**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 × 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

= \(\frac{( 28 × 84)}{(28 + 84)}\)

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

= 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 × 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 {(\(\frac{12}{100%}\)) × 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**:

X’s 1 day’s work = \(\frac{1}{20}\)

X’s 4 day’s work = \(\frac{1}{20}\) × 4 = \(\frac{1}{5}\)

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

X and Y’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

= \(\frac{4}{5}\)× \(\frac{15}{2}\) = 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 ‘s 1 day’s work = \(\frac{1}{30}\)

Their 20 day’s work = \(\frac{1}{30}\) × 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 × 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 + B’s 1 day’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. B’s 1 day’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 + B’s 1 day’s work = \(\frac{1}{12}\)

B+ C’s 1 days’ 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. A’s 5 day’s work + B’s 7 day’s work + C’s 13 day’s work = 1

(A+ B)’s 5 days work + (B + C)’s 2 days work + C’s 11 day’s work = 1

\(\frac{5}{12}\) + \(\frac{2}{16}\) + C’s 11 days work = 1

So, C’s 11 day’s work = 1 – (\(\frac{5}{12}\) + \(\frac{2}{16}\)) = \(\frac{11}{24}\)

C’s 1 days’ 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 × 40/85 = \(\frac{360}{17}\) days

A and B’s 1 day’s work = \(\frac{17}{360}\)

A’s 1 day’s work = \(\frac{1}{45}\)

B’s 1 day’s work = \(\frac{1}{40}\)

B’s 23 day’s work \(\frac{1}{40}\) × 23 = \(\frac{23}{40}\)

Remaining work = 1 – 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 \(\frac{360}{17}\) = \(\frac{17}{40}\) × \(\frac{360}{17}\) 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**:

A’ s 1 day’s work = \(\frac{1}{20}\)

B’s 1 day’s work = latex]\frac{1}{13}[/latex]

B’s 9 days work = latex]\frac{1}{12}[/latex] × 9 = \(\frac{3}{4}\)

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

A can finish the remaining work in = \(\frac{1}{4}\) X 20 = \(\frac{1}{4}\) ×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 B’s 1 day’s work = \(\frac{1}{3}\)

Their 2 day’s work = \(\frac{1}{3}\) × 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 × 2 = 6 days

So, A’s 1 day’s work = \(\frac{1}{6}\)

Therefore, B’s 1 day’s work = \(\frac{1}{3}\) – \(\frac{1}{6}\) = \(\frac{1}{6}\)

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)’s 1 day’s work = \(\frac{1}{3}\)

Man’s 1day’s work = \(\frac{1}{6}\)

Boy’s 1 day’s work = \(\frac{1}{18}\)

Then, Woman’ 1 day’s work = \(\frac{1}{3}\) – ( \(\frac{1}{6}\) + \(\frac{1}{18}\) ) = \(\frac{1}{3}\) – \(\frac{4}{18}\) = \(\frac{2}{18}\) = \(\frac{1}{9}\)

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**:

A’s 1 day’s work = A + B + C’s 1 day’s work – B + C’s 1 day’s work = \(\frac{1}{6}\) – \(\frac{1}{12}\) = \(\frac{1}{12}\)

B’s 1 day’s work = A + B’s 1 day’s work – A’s 1 day’s work = \(\frac{1}{8}\) – \(\frac{1}{12}\) = \(\frac{1}{24}\)

C’s 1 day’s work = B+ C’s 1 day’s work – B’s 1 day’s work = \(\frac{1}{12}\) – \(\frac{1}{24}\) = \(\frac{1}{24}\)

A +B’ 1 day’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