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

a) $$\frac{1}{6}$$

b) $$\frac{1}{9}$$

c) $$\frac{2}{5}$$

d) $$\frac{2}{7}$$

Explanation:

$$\eqalign{ & {\text{A's 1 day's work}} = \frac{1}{{18}} \cr & {\text{B's 1 day's work}} = \frac{1}{9} \cr & \left( {{\text{A}} + {\text{B}}} \right)'{\text{s 1 day's work}} \cr & = \left( {\frac{1}{{18}} + \frac{1}{9}} \right) \cr & = \frac{1}{6} \cr} $$

2. A can knit a pair of socks in 3 days. B can knit the same pair of socks in 9 days. If they are knitting together, then in how many days will they knit two pairs of socks ?

a) 3 days

b) 4 days

c) $${\text{4}}\frac{1}{2}$$ days

d) 5 days

Explanation: Number of pairs knit by A and B together in 1 day

$$\eqalign{ & = \left( {\frac{1}{3} + \frac{1}{9}} \right) \cr & = \frac{4}{9}{\text{ }} \cr} $$

Required number of days,

$$\eqalign{ & = \left( {2 \div \frac{4}{9}} \right) \cr & = \left( {2 \times \frac{9}{4}} \right) \cr & = \frac{9}{2} \cr & = 4\frac{1}{2} \text{ days} \cr} $$

3. P can complete $$\frac{1}{4}$$ of a work in 10 days, Q can complete 40% of the same work in 145 days. R, complete $$\frac{1}{3}$$ of the work in 13 days and S, $$\frac{1}{6}$$ of the work in 7 days. Who will be able complete the work first ?

a) P

b) Q

c) R

d) S

Explanation: P completes $$\frac{1}{4}$$ of work in 10 days

P completes full of work in

$$\eqalign{ & = \frac{{10}}{1} \times 4 \cr & = 40{\text{ days}} \cr} $$

Q completes 40% of work in 145 days

Q completes full 100% of work in

$$\eqalign{ & = \frac{{145}}{{40}} \times 100 \cr & = 362.5{\text{ days}} \cr} $$

R completes $$\frac{1}{3}$$ of work in 13 days

R completes full of work in

$$\eqalign{ & = \frac{{13}}{1} \times 3 \cr & = 39{\text{ days}} \cr} $$

S completes $$\frac{1}{6}$$ of work in 7 days

S completes full of work in

$$\eqalign{ & = \frac{7}{1} \times 6 \cr & = 42{\text{ days}} \cr} $$

We can see R completes the work first

4. George takes 8 hours to copy a 50-page manuscript while Sonia can copy the same manuscript in 6 hours. How many hours would it take them to copy a 100-page manuscript, if they work together ?

a) $$6\frac{6}{7}\,\,{\text{hours}}$$

b) $${\text{9 hours}}$$

c) $$9\frac{5}{7}\,\,{\text{hours}}$$

d) $${\text{14 hours}}$$

Explanation: Number of pages typed by Gorge in 1 hour

$$\eqalign{ & = \frac{{50}}{8} \cr & = \frac{{25}}{4} \cr} $$

Number of pages typed by Sonia in 1 hour

$$\eqalign{ & = \frac{{50}}{6} \cr & = \frac{{25}}{3} \cr} $$

Number of pages typed by Gorge and Sonia together in 1 hour

$$\eqalign{ & = \left( {\frac{{25}}{4} + \frac{{25}}{3}} \right) \cr & = \left( {\frac{{75 + 100}}{{12}}} \right) \cr & = \frac{{175}}{{12}} \cr & {\text{Required time}} \cr & = \left( {100 \div \frac{{175}}{{12}}} \right){\text{hours}} \cr & = \left( {\frac{{100 \times 12}}{{175}}} \right){\text{ hours}} \cr & = \frac{{48}}{7}{\text{ hours}} \cr & = 6\frac{6}{7}{\text{ hours}} \cr} $$

5. A and B together complete a piece of work in T days. If A alone completes the work in T + 3 days and B alone completes the piece of work in T + 12 days, what is T ?:

a) 3 days

b) 9 days

c) 12 days

d) None of these

Explanation:

$$\eqalign{ & {\text{A's 1 day's work}} \cr & = \frac{1}{{{\text{T}} + 3}} \cr & {\text{B's 1 day's work}} \cr & = \frac{1}{{{\text{T}} + 12}} \cr & \left( {{\text{A}} + {\text{B}}} \right){\text{'s 1 day's work}} = \frac{1}{{\text{T}}} \cr & \therefore \frac{1}{{{\text{T}} + 3}} + \frac{1}{{{\text{T}} + 12}} = \frac{1}{{\text{T}}} \cr & \Rightarrow \frac{{2{\text{T}} + 15}}{{\left( {{\text{T}} + 3} \right)\left( {{\text{T}} + 12} \right)}} = \frac{1}{{\text{T}}} \cr & \Rightarrow 2{{\text{T}}^2} + 15{\text{T}} = {{\text{T}}^2} + 15{\text{T}} + 36 \cr & {{\text{T}}^2} = 36 \cr & {\text{T}} = 6 \cr} $$

6. X can do a piece of work in 40 days. He works at it for 8 days and then Y finished it in 16 days. How long will they together take to complete the work?

a) $$13\frac{1}{3}$$ days

b) 15 days

c) 20 days

d) 26 days

Explanation: Work done by X in 8 days = $$ {\frac{1}{{40}} \times 8} $$ = $$\frac{1}{5}$$

Remaining work = $$ {1 - \frac{1}{5}} $$ = $$\frac{4}{5}$$

Now, $$\frac{4}{5}$$ work is done by Y in 16 days

Whole work will be done by Y in = $$ {16 \times \frac{5}{4}} $$ = 20 days

X's 1 day's work = $$\frac{1}{{40}}$$

Y's 1 day's work = $$\frac{1}{{20}}$$

(X + Y)'s 1 day's work

$$\eqalign{ & = {\frac{1}{{40}} + \frac{1}{{20}}} \cr & = \frac{3}{{40}} \cr} $$

Hence, X and Y will together complete the work in

$$\eqalign{ & = {\frac{{{\text{40}}}}{{\text{3}}}} \cr & {\text{ = 13}}\frac{{\text{1}}}{{\text{3}}}\,\,{\text{days}} \cr} $$

7. A and B can do a job together in 7 days. A is $$1\frac{3}{4}$$ times as efficient as B. The same job can be done by A alone in :

a) $$9\frac{1}{3}$$ days

b) 11 days

c) $$12\frac{1}{4}$$ days

d) $$16\frac{1}{3}$$ days

Explanation:

$$\eqalign{ & \left( {{\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}}} \right){\text{:}}\left( {{\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}}} \right) \cr & = \frac{7}{4}:1 = 7:4 \cr & {\text{Let}}\,{\text{A's}}\,{\text{and}}\,{\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}}\,{\text{be}} \cr & 7x\,{\text{and}}\,4x\,{\text{respectively}} \cr & {\text{Then}},\,7x + 4x = \frac{1}{7} \cr & 11x = \frac{1}{7} \cr & x = \frac{1}{{77}} \cr & {\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{{77}} \times {\text{7}}} = \frac{1}{{11}} \cr} $$

So, A will do the work in 11 days

8. 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?

a) 30 days

b) 40 days

c) 60 days

d) 70 days

Explanation:

$$\eqalign{ & {\text{Let}}\,{\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = x\,{\text{and}} \cr & {\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = y \cr & {\text{Then}},\,x + y = \frac{1}{{30}} \cr & 16x + 44y = 1 \cr & {\text{Solving}}\,{\text{these}}\,{\text{two}}\,{\text{equations,}} \cr & x = \frac{1}{{60}}\,{\text{and}}\,y = \frac{1}{{60}} \cr & {\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{60}} \cr & {\text{Hence,}}\,{\text{B}}\,{\text{alone}}\,{\text{shall}}\,{\text{finish}}\,{\text{the}}\,{\text{whole}}\,{\text{work}}\,{\text{in}}\,{\text{60}}\,{\text{days}} \cr} $$

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

a) $$\frac{1}{6}$$

b) $$\frac{2}{5}$$

c) $$\frac{1}{9}$$

d) $$\frac{2}{7}$$

Explanation: A's 1 day work = $$\frac{1}{{18}}$$

B's 1 day work = $$\frac{1}{9}$$ [because B take half time than A]

(A + B)'s one day work

$$\eqalign{ & = {\frac{1}{{18}} + \frac{1}{9}} \cr & = {\frac{{1 + 2}}{{18}}} \cr & = \frac{1}{6} \cr} $$

10. A and B can do a piece of work in 72 days, B and C can do it in 120 days and A and C can do it in 90 days. In how many days all three together can do the work ?

a) 80 days

b) 100 days

c) 60 days

d) 150 days

Explanation: In these type of questions, always take total work as L.C.M. of number of days.

Here L.C.M. of Total Work = 360

One day work of A + B = $$\frac{{360}}{{72}}$$ = 5 unit/day

One day work of B + C = $$\frac{{360}}{{120}}$$ = 3 unit/day

One day work of C + A = $$\frac{{360}}{{90}}$$ = 4 unit/day

Total units/day = 5 + 3 + 4 = 12

(Here , 12 unit represents twice of the work done by A, B and C. So we will divide it by 2)

Work done by (A + B + C) per day = $$\frac{{12}}{2}$$ = 6 units/day

Total time taken by (A + B + C)

$$\eqalign{ & = \frac{{360}}{6} \cr & = 60{\text{ days}} \cr} $$