\displaystyle \textbf{Question 1: }A\text{ can do a piece of work in }15\text{ days, while }B \text{ can do it in }10\text{ days}.
\displaystyle \text{How long will they take together to complete the work?}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{15}.
\displaystyle \text{B's one day's work}=\frac{1}{10}.
\displaystyle \text{A's and B's combined one day's work} =\frac{1}{15}+\frac{1}{10}.
\displaystyle =\frac{2+3}{30}=\frac{5}{30}=\frac{1}{6}.
\displaystyle \text{Time taken together}=\frac{1}{1/6}=6\text{ days}.
\displaystyle \therefore \text{A and B together can complete the work in }6\text{ days}.
\displaystyle \\

\displaystyle \textbf{Question 2: }A,\ B\text{ and }C\text{ can do a piece of work in }12\text{ days,}
\displaystyle 15\text{ days and }10\text{ days respectively. In what time will they together}
\displaystyle \text{finish the work?}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{12}.
\displaystyle \text{B's one day's work}=\frac{1}{15}.
\displaystyle \text{C's one day's work}=\frac{1}{10}.
\displaystyle \text{Their combined one day's work} =\frac{1}{12}+\frac{1}{15}+\frac{1}{10}.
\displaystyle =\frac{5+4+6}{60}=\frac{15}{60}=\frac{1}{4}.
\displaystyle \text{Time taken together}=\frac{1}{1/4}=4\text{ days}.
\displaystyle \therefore \text{A, B and C together can complete the work in }4\text{ days}.
\displaystyle \\

\displaystyle \textbf{Question 3: }A\text{ and }B\text{ together can do a piece of work in }35 \text{ days, while }A
\displaystyle \text{alone can do it in }60\text{ days. How long would }B\text{ alone take to do it?}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{60}.
\displaystyle \text{A's and B's combined one day's work}=\frac{1}{35}.
\displaystyle \text{B's one day's work}=\frac{1}{35}-\frac{1}{60}.
\displaystyle =\frac{12-7}{420}=\frac{5}{420}=\frac{1}{84}.
\displaystyle \text{Time taken by B alone}=\frac{1}{1/84}=84\text{ days}.
\displaystyle \therefore B\text{ alone would take }84\text{ days to complete the work}.
\displaystyle \\

\displaystyle \textbf{Question 4: }A\text{ can do a piece of work in }20\text{ days, while }B \text{ can do it in }15\text{ days}.
\displaystyle \text{With the help of }C,\text{ they finish the work in }5\text{ days. In what time}
\displaystyle \text{would }C\text{ alone do it?}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{20}.
\displaystyle \text{B's one day's work}=\frac{1}{15}.
\displaystyle \text{A's, B's and C's combined one day's work}=\frac{1}{5}.
\displaystyle \text{C's one day's work} =\frac{1}{5}-\frac{1}{20}-\frac{1}{15}.
\displaystyle =\frac{12-3-4}{60}=\frac{5}{60}=\frac{1}{12}.
\displaystyle \text{Time taken by C alone}=\frac{1}{1/12}=12\text{ days}.
\displaystyle \therefore C\text{ alone would take }12\text{ days to complete the work}.
\displaystyle \\

\displaystyle \textbf{Question 5: }A\text{ can do a piece of work in }12\text{ days and }B \text{ alone can do it in }16\text{ days}.
\displaystyle \text{They worked together for }3\text{ days and then }A\text{ left. How long did }B
\displaystyle \text{take to finish the remaining work?}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{12}.
\displaystyle \text{B's one day's work}=\frac{1}{16}.
\displaystyle \text{A's and B's combined one day's work} =\frac{1}{12}+\frac{1}{16}.
\displaystyle =\frac{4+3}{48}=\frac{7}{48}.
\displaystyle \text{Work completed by A and B in }3\text{ days} =3\times\frac{7}{48}=\frac{7}{16}.
\displaystyle \text{Remaining work}=1-\frac{7}{16}=\frac{9}{16}.
\displaystyle \text{Time taken by B to complete the remaining work} =\frac{9/16}{1/16}.
\displaystyle =\frac{9}{16}\times16=9\text{ days}.
\displaystyle \therefore B\text{ will take }9\text{ days to finish the remaining work}.
\displaystyle \\

\displaystyle \textbf{Question 6: }A\text{ can do }\frac{1}{4}\text{ of a work in }5\text{ days, while }B
\displaystyle \text{can do }\frac{1}{5}\text{ of the work in }6\text{ days. In how many days can both}
\displaystyle \text{complete the work together?}
\displaystyle \text{Answer:}
\displaystyle A\text{ can do }\frac{1}{4}\text{ of the work in }5\text{ days}.
\displaystyle \therefore A\text{ can complete the whole work in }5\times4=20\text{ days}.
\displaystyle \text{A's one day's work}=\frac{1}{20}.
\displaystyle B\text{ can do }\frac{1}{5}\text{ of the work in }6\text{ days}.
\displaystyle \therefore B\text{ can complete the whole work in }6\times5=30\text{ days}.
\displaystyle \text{B's one day's work}=\frac{1}{30}.
\displaystyle \text{A's and B's combined one day's work} =\frac{1}{20}+\frac{1}{30}.
\displaystyle =\frac{3+2}{60}=\frac{5}{60}=\frac{1}{12}.
\displaystyle \text{Time taken together}=\frac{1}{1/12}=12\text{ days}.
\displaystyle \therefore A\text{ and }B\text{ together can complete the work in }12\text{ days}.
\displaystyle \\

\displaystyle \textbf{Question 7: }A\text{ can dig a trench in }6\text{ days, while }B\text{ can dig it}
\displaystyle \text{in }8\text{ days. They worked together and received Rs. }1120\text{ for it.}
\displaystyle \text{Find the share of each.}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{6}.
\displaystyle \text{B's one day's work}=\frac{1}{8}.
\displaystyle \text{Since they work for the same time, their shares are in the ratio of their}
\displaystyle \text{one-day work rates}.
\displaystyle A:B=\frac{1}{6}:\frac{1}{8}=8:6=4:3.
\displaystyle \text{Total number of parts}=4+3=7.
\displaystyle \text{A's share}=\frac{4}{7}\times1120=\text{Rs. }640.
\displaystyle \text{B's share}=\frac{3}{7}\times1120=\text{Rs. }480.
\displaystyle \therefore \text{A's share is Rs. }640\text{ and B's share is Rs. }480.
\displaystyle \\

\displaystyle \textbf{Question 8: }A\text{ can mow a field in }9\text{ days, }B\text{ can mow it in }12
\displaystyle \text{days, while }C\text{ can mow it in }8\text{ days. They all mowed the field}
\displaystyle \text{together and received Rs. }1610\text{ for it. How will the money be shared?}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{9}.
\displaystyle \text{B's one day's work}=\frac{1}{12}.
\displaystyle \text{C's one day's work}=\frac{1}{8}.
\displaystyle \text{Since they work for the same time, their shares are in the ratio of their}
\displaystyle \text{one-day work rates}.
\displaystyle A:B:C=\frac{1}{9}:\frac{1}{12}:\frac{1}{8}.
\displaystyle =8:6:9.
\displaystyle \text{Total number of parts}=8+6+9=23.
\displaystyle \text{A's share}=\frac{8}{23}\times1610=\text{Rs. }560.
\displaystyle \text{B's share}=\frac{6}{23}\times1610=\text{Rs. }420.
\displaystyle \text{C's share}=\frac{9}{23}\times1610=\text{Rs. }630.
\displaystyle \therefore \text{the shares of A, B and C are Rs. }560,\text{ Rs. }420 \text{ and Rs. }630\text{ respectively}.
\displaystyle \\

\displaystyle \textbf{Question 9: }A\text{ and }B\text{ can do a piece of work in }30\text{ days;}
\displaystyle B\text{ and }C\text{ in }24\text{ days; }C\text{ and }A\text{ in }40\text{ days. How long}
\displaystyle \text{will they take together? In what time can each finish it working alone?}
\displaystyle \text{Answer:}
\displaystyle \text{A's and B's combined one day's work}=\frac{1}{30}.
\displaystyle \text{B's and C's combined one day's work}=\frac{1}{24}.
\displaystyle \text{C's and A's combined one day's work}=\frac{1}{40}.
\displaystyle 2\times\text{(A's + B's + C's one day's work)}
\displaystyle =\frac{1}{30}+\frac{1}{24}+\frac{1}{40}.
\displaystyle =\frac{4+5+3}{120}=\frac{12}{120}=\frac{1}{10}.
\displaystyle \therefore \text{A's + B's + C's one day's work}=\frac{1}{20}.
\displaystyle \therefore \text{A, B and C together can complete the work in }20\text{ days}.

\displaystyle \text{A's one day's work} =\frac{1}{20}-\frac{1}{24}=\frac{1}{120}.
\displaystyle \therefore A\text{ alone can complete the work in }120\text{ days}.

\displaystyle \text{B's one day's work} =\frac{1}{20}-\frac{1}{40}=\frac{1}{40}.
\displaystyle \therefore B\text{ alone can complete the work in }40\text{ days}.

\displaystyle \text{C's one day's work} =\frac{1}{20}-\frac{1}{30}=\frac{1}{60}.
\displaystyle \therefore C\text{ alone can complete the work in }60\text{ days}.
\displaystyle \\

\displaystyle \textbf{Question 10: }A\text{ can do a piece of work in }80\text{ days. He works at it}
\displaystyle \text{for }10\text{ days and then }B\text{ alone finishes the remaining work in }42
\displaystyle \text{days. In how many days could both complete the work together?}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{80}.
\displaystyle \text{Work completed by A in }10\text{ days} =10\times\frac{1}{80}=\frac{1}{8}.
\displaystyle \text{Remaining work}=1-\frac{1}{8}=\frac{7}{8}.
\displaystyle B\text{ completes }\frac{7}{8}\text{ of the work in }42\text{ days}.
\displaystyle \text{B's one day's work}=\frac{7/8}{42} =\frac{7}{336}=\frac{1}{48}.
\displaystyle \therefore B\text{ alone can complete the whole work in }48\text{ days}.
\displaystyle \text{A's and B's combined one day's work} =\frac{1}{80}+\frac{1}{48}.
\displaystyle =\frac{3+5}{240}=\frac{8}{240}=\frac{1}{30}.
\displaystyle \therefore A\text{ and }B\text{ together can complete the work in }30\text{ days}.
\displaystyle \\

\displaystyle \textbf{Question 11: }A\text{ and }B\text{ can together finish a work in }30\text{ days.}
\displaystyle \text{They worked at it for }20\text{ days and then }B\text{ left. The remaining work}
\displaystyle \text{was done by }A\text{ alone in }20\text{ more days. In how many days can }A
\displaystyle \text{alone do it?}
\displaystyle \text{Answer:}
\displaystyle \text{A's and B's combined one day's work}=\frac{1}{30}.
\displaystyle \text{Work completed by A and B in }20\text{ days} =20\times\frac{1}{30}=\frac{2}{3}.
\displaystyle \text{Remaining work}=1-\frac{2}{3}=\frac{1}{3}.
\displaystyle A\text{ completes }\frac{1}{3}\text{ of the work in }20\text{ days}.
\displaystyle \text{A's one day's work}=\frac{1/3}{20}=\frac{1}{60}.
\displaystyle \text{Time taken by A alone}=\frac{1}{1/60}=60\text{ days}.
\displaystyle \therefore A\text{ alone can complete the work in }60\text{ days}.
\displaystyle \\

\displaystyle \textbf{Question 12: }A\text{ can do a certain job in }25\text{ days, while }B \text{ alone can do it in }20\text{ days}.
\displaystyle A\text{ started the work and was joined by }B\text{ after }10\text{ days. In how many}
\displaystyle \text{days was the whole work completed?}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{25}.
\displaystyle \text{B's one day's work}=\frac{1}{20}.
\displaystyle \text{Work completed by A in }10\text{ days} =10\times\frac{1}{25}=\frac{2}{5}.
\displaystyle \text{Remaining work}=1-\frac{2}{5}=\frac{3}{5}.
\displaystyle \text{A's and B's combined one day's work} =\frac{1}{25}+\frac{1}{20}.
\displaystyle =\frac{4+5}{100}=\frac{9}{100}.
\displaystyle \text{Time taken by A and B to complete the remaining work} =\frac{3/5}{9/100}.
\displaystyle =\frac{3}{5}\times\frac{100}{9} =\frac{20}{3}=6\frac{2}{3}\text{ days}.
\displaystyle \text{Total time taken} =10+6\frac{2}{3}=16\frac{2}{3}\text{ days}.
\displaystyle \therefore \text{the whole work was completed in }16\frac{2}{3}\text{ days}.
\displaystyle \\

\displaystyle \textbf{Question 13: }A\text{ can do a piece of work in }14\text{ days, while }B \text{ can do it in }21\text{ days}.
\displaystyle \text{They begin together, but }3\text{ days before the completion of the work, }A
\displaystyle \text{leaves. Find the total number of days taken to complete the work.}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{14}.
\displaystyle \text{B's one day's work}=\frac{1}{21}.
\displaystyle \text{A's and B's combined one day's work} =\frac{1}{14}+\frac{1}{21}.
\displaystyle =\frac{3+2}{42}=\frac{5}{42}.
\displaystyle \text{Since A leaves }3\text{ days before completion, B works alone for the last}
\displaystyle 3\text{ days}.
\displaystyle \text{Work completed by B in the last }3\text{ days} =3\times\frac{1}{21}=\frac{1}{7}.
\displaystyle \text{Work completed earlier by A and B together} =1-\frac{1}{7}=\frac{6}{7}.
\displaystyle \text{Time taken by A and B to complete }\frac{6}{7}\text{ of the work} =\frac{6/7}{5/42}.
\displaystyle =\frac{6}{7}\times\frac{42}{5} =\frac{36}{5}=7\frac{1}{5}\text{ days}.
\displaystyle \text{Total time taken} =7\frac{1}{5}+3=10\frac{1}{5}\text{ days}.
\displaystyle \therefore \text{the work was completed in }10\frac{1}{5}\text{ days}.
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{A is thrice as good a workman as B and B is twice}
\displaystyle \text{as good a workman as C. All three took up a job and received Rs. }1800
\displaystyle \text{as remuneration. Find the share of each.}
\displaystyle \text{Answer:}
\displaystyle \text{Let C's efficiency be }1.
\displaystyle \text{Since B is twice as efficient as C, B's efficiency}=2.
\displaystyle \text{Since A is thrice as efficient as B, A's efficiency}=3\times2=6.
\displaystyle \therefore \text{Efficiency ratio of A, B and C}=6:2:1.
\displaystyle \text{Since they work for the same time, their shares are in the ratio of}
\displaystyle \text{their efficiencies}.
\displaystyle \text{Total number of parts}=6+2+1=9.
\displaystyle \text{A's share}=\frac{6}{9}\times1800=\text{Rs. }1200.
\displaystyle \text{B's share}=\frac{2}{9}\times1800=\text{Rs. }400.
\displaystyle \text{C's share}=\frac{1}{9}\times1800=\text{Rs. }200.
\displaystyle \therefore \text{A's share is Rs. }1200,\text{ B's share is Rs. }400\text{ and}
\displaystyle \text{C's share is Rs. }200.
\displaystyle \\

\displaystyle \textbf{Question 15: }A\text{ can do a certain job in }12\text{ days. }B\text{ is }60\% \text{ more efficient than }A.
\displaystyle \text{Find the number of days taken by }B\text{ to finish the job.}
\displaystyle \text{Answer:}
\displaystyle \text{A's one day's work}=\frac{1}{12}.
\displaystyle B\text{ is }60\%\text{ more efficient than }A.
\displaystyle \text{B's efficiency}=160\%\text{ of A's efficiency} =\frac{160}{100}=\frac{8}{5}.
\displaystyle \text{B's one day's work} =\frac{8}{5}\times\frac{1}{12}=\frac{2}{15}.
\displaystyle \text{Time taken by B}=\frac{1}{2/15} =\frac{15}{2}=7\frac{1}{2}\text{ days}.
\displaystyle \therefore B\text{ will take }7\frac{1}{2}\text{ days to finish the job}.
\displaystyle \\

\displaystyle \textbf{Question 16: }A\text{ is twice as good a workman as }B\text{ and together they}
\displaystyle \text{finish a piece of work in }14\text{ days. In how many days can }A\text{ alone}
\displaystyle \text{do it?}
\displaystyle \text{Answer:}
\displaystyle \text{Since A is twice as efficient as B, their efficiency ratio is }2:1.
\displaystyle \text{Let B's one day's work}=x.
\displaystyle \therefore \text{A's one day's work}=2x.
\displaystyle \text{Their combined one day's work}=x+2x=3x.
\displaystyle \text{Since they complete the work together in }14\text{ days,}
\displaystyle 3x=\frac{1}{14}.
\displaystyle x=\frac{1}{42}.
\displaystyle \therefore \text{A's one day's work}=2x=\frac{2}{42}=\frac{1}{21}.
\displaystyle \therefore A\text{ alone can complete the work in }21\text{ days}.
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{Two pipes A and B can separately fill a tank in }36
\displaystyle \text{minutes and }45\text{ minutes respectively. If both pipes are opened}
\displaystyle \text{simultaneously, how much time will be taken to fill the tank?}
\displaystyle \text{Answer:}
\displaystyle \text{A's one-minute fill rate}=\frac{1}{36}.
\displaystyle \text{B's one-minute fill rate}=\frac{1}{45}.
\displaystyle \text{Combined fill rate}=\frac{1}{36}+\frac{1}{45}.
\displaystyle =\frac{5+4}{180}=\frac{9}{180}=\frac{1}{20}.
\displaystyle \text{Time taken to fill the tank}=\frac{1}{1/20}=20\text{ minutes}.
\displaystyle \therefore \text{the tank will be filled in }20\text{ minutes}.
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{One tap can fill a cistern in }3\text{ hours and the waste}
\displaystyle \text{pipe can empty the full cistern in }5\text{ hours. In what time will the empty}
\displaystyle \text{cistern be full if the tap and the waste pipe are kept open together?}
\displaystyle \text{Answer:}
\displaystyle \text{Tap's one-hour fill rate}=\frac{1}{3}.
\displaystyle \text{Waste pipe's one-hour emptying rate}=\frac{1}{5}.
\displaystyle \text{Net fill rate}=\frac{1}{3}-\frac{1}{5}.
\displaystyle =\frac{5-3}{15}=\frac{2}{15}.
\displaystyle \text{Time taken to fill the cistern}=\frac{1}{2/15} =\frac{15}{2}=7\frac{1}{2}\text{ hours}.
\displaystyle \therefore \text{the cistern will be filled in }7\frac{1}{2}\text{ hours}.
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{Two pipes A and B can separately fill a cistern in }20
\displaystyle \text{minutes and }30\text{ minutes respectively, while a third pipe C can empty}
\displaystyle \text{the full cistern in }15\text{ minutes. If all the pipes are opened together,}
\displaystyle \text{in what time will the empty cistern be filled?}
\displaystyle \text{Answer:}
\displaystyle \text{A's one-minute fill rate}=\frac{1}{20}.
\displaystyle \text{B's one-minute fill rate}=\frac{1}{30}.
\displaystyle \text{C's one-minute emptying rate}=\frac{1}{15}.
\displaystyle \text{Net fill rate}=\frac{1}{20}+\frac{1}{30}-\frac{1}{15}.
\displaystyle =\frac{3+2-4}{60}=\frac{1}{60}.
\displaystyle \text{Time taken to fill the cistern}=\frac{1}{1/60}=60\text{ minutes}.
\displaystyle \therefore \text{the cistern will be filled in }60\text{ minutes, or }1\text{ hour}.
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{A pipe can fill a tank in }16\text{ hours. Due to a leak in}
\displaystyle \text{the bottom, the tank is filled in }24\text{ hours. If the tank is full, how much}
\displaystyle \text{time will the leak take to empty it?}
\displaystyle \text{Answer:}
\displaystyle \text{Pipe's one-hour fill rate}=\frac{1}{16}.
\displaystyle \text{Net one-hour fill rate with the leak}=\frac{1}{24}.
\displaystyle \text{Leak's one-hour emptying rate} =\frac{1}{16}-\frac{1}{24}.
\displaystyle =\frac{3-2}{48}=\frac{1}{48}.
\displaystyle \text{Time taken by the leak to empty the full tank} =\frac{1}{1/48}=48\text{ hours}.
\displaystyle \therefore \text{the leak alone will empty the full tank in }48\text{ hours}.
\displaystyle \\

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