## Time and Work Questions - 1

A can do a piece of work in 12 days; B can do it in 18 days. Then how many days both working together can complete the work ?
• 7 $\displaystyle\frac{1}{5}$ days
• 8 $\displaystyle\frac{1}{2}$ days
• 8 days
• 7 $\displaystyle\frac{1}{2}$ days
• 7 days
Explanation

One day work of A = $\displaystyle \frac{1}{12}$, B = $\displaystyle \frac{1}{18}$

Working together A and B can complete = $\displaystyle \frac{1}{12} + \frac{1}{18} = \frac{3 + 2}{36} = \frac{5}{36}$ = 7 $\displaystyle \frac{1}{5}$ days.

Or

$\displaystyle \frac{12 \times 18}{12 + 18} = \frac{12 \times 18}{30} = \frac{36}{5}$ = 7 $\displaystyle\frac{1}{5}$ days.

Workspace
Working together P and Q can complete a work in â€˜xâ€™ no of days if P alone can complete the work in y days then in how many days Q alone can complete the work ?
• $\displaystyle \frac{ \ \ \ xy}{x + y}$
• $\displaystyle \frac{ \ \ \ xy}{x - y}$
• $\displaystyle \frac{y - x}{ \ \ xy}$
• $\displaystyle \frac{x + y}{ \ \ xy}$
• $\displaystyle \frac{ \ \ 2xy}{x + y}$
Explanation

$\displaystyle \frac{xy}{x + y}$

Workspace
Somu is twice as good work man as Jai, working together both can finish a work in 18 days. Then in how many days Jai alone can complete the work ?
• 27 days
• 36 days
• 54 days
• 57 days
• 42 days
Explanation

As Somu is twice as efficient as Jai then efficiency ratio will be = 2 : 1 ratio or [units]

Working together Somu and Jai can complete 2 + 1 = 3 units in a day

Total work = 18 x 3 = 54 units.

Somu = $\displaystyle \frac{54 \ \text{units}}{2 \ \text{units}}$ = 27 days

Jai = $\displaystyle \frac{54 \ \text{units}}{1 \ \text{unit}}$ = 54 days

Workspace
Rekha is thrice as capable worker as Jaya. If both working together can finish a work in 24 days, then in how many more days Jaya take to complete the work then Rekha ?
• 32 days
• 96 days
• 72 days
• 64 days
• 69 days
Explanation

Let one day work of Rekha can complete 3 units of work, while Jaya can 1 unit

Rekha + Jaya = 3 + 1 = 4 units in a day, total = 24 x 4 = 96 units.

Rekha alone $\displaystyle \frac{96}{3}$ = 32 days, Jaya = $\displaystyle \frac{96}{1}$ = 96 days.

Therefore Jaya takes 96 â€“ 32 = 64 days more then Rekha.

Workspace
Working together A and B can complete a work in 15 days. Or B and C can complete it in 18 days. Or A and C in 24 days in how many days B alone can complete the work ?
• 26 $\displaystyle \frac{24}{29}$
• 24 $\displaystyle \frac{24}{29}$
• 32 $\displaystyle \frac{8}{11}$
• 51 $\displaystyle \frac{3}{19}$
• 24 $\displaystyle \frac{23}{29}$
Explanation

B alone = $\displaystyle \frac{1}{2} \left((A + B)+(B + C)â€“(A + C)\right) = \frac{1}{2} \left(\frac{1}{15} + \frac{1}{18} - \frac{1}{24} \right)$

= $\displaystyle \frac{1}{2} \left(\frac{24 + 20 - 15}{360} \right) = \frac{1}{2} \left(\frac{29}{360} \right) = \frac{29}{720} = 24 \frac{24}{29}$ days.

Workspace
Jai can do a piece of work in 15 days. Veeru can finish the same work in 20 days. If they work at it on alternate days in how many days work would be completed, if the work is started with Veeru.
• 17 days
• 18 days
• 17 $\displaystyle \frac{1}{2}$ days
• 16 days
• 17 $\displaystyle \frac{1}{4}$ days
Explanation

If the work starts with Veeru.

Two days work of Jai and Veeru = $\displaystyle \frac{1}{15} + \frac{1}{20} = \frac{7}{60}$ th part.

In 16 days 8 $\displaystyle \frac{7}{60} = \frac{56}{60}$ th part.

On 17th day Veeruâ€™s turn $\displaystyle \frac{56}{60} + \frac{1}{20} = \frac{59}{60}$ th part.

Remaining work = 1 â€“ $\displaystyle \frac{59}{60} = \frac{1}{60}$ th part.

On 18th day Jaiâ€™s turn as Jai can complete the work in 15 days then $\displaystyle \frac{1}{60}$ th part in.

$\displaystyle \frac{1}{60} \times 15 = \frac{1}{4}$ th day, therefore the work will be completed in = 17 $\displaystyle \frac{1}{4}$ days.

Workspace
A man can do a piece of work in 12 days, with the help of a woman he can complete the work in 10 days, if they get Rs.12,000 for the work, then the share of the man is ?
• Rs.8000
• Rs.6000
• Rs.10,000
• Rs.7500
• Rs.8500
Explanation

One day work of Man = $\displaystyle \frac{1}{12}$, working together by Man and Woman = $\displaystyle \frac{1}{10}$

Woman alone = $\displaystyle \frac{1}{12} - \frac{1}{10} = \frac{5 - 6}{60} = \frac{1}{60}$ = 60 days.

Amount sharing ratio = Efficiency ratio.

Time Ratio = Man : woman = 12 : 60 or 1 : 5.

But efficiency ratio will be in inverse proportion to the time ratio.

Therefore efficiency ratio Man to Woman = 5 : 1

Man = 12,000 $\displaystyle \times \frac{5}{6}$ = Rs.10,000; Women = 12,000 $\displaystyle \times \frac{1}{6}$ = Rs.2,000.

Workspace
45 persons can complete a work in 72 days. How many persons with 1 $\displaystyle \frac{1}{2}$ times efficiency will complete the same piece of work in 24 days ?
• 72 persons
• 96 persons
• 64 persons
• 56 persons
• 90 persons
Explanation

Number of personâ€™s required to complete the work in 24 days, with as equally capable as 45 is

$\displaystyle M_2 = \frac{M_1 \times D_1}{D_2} => \frac{45 \times 72}{24}$ = 135 persons are required, but with 1 $\displaystyle \frac{1}{2}$ efficiency

135 Ã· 1 $\displaystyle \frac{1}{2}$ = 90 or 90 persons.

Workspace
If 12 persons can complete a work in 24 days. Then how many persons are required to complete the same work in 18 days ?
• 18 persons
• 17 persons
• 16 persons
• 15 persons
• 14 persons
Explanation

=> $\displaystyle M_2 = \frac{M_1 \times D_1}{D_2} = M_2 = \frac{12 \times 24}{18}$ = 16 or 16 persons.

Workspace
A can do a piece of work in 12 days B in 16 days. A left the job after 3 days. In how many days B can complete the remaining work ?
• 14 days
• 9 days
• 10 days
• 12 days
• 16 days
Explanation

The work completed by A and B in one day is = $\displaystyle \frac{1}{12} + \frac{1}{16} = \frac{4 + 3}{48} = \left(\frac{7}{48} \right)^{th}$ part.

In 3 days = $\displaystyle 3 \left(\frac{7}{48} \right) = \left(\frac{21}{48} \right)^{th}$ part.

Remaining = 1 - $\displaystyle \frac{21}{48} = \left(\frac{27}{48} \right)^{th}$ part.

B can complete in $\displaystyle \frac{27}{48} \times$ 16 = 9 days.

Workspace

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## Time and Work Formulas

1. If A can complete a work in â€˜nâ€™ number of days then in one day the work completed by A is $\displaystyle \frac{1}{n}$ part of work.

2. If, in one day A can complete $\displaystyle \frac{1}{n}$ th part of work then total work would be completed in â€˜nâ€™ number of days.

Note: Here â€˜nâ€™ denotes time, it can be Days/Hours/Minutes.

3. If A can complete a work in â€˜xâ€™ days while B can complete the same work in â€˜yâ€™ days. Then working together A and B can finish the work in one day = $\displaystyle \left(\frac{x + y}{xy}\right)^ {th}$ part, then total work = $\displaystyle \left(\frac{xy}{x + y}\right)$ days.

4. If â€˜mâ€™ number of personâ€™s can complete a work in â€˜nâ€™ number of days. Then working alone 1 person can complete the work in m x n = mn days.

5. If A and B working together can complete a work in â€˜xâ€™ days, if A alone can complete the work in â€˜yâ€™ days, then B alone can complete the work in = $\displaystyle \frac{1}{x} - \frac{1}{y} = \frac{xy}{x - y}$ days.

6. A and B can complete a work in â€˜xâ€™ days, B and C can complete the same work in â€˜yâ€™ days, A and C can complete the same work in â€˜zâ€™ days. Then working together A, B and C can complete the work in 1 day = $\displaystyle \frac{1}{2} \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)$ = $\displaystyle \frac{1}{2} \left(\frac{yz + xz + xy}{xyz}\right)$

Note: The same rules are also applicable to Pipes & Cisterns.

7. M1 x D1 = M2 x D2

M = Number of Persons

D = Number of days / hours / Minutes

M2 = $\displaystyle \frac{M_1 \times D_1}{D_2}$

D2 = $\displaystyle \frac{M_1 \times D_1}{M_2}$

The workerâ€™s left after

$\displaystyle \frac{M_1 \times D_1 - M_2 \times D_2}{M_1 - M_2}$

Time: