Time And Work - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & {\text{Number}}\,{\text{of}}\,{\text{pages}}\,{\text{typed}}\,{\text{by}}\,{\text{Ravi}}\,{\text{in}}\,{\text{1}}\,{\text{hour}} \cr & = \frac{{32}}{6} = \frac{{16}}{3} \cr & {\text{Number}}\,{\text{of}}\,{\text{pages}}\,{\text{typed}}\,{\text{by}}\,{\text{Kumar}}\,{\text{in}}\,{\text{1}}\,{\text{hour}} \cr & = \frac{{40}}{5} = 8 \cr & {\text{Number}}\,{\text{of}}\,{\text{pages}}\,{\text{typed}}\,{\text{by}}\,{\text{both}}\,{\text{in}}\,{\text{1}}\,{\text{hour}} \cr & = {\frac{{16}}{3} + 8} = \frac{{40}}{3} \cr & \therefore {\text{Time}}\,{\text{taken}}\,{\text{by}}\,{\text{both}}\,{\text{to}}\,{\text{type}}\,{\text{110}}\,{\text{pages}} \cr & = {110 \times \frac{3}{{40}}} {\text{hours}} \cr & = 8\frac{1}{4}\,{\text{hours}}\,{\text{(or)}}\,{\text{8}}\,{\text{hours}}\,{\text{15}}\,{\text{minutes}} \cr} $$

Formula: If A can do a piece of work in n days, then A's 1 day's work = $$\frac{1}{{\text{n}}}$$
$$\eqalign{ & (A + B + C)'s\,1\,{\text{day's work}} \cr & = {\frac{1}{{24}} + \frac{1}{6} + \frac{1}{{12}}} = \frac{7}{{24}} \cr} $$
Formula: If A's 1 day's work = $$\frac{1}{{\text{n}}}$$ , then A can finish the work in n days
So, all the three together will complete the job in
$$ {\frac{{24}}{7}} {\text{ days}} = 3\frac{3}{7}{\text{ days}}$$

Ratio of times taken by Sakshi and Tanya
= 125 : 100
= 5 : 4
Suppose Tanya takes x days to do the work
5 : 4 :: 20 : x
⇒ $$x = {\frac{{4 \times 20}}{5}} $$
⇒ x = 16 days
Hence, Tanya takes 16 days to complete the work

$$\eqalign{ & {\text{Suppose}}\,{\text{A,}}\,{\text{B}}\,{\text{and}}\,{\text{C}}\,{\text{take}} \cr & x,\,\frac{x}{2},\,\frac{x}{3}\,{\text{days}}\,{\text{respectively}}\,{\text{to}}\,{\text{finish}}\,{\text{the}}\,{\text{work}} \cr & {\text{Then}},\, {\frac{1}{x} + \frac{2}{x} + \frac{3}{x}} = \frac{1}{2} \cr & \Rightarrow \frac{6}{x} = \frac{1}{2} \cr & \Rightarrow x = 12 \cr & {\text{So,}}\,{\text{B}}\,{\text{takes}}\, {\frac{{12}}{2}} \cr & = 6\,{\text{days}}\,{\text{to}}\,{\text{finish}}\,{\text{the}}\,{\text{work}} \cr} $$

$$\eqalign{ & \left( {{\text{A + B}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{{15}} + \frac{1}{{10}}} = \frac{1}{6} \cr & {\text{Work}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{and}}\,{\text{B}}\,{\text{in}}\,{\text{2}}\,{\text{days}} \cr & = {\frac{1}{6} \times 2} = \frac{1}{3} \cr & {\text{Remaining}}\,{\text{work}} \cr & = {1 - \frac{1}{3}} = \frac{2}{3} \cr & {\text{Now}},\,\frac{1}{{15}}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & \therefore \frac{2}{3}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{a}}\,{\text{in}} \cr & {15 \times \frac{2}{3}} = 10\,{\text{days}} \cr & {\text{Hence,}}\,{\text{the}}\,{\text{total}}\,{\text{time}}\,{\text{taken}} \cr & = {10 + 2} = 12\,{\text{days}} \cr} $$

$$\eqalign{ & {\text{2(A + B + C)'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{{30}} + \frac{1}{{24}} + \frac{1}{{20}}} \cr & = \frac{{15}}{{120}} = \frac{1}{8} \cr & \therefore \left( {{\text{A + B + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \frac{1}{{2 \times 8}} = \frac{1}{{16}} \cr & {\text{Work}}\,{\text{done}}\,{\text{by}}\,{\text{A,}}\,{\text{B,}}\,{\text{C}}\,{\text{in}}\,{\text{10}}\,{\text{days}} \cr & = \frac{{10}}{{16}} = \frac{5}{8} \cr & {\text{Remaining}}\,{\text{work}} \cr & = {1 - \frac{5}{8}} = \frac{3}{8} \cr & {\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{{16}} - \frac{1}{{24}}} = \frac{1}{{48}} \cr & {\text{Now}},\,\frac{1}{{48}}\,{\text{work}}\,{\text{isdone}}\,{\text{by}}\,{\text{A}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & {\text{So}},\,\frac{3}{8}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}} \cr & {48 \times \frac{3}{8}} = 18\,{\text{days}} \cr} $$

$$\eqalign{ & {\text{Ration}}\,{\text{of}}\,{\text{rates}}\,{\text{of}}\,{\text{working}}\,{\text{of}}\,{\text{A}}\,{\text{and}}\,{\text{B}} \cr & = 2:1 \cr & {\text{So,}}\,{\text{ratio}}\,{\text{of}}\,{\text{times}}\,{\text{taken}} = 1:2 \cr & {\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{12}} \cr & \therefore {\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \frac{1}{3};\,({\text{2 times}}\,{\text{of}}\,{\text{B's}}\,{\text{work}}) \cr & \left( {{\text{A + B}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{6} + \frac{1}{{12}}} = \frac{3}{{12}} = \frac{1}{4} \cr & {\text{So,}}\,{\text{A}}\,{\text{and}}\,{\text{B}}\,{\text{together}}\,{\text{can}}\,{\text{finish}}\,{\text{the}} \cr & {\text{work}}\,{\text{in}}\,{\text{4}}\,{\text{days}}{\text{.}}\, \cr} $$

$$\eqalign{ & \left( {20 \times 16} \right){\text{women}}\,{\text{can}}\,{\text{complete}}\,{\text{the}}\,{\text{work}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & \therefore {\text{1}}\,{\text{woman's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{320}} \cr & \left( {16 \times 15} \right)\,{\text{men}}\,{\text{can}}\,{\text{complete}}\,{\text{the}}\,{\text{work}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & \therefore {\text{1}}\,{\text{man's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{240}} \cr & {\text{So,}}\,{\text{required}}\,{\text{ratio}} \cr & = \frac{1}{{240}}:\frac{1}{{320}} \cr & = \frac{1}{3}:\frac{1}{4} \cr & = 4:3\,\left( {{\text{cross}}\,{\text{multiplied}}} \right) \cr} $$

$$\eqalign{ & \left( {{\text{A + B + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{6} \cr & \left( {{\text{A + B}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{8} \cr & \left( {{\text{B + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{12}} \cr & \therefore \left( {{\text{A + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \left( {2 \times \frac{1}{6}} \right) - \left( {\frac{1}{8} + \frac{1}{{12}}} \right) \cr & = {\frac{1}{3} - \frac{5}{{24}}} \cr & = \frac{3}{{24}} \cr & = \frac{1}{8} \cr }$$
So, A and C together will do the work in 8 days

$$\eqalign{ & \left( {{\text{B + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{9} + \frac{1}{{12}}} = \frac{7}{{36}} \cr & {\text{Work}}\,{\text{done}}\,{\text{by}}\,{\text{B}}\,{\text{and}}\,{\text{C}}\,{\text{in}}\,{\text{3}}\,{\text{days}} \cr & = {\frac{7}{{36}} \times 3} = \frac{7}{{12}} \cr & {\text{Remaining}}\,{\text{work}} \cr & = {1 - \frac{7}{{12}}} = \frac{5}{{12}} \cr & {\text{Now}},\,\frac{1}{{24}}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & {\text{So}},\,\frac{5}{{12}}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}} \cr & {24 \times \frac{5}{{12}}} = 10\,{\text{days}} \cr} $$