**Answer**: Option: A

**Explanation**:

Given

A + B ‘s one day work will be \(\frac{1}{12}\)

Similarly, B + C’s one day work will be \(\frac{1}{15}\)

C + A ‘s one day work be \(\frac{1}{20}\)

A + B = \(\frac{1}{12}\) —-> (1)

B + C = \(\frac{1}{15}\) —-> (2)

C + A = \(\frac{1}{20}\) —-> (3)

Adding all the above equation we get

2(A+B+C) = \(\frac{1}{12}\) + \(\frac{1}{15}\) + \(\frac{1}{20}\)

=> \(\frac{5}{60}\) + \(\frac{4}{60}\) + \(\frac{3}{60}\)

=> \(\frac{12}{60}\)

A+B+C = \(\frac{12}{120}\)

= \(\frac{1}{10}\)

If all three work together, the work will be completed in 10 days.

**2. Adam can do a job in 15 days, John can do the same job in 20 days. If they work together for 4 days on this job. What fraction of job is incomplete?**

**Answer**: Option: D

**Explanation**:

Adam can do 1/15 of the job per day

John can do 1/20 of the job per day

If they work together they can do \(\frac{7}{60}\) of the work together

Remaining job 1 – 4*\(\frac{7}{60}\) = \(\frac{32}{60}\) = \(\frac{8}{15}\)

**3. If A, B and C can do a job in 20, 30 and 60 days respectively. In how many days A can do the work if B and C help him on every third day? **

**Answer**: Option: B

**Explanation**:

Efficiency of A = \(\frac{1}{20}\) = 5% per day

Efficiency of B = \(\frac{2}{30}\) = 3.33% per day

Efficiency of C = \(\frac{1}{60}\) = 1.66% per day

In three days A can do 15% of the job himself and B and C do 5% of the job ( 1.66% + 3.33% )

In three days they can do 20% of the job, to do 100% of the job, they need 3 Ã— 5 = 15 days

**4. Annie and David can complete the work in 28 and 84 days respectively. If they work together in how many days the work will get completed (in days) ? **

**Answer**: Option: A

**Explanation**:

Given that Annie can do the work in 28 days which means Annie can complete 1/ 28th the work in 1 day

Given that David can do the work in 84 days which means David can complete 1/ 84th the work in 1 day

Let Annie one day work = \(\frac{1}{28}\)

Let David one day work = \(\frac{1}{84}\)

Both work together,

Together,they finish the work in \(\frac{XY}{X + Y}\) days

= \(\frac{( 28 Ã— 84)}{(28 + 84)}\)

= \(\frac{2352}{112}\)

= 21 days.

**5. Albert and Sam can complete a work in 12 and 36 days respectively. If both work together in how many days 50 % of the work will get completed?**

**Answer**: Option: C

**Explanation**:

Let the time taken by Albert = A days = 12 days

the time taken by Sam = B days = 36 days

Albert + Sam together can complete 100 % of the work in:

=> \(\frac{1}{(A + B)}\) = \(\frac{1}{(A )}\)(1/A) + \(\frac{1}{(B)}\)

=> \(\frac{1}{(A + B)}\) = \(\frac{1}{(12)}\) + \(\frac{1}{(36)}\)

=> \(\frac{1}{(A + B)}\) = \(\frac{(36 + 12)}{((12 * 36)}\)

=> \(\frac{1}{(A + B)}\) = \(\frac{48}{432}\)

Taking reciprocal on both sides

A + B = \(\frac{432}{48}\)

A + B = 9 days

Thus 100% of the work is completed in 9 days,

50% of the work is completed in {(\(\frac{9}{100%}\)) * 50%} days = 4.5 days

**Answer**: Option: B

**Explanation**:

Let the time taken by Somu= A = 20 days

the time taken by Ramu= B = 30 days

Then number of days to complete 100% of work by Somu+ Ramutogether:

=>\(\frac{1}{(A + B)}\) = (\(\frac{1}{a}\)) + (\(\frac{1}{b}\))

=> \(\frac{1}{(A + B)}\) = (\(\frac{1}{20}\)) + (\(\frac{1}{30}\))

=> \(\frac{1}{(A + B)}\) = (30 + 20) / (20 Ã— 30)

=> \(\frac{1}{(A + B)}\) = \(\frac{50}{600}\)

Taking reciprocal on both sides

A + B = \(\frac{600}{50}\)

A + B = 12 days

Thus 100% of the work is completed in 12 days,

50% of the work is completed in {(\(\frac{12}{100%}\)) Ã— 50%} days = 6 days

**2. X and Y can do a piece of work in 20 days and 12 days respectively. X started the work alone and then after 4 days Y joined him till the completion of the work. How long did the work last?**

**Answer**: Option: B

**Explanation**:

Xâ€™s 1 dayâ€™s work = \(\frac{1}{20}\)

Xâ€™s 4 dayâ€™s work = \(\frac{1}{20}\) Ã— 4 = \(\frac{1}{5}\)

The remaining work = \(\frac{4}{5}\)

X and Yâ€™s 1 day work = \(\frac{1}{20}\) + \(\frac{1}{12}\) = \(\frac{4}{30}\) = \(\frac{2}{15}\)

Therefore, Both together finish the remaining work in \(\frac{4/5}{2/15}\) days

= \(\frac{4}{5}\)Ã— \(\frac{15}{2}\) = 6 days

Therefore, the total number of days taken to finish the work = 4 + 6 = 10 days.

**3. A and B can together finish a work in 30 days. They worked together for 20 days and then B left. After another 20 days, A finished he remaining work. In how many days A alone can finish the job? **

**Answer**: Option: D

**Explanation**:

A + B â€˜s 1 dayâ€™s work = \(\frac{1}{30}\)

Their 20 dayâ€™s work = \(\frac{1}{30}\) Ã— 20 = \(\frac{2}{3}\)

Remaining work = \(\frac{1}{3}\)

\(\frac{1}{3}\) of the work is done by A in 20 days.

Then, whole work can be done by A in 3 Ã— 20 = 60 days

**4. A and B together can do a piece of work in 30 days. A having worked for 16 days, B finishes the remaining work alone in 44 days. In how many days shall B finish the whole work alone? **

**Answer**: Option: C

**Explanation**:

A + Bâ€™s 1 dayâ€™s work = 1/30

i.e. A + B = 1/30 ——— (i)

16 A + 44 B = 1 —— (ii) ( i.e. 1 = the whole work done)

Multiplying (i) by 16 and subtracting it from (ii), we get

i.e. 16 A + 44 B = 1

16 A + 16B = \(\frac{8}{15}\)

28 B = \(\frac{7}{15}\)

B = \(\frac{1}{60}\) i.e. Bâ€™s 1 dayâ€™s work = \(\frac{1}{60}\)

Therefore, B alone can finish the work in 60 days.

**5. A and B together can do a piece of work in 12 days, which B and C together can do in 16 days. After A has been working at it for 5 days and B for 7 days, C finishes it in 13 days. In how many days C alone will do the work?**

**Answer**: Option: B

**Explanation**:

According to the question,

A + Bâ€™s 1 dayâ€™s work = \(\frac{1}{12}\)

B+ Câ€™s 1 daysâ€™ work = \(\frac{1}{16}\)

A worked for 5 days, B for 7 days and C for 13 days. So, we can assume that,

A+ B has been working for 5 days and B+ C has been working for 2 days and C alone for 11 days.

i.e. Aâ€™s 5 dayâ€™s work + Bâ€™s 7 dayâ€™s work + Câ€™s 13 dayâ€™s work = 1

(A+ B)â€™s 5 days work + (B + C)â€™s 2 days work + Câ€™s 11 dayâ€™s work = 1

\(\frac{5}{12}\) + \(\frac{2}{16}\) + Câ€™s 11 days work = 1

So, Câ€™s 11 dayâ€™s work = 1 â€“ (\(\frac{5}{12}\) + \(\frac{2}{16}\)) = \(\frac{11}{24}\)

Câ€™s 1 daysâ€™ work = \(\frac{11}{24}\) X \(\frac{1}{11}\) = \(\frac{1}{24}\)

Therefore, C alone can finish the work in 24 days.

**Answer**: Option: C

**Explanation**:

A and B together can finish the work in 45 Ã— 40/85 = \(\frac{360}{17}\) days

A and Bâ€™s 1 dayâ€™s work = \(\frac{17}{360}\)

Aâ€™s 1 dayâ€™s work = \(\frac{1}{45}\)

Bâ€™s 1 dayâ€™s work = \(\frac{1}{40}\)

Bâ€™s 23 dayâ€™s work \(\frac{1}{40}\) Ã— 23 = \(\frac{23}{40}\)

Remaining work = 1 â€“ 23/40 = \(\frac{17}{40}\)

\(\frac{17}{40}\) of the work is done by A and B together

\(\frac{17}{40}\) of the work is done by A and B together in = \(\frac{17}{40}\) X \(\frac{360}{17}\) = \(\frac{17}{40}\) Ã— \(\frac{360}{17}\) days

= 9 days

Therefore, A left after 9 days.

**2. A can do a piece of work in 20 days which B can do in 12 days. B worked at it for 9 days. A can finish the remaining work in: **

**Answer**: Option: B

**Explanation**:

Aâ€™ s 1 dayâ€™s work = \(\frac{1}{20}\)

Bâ€™s 1 dayâ€™s work = latex]\frac{1}{13}[/latex]

Bâ€™s 9 days work = latex]\frac{1}{12}[/latex] Ã— 9 = \(\frac{3}{4}\)

Remaining work = latex]\frac{1}{4}[/latex]

A can finish the remaining work in = \(\frac{1}{4}\) X 20 = \(\frac{1}{4}\) Ã—20 = 5 days.

**3. A and B together can complete a work in 3 days. They start together. But, after 2 days, B left the work. If the work is completed after 2 more days, B alone could do the work in **

**Answer**: Option: B

**Explanation**:

A and Bâ€™s 1 dayâ€™s work = \(\frac{1}{3}\)

Their 2 dayâ€™s work = \(\frac{1}{3}\) Ã— 2 = \(\frac{2}{3}\)

Remaining work = \(\frac{1}{3}\)

\(\frac{1}{3}\) of the work is finished by A in 2 days

Then, the whole work can be finished by A alone in 3 Ã— 2 = 6 days

So, Aâ€™s 1 dayâ€™s work = \(\frac{1}{6}\)

Therefore, Bâ€™s 1 dayâ€™s work = \(\frac{1}{3}\) â€“ \(\frac{1}{6}\) = \(\frac{1}{6}\)

Therefore, the whole work can be done B alone in 6 days

**4. A man, a woman and a boy together complete a piece of work in 3 days. If a man alone can do it in 6 days and a boy alone in 18 days, how long will a woman take to complete the work? **

**Answer**: Option: A

**Explanation**:

Man + Boy + Woman)â€™s 1 dayâ€™s work = \(\frac{1}{3}\)

Manâ€™s 1dayâ€™s work = \(\frac{1}{6}\)

Boyâ€™s 1 dayâ€™s work = \(\frac{1}{18}\)

Then, Womanâ€™ 1 dayâ€™s work = \(\frac{1}{3}\) â€“ ( \(\frac{1}{6}\) + \(\frac{1}{18}\) ) = \(\frac{1}{3}\) â€“ \(\frac{4}{18}\) = \(\frac{2}{18}\) = \(\frac{1}{9}\)

Therefore, the Woman alone can finish the work in 9 days

**5. A and B together can complete a piece of work in 8 days while B and C together can do it in 12 days. All the three together can complete the work in 6 days. In how much time will A and C together complete the work? **

**Answer**: Option: A

**Explanation**:

Aâ€™s 1 dayâ€™s work = A + B + Câ€™s 1 dayâ€™s work â€“ B + Câ€™s 1 dayâ€™s work = \(\frac{1}{6}\) â€“ \(\frac{1}{12}\) = \(\frac{1}{12}\)

Bâ€™s 1 dayâ€™s work = A + Bâ€™s 1 dayâ€™s work â€“ Aâ€™s 1 dayâ€™s work = \(\frac{1}{8}\) â€“ \(\frac{1}{12}\) = \(\frac{1}{24}\)

Câ€™s 1 dayâ€™s work = B+ Câ€™s 1 dayâ€™s work â€“ Bâ€™s 1 dayâ€™s work = \(\frac{1}{12}\) â€“ \(\frac{1}{24}\) = \(\frac{1}{24}\)

A +Bâ€™ 1 dayâ€™s work = \(\frac{1}{12}\) + \(\frac{1}{24}\) = \(\frac{3}{24}\)

Therefore, A and C together will finish the whole work in \(\frac{24}{3}\) = 8 days