, Q, R can do that work in $\frac{\mathrm{xyz}}{\mathrm{xy}+\mathrm{yz}+\mathrm{zx}}$days.

So, $\frac{6\times 12\times 5}{(6\times 12)+(12\times 5)+(5\times 6)}{=}\frac{360}{72+60+30}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{=}\frac{360}{162}{=}{2}{.}{22}{}{\mathrm{days}}$

Hence, if all of them work together, the work will be completed in 2.22 days

**Formula 4:**

Consider P and Q two persons doing a work. If P takes x days more to complete a work than the time taken by (P+Q) to do the same work and Q takes y days more than the time taken by (P+Q) to do the same work, then (P+Q) together complete the same work in √xy days.

**Example: **Diya and Piya a completing some work. If Diya takes 9 days more to complete a work than the time taken by Piya and DIya together; Piya takes 4 days more than the time taken by DIya and Piya together, Then the time taken by both to finish the work is?

**Solution: **According to the above given formula, If P takes x days more to complete a work than the time taken by (P+Q) to do the same work and Q takes y days more than the time taken by (P+Q) to do the same work, then (P+Q) together complete the same work in √xy days..

so the time taken by both Diya and Piya to complete the work is

$\sqrt{9\times 4}=\sqrt{36}=6\mathrm{days}$

**Formula 5:**

Consider P and Q are two persons doing a work in a and b days respectively. Both begin together but if,

- P leaves the work x days before its completion, then total time taken for completion of work is, ${\mathrm{T}}{=}\frac{\left(\mathrm{a}+\mathrm{x}\right)\mathrm{b}}{\mathrm{a}+\mathrm{b}}$ days.
- Q leaves the work x days before the completion, the total time taken for completion of work is, ${\mathrm{T}}{=}\frac{\left(\mathrm{b}+\mathrm{x}\right)\mathrm{a}}{\mathrm{a}+\mathrm{b}}$ days.

**Example:** Rani and Neetu are doing some work. Rani can complete a piece of work in 10 days while Neetu can complete the same work in 15 days. Initially they begin together but 5 days before the completion of the work Rani has to leave. Find the total number of days for the completion of the work.

**Solution: **According to the above given formula, when two perons begin together but on person has to leave , then the total time for completion of the work can be calculated by ${\mathrm{T}}{=}\frac{\left(\mathrm{a}+\mathrm{x}\right)\mathrm{b}}{\mathrm{a}+\mathrm{b}}$ days

Here a= 10 days; b=15 days; x=5; T=?

${\mathrm{T}}{=}\frac{\left(10+5\right)15}{10+15}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{=}\frac{15\times 15}{25}{=}\frac{225}{25}{=}{9}{}{\mathrm{days}}$

Hence, total number of days to complete the work is 9 days.

**Formula 6:**

P and Q do a work in a and b days respectively. Both begin together, but after some days P leaves and the remaining work is completed by Q alone in x days. The time after which P left is, ${\mathrm{T}}{=}\frac{\left(b-\mathrm{x}\right)\mathrm{a}}{\mathrm{a}+\mathrm{b}}$ days.

**Example: **A and B do a piece of work in 40 days and 60 days respectively. Both begin together but after few days A has to leave. The remaining work is ompleted by B alone in 25 days. After how many days did A leave?

**Solution: **Given a=40 days, b=60 days, x=25 days. t=?

According to the formula, ${\mathrm{T}}{=}\frac{\left(b-\mathrm{x}\right)\mathrm{a}}{\mathrm{a}+\mathrm{b}}$ days.

$\frac{(60-25)\times 40}{40+60}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{=}\frac{35\times 40}{100}{=}{14}{}{\mathrm{days}}$

Hence, A left after 14 days

**Formula 7:**

If A and B can complete the work in x days and A alone can do the same work in y days, the the time taken by B to complete the same work will be given by $\frac{xy}{y-x}$days.

**Example: **A can complete a piece of work in 12 days. A and B together can complete the same work in 8 days. In how many days, can B alone complete the same peice of work?

**Solution: **Given x=8 days, y=12 days

According to the formula, $\frac{\mathrm{xy}}{\mathrm{y}-\mathrm{x}}$ days

$\frac{8\times 12}{12-8}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{=}\frac{96}{4}{=}{24}{}{\mathrm{days}}$

Hence, B alone can complete the work in 24 days.

**Formula 8:**

If A and B working together, can finish a piece of work in x days, B and C can finish in y days, C and A can finish in z days, then

- A, B and C working together, will finish the work in $\frac{2xyz}{xy+yz+zx}$days

- A alone can finish the work in $\frac{2xyz}{xy+yz-zx}$days

- B alone can finish the work in $\frac{2xyz}{yz+zx-xy}$days

**Formula 9:**

If A is *k* times more efficient than B and so able to finish the work in *l* days then,

- A and B working together, will finish the work in $\frac{kl}{{k}^{2}-1}$days

- A alone can finish the work in $\frac{l}{k-l}$days

- B alone can finish the work in $\frac{kl}{k-l}$days

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