## Time and Work

1. A can do a work in 20 days and B in 30 days. In how many days they can complete the work if they work together.

A. 12 days

B. 15 days

C. 18 days

D. 20 days

Option A

Explanation:

$\par&space;A's&space;one&space;day's&space;work&space;=$  $\frac{1}{20}$

$\par&space;B's&space;one&space;day's&space;work&space;=$  $\frac{1}{30}$

$\par&space;(A&space;+&space;B)'s&space;one&space;day&space;work&space;=$ $\frac{1}{20}&space;+&space;\frac{1}{30}&space;=&space;\frac{3&space;+&space;2}{60}&space;=&space;\frac{5}{60}&space;=&space;\frac{1}{12}$

$\par&space;A&space;and&space;B&space;can&space;complete&space;the&space;work&space;in&space;12&space;days$

$\par&space;\underline{Shortcut&space;Method}$

$\par&space;If&space;A&space;can&space;do&space;a&space;work&space;in$ $t_{1}$ $\par&space;days&space;and&space;B&space;can&space;do&space;it&space;in$ $t_{2}&space;days$

$\par&space;then&space;A&space;and&space;B&space;Together&space;can&space;do&space;same&space;work&space;in$ $\frac{t_{1&space;\times&space;}&space;t&space;_&space;{2}}{t_{1}+&space;t_{2}}$ days.

$\frac{20&space;\times&space;30}{20+50}$ $=&space;\frac{600}{50}&space;=&space;12$

2. A, B and C complete a work in 6, 8 and 12 days respectively. In how many days they can complete the work together.

A. 2 2/3 days

B. 8/2 days

C. 2 3/2 days

D. 3 3/2 days

Option A

Explanation:

$\par&space;A's&space;one&space;day's&space;work&space;=$  $\frac{1}{6}$

$\par&space;B's&space;one&space;day's&space;work&space;=$  $\frac{1}{8}$

$\par&space;C's&space;one&space;day's&space;work&space;=$  $\frac{1}{12}$

$\par&space;(A&space;+&space;B&space;+&space;C)'s&space;one&space;day&space;work&space;=$ $\frac{1}{6}&space;+&space;\frac{1}{8}+\frac{1}{12}&space;=&space;\frac{4+3+2}{24}&space;=&space;\frac{9}{24}&space;=&space;\frac{3}{8}$

$\par&space;(A+B+C)&space;will&space;complete&space;the&space;work&space;=$  $1\div&space;\frac{3}{8}&space;=&space;\frac{8}{3}&space;=&space;2\frac{2}{3}$

$\par&space;\underline{Shortcut&space;Method}$

$\frac{6\times&space;8\times&space;12}{6\times&space;8+6\times&space;12+8\times&space;12}&space;=&space;\frac{8}{3}&space;=&space;2\frac{2}{3}$                                          $\lbrack&space;\frac{t_{1&space;\times&space;}t_{2&space;\times&space;}t_{3}}{t_{1}&space;t_{2}+t_{1}&space;t_{3}+t_{2}t_{3}}&space;\rbrack$

3. A and B together complete a work in 12 days. B alone can complete the work in 28 days. A alone can do the work in how many days.

A. 18 days

B. 20 days

C. 21 days

D. 22 days

Option C

Explanation:

$\par&space;(A+B)'s&space;one&space;day's&space;work&space;=$ $\frac{1}{12}$

$\par&space;B's&space;one&space;day&space;work&space;=$ $\frac{1}{28}$                                                                       $\lbrack&space;(A+B)-&space;(B)&space;=&space;A\rbrack$

$\par&space;A's&space;one&space;day&space;work&space;=$ $\frac{1}{12}&space;-&space;\frac{1}{28}&space;=&space;\frac{7-3}{84}&space;=&space;\frac{4}{84}&space;=&space;\frac{1}{21}$

$\par&space;So,&space;A&space;can&space;do&space;it&space;in&space;21&space;days$

$\par&space;\underline{Shortcut&space;Method}$

$\frac{12\times&space;28}{28-12}&space;=&space;\frac{336}{16}&space;=&space;21$                                                                                 $t_{2}&space;=&space;\frac{T&space;\times&space;t_{1}}{t_{1}-T}$

4. A, B complete a work in 25, 20 days respectively. after working together for 5 days A left. B complete the work in

A. 10 days

B. 11 days

C. 12 days

D. 15 days

Option B

Explanation:

$\par&space;(A+B)'s&space;5&space;days&space;work&space;=$  $5\left&space;[\frac{1}{25}+\frac{1}{20}\right]$ $=$  $5&space;\left&space;[\frac{4+5}{100}\right]$ $=$  $5&space;\left&space;[&space;\frac{9}{100}\right]$ $=$  $\frac{9}{20}$

$\par&space;Remaining&space;work&space;=$  $1&space;-&space;\frac{9}{20}&space;=&space;\frac{11}{20}$

$\par&space;B&space;can&space;complete$ $\frac{11}{20}$  $\par&space;work&space;=$  $\frac{11}{20}&space;\times&space;20&space;=&space;11s$

5. A, B can dig a well in 10 days and 12 days respectively. If Rs.11 is for total work what is A share.

A. Rs. 5

B. Rs. 6

C. Rs. 7

D. Rs. 8

Option B

Explanation:

$\par&space;A's&space;one&space;day's&space;work&space;=$  $\frac{1}{10}$

$\par&space;B's&space;one&space;day's&space;work&space;=$  $\frac{1}{12}$

$\par&space;Ratio&space;of&space;(A+B)'s&space;one&space;day&space;work&space;=$  $\frac{1}{10}&space;:&space;\frac{1}{12}&space;=&space;6&space;:&space;5$

$\par&space;Amount&space;should&space;be&space;shared&space;in&space;6&space;:&space;5&space;ratio$

$\par&space;A's&space;share&space;=$  $\frac{6}{11}&space;\times&space;11$ $\par&space;=&space;Rs.&space;6$

6. A does a work in 10 days and B does the same work in 15 Days. In how many days they together will do the same work.

A. 5 days

B. 6 days

C. 8 days

D. 9 days

Option B

Explanation:

$\par&space;A's&space;one&space;day's&space;work&space;=$  $\frac{1}{10}$

$\par&space;B's&space;one&space;day's&space;work$ $=&space;\frac{1}{15}$

$\par&space;(A&space;+&space;B)'s&space;one&space;days&space;work&space;=$ $\left&space;[&space;\frac{1}{10}+\frac{1}{15}&space;\right&space;]$  $=&space;\frac{1}{6}$

$\par&space;So,&space;both&space;together&space;will&space;finish&space;the&space;work&space;in&space;6&space;days.$

$\par&space;\underline{Shortcut&space;method}$

$\frac{10&space;\times&space;15}{10&space;+&space;15}&space;=&space;\frac{150}{25}&space;=&space;6$

7. A can finish a work in 18 days and B can do the same work in half the time taken by A. Then, working together, what part of the same work they can finish in a day ?

A. 1/6

B. 1/9

C. 2/5

D. 2/7

Option A

Explanation:

$\par&space;A's&space;one&space;day's&space;work&space;=$  $\frac{1}{18}$

$\par&space;B's&space;one&space;day's&space;work&space;=$  $\frac{1}{9}$

$\par&space;(A+B)'s&space;one&space;day's&space;work&space;=$ $\frac{1}{18}+\frac{1}{9}&space;=&space;\frac{1}{6}$

8. A tyre has two punctures. The first puncture alone would have made the tyre flat in 9 minutes and the second alone would have done it in 6 minutes. If air leaks out at a constant rate, how long does it take both the punctures together to make it flat ?

A. 1 1/2

B. 3 1/2

C. 3 3/5

D. 4 1/4

Option C

Explanation:

$\par&space;One&space;minute's&space;work&space;of&space;both&space;the&space;punctures&space;=$  $\frac{1}{9}+\frac{1}{6}&space;=&space;\frac{5}{18}$

$\par&space;So,&space;both&space;punctures&space;will&space;make&space;the&space;tyre&space;flat&space;in$ $\frac{18}{5}&space;=&space;3\frac{3}{5}$ minutes

9. A, B and C can complete a piece of work in 24, 6 and 12 days respectively. Working together, they will complete the same work in

A. 2 1/2 days

B. 2 4/5 days

C. 3 3/7 days

D. 4 days

Option C

Explanation:

$\par&space;(A&space;+&space;B&space;+&space;C)'s&space;one&space;day's&space;work&space;=$  $\frac{1}{24}+\frac{1}{6}+\frac{1}{12}&space;=&space;\frac{7}{24}$

$\par&space;So,&space;A,&space;B&space;and&space;C&space;together&space;will&space;complete&space;the&space;job&space;in$ $\frac{24}{7}&space;=&space;3\frac{3}{7}$ $\par&space;days$

10. A man can do a job in 15 days. His father takes 20 days and his son finishes it in 25 days. How long will they take to complete the job if they all work together ?

A. Less than 6 days

B. Exactly 6 days

C. Approximately 6.4 days

D. More than 10 days

$\par&space;One&space;day's&space;work&space;of&space;the&space;three&space;persons&space;=$ $\frac{1}{15}+\frac{1}{20}+\frac{1}{25}&space;=&space;\frac{47}{300}$
$\par&space;So,&space;all&space;the&space;three&space;together&space;will&space;complete&space;the&space;work&space;in$ $\frac{300}{47}&space;=&space;6.4$  $\par&space;days$