Time And Work - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & \left( {{\text{A + B}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{10}} \cr & {\text{C's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{50}} \cr & \left( {{\text{A + B + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{{10}} + \frac{1}{{50}}} = \frac{6}{{50}} = \frac{3}{{25}}........\left( {\text{i}} \right) \cr & {\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \left( {{\text{B + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}}\,........\left( {{\text{ii}}} \right) \cr & {\text{From}}\,\left( {\text{i}} \right)\,{\text{and}}\,\left( {{\text{ii}}} \right){\text{,we}}\,{\text{get}}:2 \times \left( {{\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}}} \right) \cr & = \frac{3}{{25}} \cr & \Rightarrow {\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{3}{{50}} \cr & \therefore {\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}}\left( {\frac{1}{{10}} - \frac{3}{{50}}} \right) \cr & = \frac{2}{{50}} = \frac{1}{{25}} \cr & {\text{So,}}\,{\text{B}}\,\,{\text{alone}}\,{\text{could}}\,{\text{do}}\,{\text{the}}\,{\text{work}}\,{\text{in}}\,{\text{25}}\,{\text{days}} \cr} $$

$$\eqalign{ & {\text{Whole}}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}} \cr & = {20 \times \frac{5}{4}} = 25\,{\text{days}} \cr & {\text{Now}},\,\left( {1 - \frac{4}{5}} \right)\,\,i.e., \cr & \frac{1}{5}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{and}}\,{\text{B}}\,{\text{in}}\,{\text{3}}\,{\text{days}} \cr & {\text{Whole}}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{and}}\,{\text{B}}\,{\text{in}} \cr & = \left( {3 \times 5} \right) = 15\,{\text{days}} \cr & {\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{25}}, \cr & \left( {{\text{A + B}}} \right)\,{\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{15}} \cr & \therefore {\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{{15}} - \frac{1}{{25}}} = \frac{4}{{150}} = \frac{2}{{75}} \cr & {\text{So,}}\,{\text{B}}\,\,{\text{alone}}\,{\text{would}}\,{\text{do}}\,{\text{the}}\,{\text{work}}\,{\text{in}} \cr & \frac{{75}}{2} = 37\frac{1}{2}\,{\text{days}} \cr} $$

$$\eqalign{ & \left( {{\text{P + Q + R}}} \right){\text{'s}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} \cr & = {\frac{1}{8} + \frac{1}{{10}} + \frac{1}{{12}}} = \frac{{37}}{{120}} \cr & {\text{Work}}\,{\text{done}}\,{\text{by}}\,{\text{P,}}\,{\text{Q}}\,{\text{and}}\,{\text{R}}\,{\text{in}}\,{\text{2}}\,{\text{hours}} \cr & = {\frac{{37}}{{120}} \times 2} = \frac{{37}}{{60}} \cr & {\text{Remaining}}\,{\text{work}} = {1 - \frac{{37}}{{60}}} = \frac{{23}}{{60}} \cr & \left( {{\text{Q + R}}} \right){\text{'s}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} \cr & = {\frac{1}{{10}} + \frac{1}{{12}}} = \frac{{11}}{{60}} \cr & {\text{Now}},\frac{{11}}{{60}}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{Q}}\,\,{\text{and}}\,{\text{R}}\,{\text{in}}\,{\text{1}}\,{\text{hour}} \cr & {\text{So}},\frac{{23}}{{60}}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{Q}}\,{\text{and}}\,{\text{R}}\,{\text{in}} \cr & = {\frac{{60}}{{11}} \times \frac{{23}}{{60}}} = \frac{{23}}{{11}}\,{\text{hours}} \approx 2\,{\text{hours}} \cr & {\text{So,}}{\text{the}}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{finished}}\,{\text{approximately}} \cr & {\text{2}}\,{\text{hours}}\,{\text{after}}\,{\text{11}}\,{\text{A}}{\text{.M}}{\text{.,}}\,{\text{i}}{\text{.e}}{\text{.,}}\,{\text{around}}\,{\text{1}}\,{\text{P}}{\text{.M}}{\text{.}} \cr} $$

$$\eqalign{ & {\text{B's}}\,{\text{10}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{{15}} \times 10} = \frac{2}{3} \cr & {\text{Remaining}}\,{\text{work}} \cr & = {1 - \frac{2}{3}} = \frac{1}{3} \cr & {\text{Now}},\frac{1}{{18}}{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & \therefore \frac{1}{3}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}} \cr & {18 \times \frac{1}{3}} = 6\,{\text{days}} \cr} $$

Let 1 man's 1 day's work = x and 1 woman's 1 day's work = y
Then, 4x + 6y = $$\frac{1}{8}$$ and 3x + 7y = $$\frac{1}{{10}}$$
Solving the two equations, we get
$$x = \frac{{11}}{{400}},\,\,\,y = \frac{1}{{400}}$$
∴ 1 woman's 1 day's work = $$\frac{1}{{400}}$$
⇒ 10 women's 1 day's work = $$ {\frac{1}{{400}} \times 10} $$   = $$\frac{1}{{40}}$$
Hence, 10 women will complete the work in 40 days

$$\eqalign{ & \left( {{\text{A + B}}} \right){\text{'s}}\,{\text{20}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{{30}} \times 20} = \frac{2}{3} \cr & {\text{Remaining}}\,{\text{work}} \cr & = {1 - \frac{2}{3}} = \frac{1}{3} \cr & {\text{Now}},\frac{1}{3}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}}\,{\text{20}}\,{\text{days}} \cr & \therefore {\text{The}}\,{\text{whole}}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}} \cr & {20 \times 3} = 60\,days \cr} $$

$$\eqalign{ & {\text{P}}\,{\text{can}}\,{\text{complete}}\,{\text{the}}\,{\text{work}} \cr & = \,\left( {12 \times 8} \right){\text{hrs}}{\text{.}} = 96\,{\text{hrs}}{\text{.}} \cr & {\text{Q}}\,{\text{can}}\,{\text{complete}}\,{\text{the}}\,{\text{work}} \cr & = \left( {8 \times 10} \right){\text{hrs}}{\text{.}} = 80\,{\text{hrs}}{\text{.}} \cr & \therefore {\text{P's}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} = \frac{1}{{96}}\,{\text{and}} \cr & \therefore {\text{Q's}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} = \frac{1}{{80}} \cr & \left( {{\text{P + Q}}} \right){\text{'s}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} \cr & = {\frac{1}{{96}} + \frac{1}{{80}}} = \frac{{11}}{{480}} \cr & {\text{So,}}\,{\text{both}}\,{\text{P}}\,{\text{and}}\,{\text{Q}}\,{\text{will}}\,{\text{finish}}\,{\text{the}}\,{\text{work}} \cr & = {\frac{{480}}{{11}}} {\text{ hrs}}{\text{.}} \cr & \therefore {\text{Number}}\,{\text{of}}\,{\text{days}}\,{\text{of}}\,{\text{8}}\,{\text{hours}}\,{\text{each}} \cr & {\frac{{480}}{{11}} \times \frac{1}{8}} = \frac{{60}}{{11}}{\text{days}} = 5\frac{5}{{11}}{\text{days}} \cr} $$

$$\eqalign{ & 1\,woman's\,1\,day's\,work = \frac{1}{{70}} \cr & {\text{1}}\,{\text{child's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{140}} \cr & \left( {{\text{5}}\,{\text{women + 10}}\,{\text{children}}} \right){\text{'s}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{5}{{70}} + \frac{{10}}{{140}}} = {\frac{1}{{14}} + \frac{1}{{14}}} = \frac{1}{7} \cr & \therefore {\text{5}}\,{\text{women}}\,{\text{and}}\,{\text{10}}\,{\text{chidren}}\,{\text{will}}\,{\text{complete}} \cr & {\text{the}}\,{\text{work}}\,{\text{in}}\,{\text{7}}\,{\text{days}} \cr} $$

$$\eqalign{ & {\text{work}}\,{\text{done}}\,{\text{by}}\,{\text{X}}\,{\text{in}}\,{\text{4}}\,{\text{days}} \cr & = {\frac{1}{{20}} \times 4} = \frac{1}{5} \cr & {\text{Remaining}}\,{\text{work}} \cr & = {1 - \frac{1}{5}} = \frac{4}{5} \cr & \left( {{\text{X + Y}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{{20}} + \frac{1}{{12}}} = \frac{8}{{60}} = \frac{2}{{15}} \cr & {\text{Now}},\frac{2}{{15}}{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{X}}\,{\text{and}}\,{\text{Y}}\,{\text{in}}\,{\text{1}}\,{\text{day}}. \cr & {\text{So}},\,\frac{4}{5}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{X}}\,{\text{and}}\,{\text{Y}}\,{\text{in}} \cr & {\frac{{15}}{2} \times \frac{4}{5}} = 6\,{\text{days}} \cr & {\text{Hence,}}\,{\text{total}}\,{\text{time}}\,{\text{taken}} \cr & = \left( {6 + 4} \right)\,{\text{days}} \cr & = 10\,{\text{days}} \cr} $$

$$\eqalign{ & {\text{Ratio}}\,{\text{of}}\,{\text{times}}\,{\text{taken}}\,{\text{by}}\,{\text{A}}\,{\text{and}}\,{\text{B}} \cr & = 100:130 = 10:13 \cr & {\text{Suppose}}\,{\text{B}}\,{\text{takes}}\,x\,{\text{days}}\,{\text{to}}\,{\text{do}}\,{\text{the}}\,{\text{work}} \cr & {\text{Then}},10:13::23:x \cr & \Rightarrow x = {\frac{{23 \times 13}}{{10}}} \cr & \Rightarrow x = \frac{{299}}{{10}} \cr & {\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{23}} \cr & {\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{{10}}{{299}} \cr & \left( {{\text{A + B}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = {\frac{1}{{23}} + \frac{{10}}{{299}}} \cr & = \frac{{23}}{{299}} \cr & = \frac{1}{{13}} \cr & \therefore A\,{\text{and}}\,{\text{B}}\,{\text{together}}\,{\text{can}}\,{\text{complete}} \cr & \,{\text{the}}\,{\text{work}}\,{\text{in}}\,{\text{13}}\,{\text{days}}{\text{.}} \cr} $$