Time And Work - Aptitude MCQ Questions and Solutions with Explanations

Let the daily wages of A, B and C be Rs. 5x, Rs. 6x and Rs. 4x respectively.
Then, ratio of their amounts
$$\eqalign{ & = \left( {5x \times 6} \right):\left( {6x \times 4} \right):\left( {4x:9} \right) \cr & = 30:24:36 \cr & = 5:4:6 \cr & \therefore {\text{A's amount}} \cr & = {\text{Rs}}{\text{.}}\left( {1800 \times \frac{5}{{15}}} \right) \cr & = {\text{Rs}}{\text{.600}} \cr} $$

L.C.M. of Total Work = 60
One day work of P + Q = $$\frac{{60}}{{6}}$$ = 10 unit/day efficiency
One day work of Q + R = $$\frac{{60}}{{\frac{{60}}{7}}}$$ = 7 unit/day efficiency
$$\eqalign{ & \left( {{\text{Q}} + {\text{R}}} \right){\text{ 6 days work}} \cr & = 7 \times 6 \cr & = 42{\text{ units}} \cr} $$
Then in 3 days = total work = 18
$$\eqalign{ & {\text{P completes}} \cr & = 60 - 42 \cr & = 18{\text{ units}} \cr & {\text{P's efficiency}} \cr & = \frac{{18}}{3} \cr & = 6{\text{ units/day}} \cr & {\text{Q's efficiency}} \cr & = 10 - 6 \cr & = 4{\text{ units/day}} \cr & {\text{R's efficiency}} \cr & = 7 - 4 \cr & = 3{\text{ units/day}} \cr & {\text{P completes whole work in}} \cr & = \frac{{60}}{6} \cr & = {\text{10 days}} \cr & {\text{R completes whole work in}} \cr & = \frac{{60}}{3} \cr & = {\text{20 days}} \cr & {\text{Difference is}} \cr & = \left( {20 - 10} \right) \cr & = 10{\text{ days }} \cr} $$

Shortest method of doing this question is ,
$${\text{Put in }}\frac{{{{\text{M}}_1} \times {{\text{T}}_1} \times {{\text{D}}_1}}}{{{{\text{W}}_1}}}$$     = $$\frac{{{{\text{M}}_2} \times {{\text{T}}_2} \times {{\text{D}}_2}}}{{{{\text{W}}_2}}}$$
$$\eqalign{ & {\text{Formula}} \cr & {\text{So, }} \cr & {\text{Let number of burners}} \to {\text{B}} \cr & \frac{{{\text{6B}} \times {6_{{\text{hours}}}} \times {8_{{\text{days}}}}}}{{450}} = \frac{{{\text{B}} \times 10 \times 5}}{{625}} \cr & {\text{After solving }} \to {\text{B}} = 8 \cr} $$

$$\eqalign{ & {\text{According to the question,}} \cr & {\text{X}} \to {\text{24 days}} \cr & \Rightarrow {\text{Work done X in 4 days alone}} \cr & = 4 \times 1 \cr & = 4{\text{ units}} \cr & \Rightarrow {\text{Remaining work}} \cr & = 24 - 4 = 20{\text{ units}} \cr & \Rightarrow {\text{20 units done by both together in}} \cr & = \left( {16 - 4{\text{ days}}} \right) \cr & = 12{\text{ days}} \cr & {\text{Then efficiencies of }}\left( {{\text{X}} + {\text{Y}}} \right) \cr & = \frac{{{\text{Work}}}}{{{\text{Days}}}} \cr & = \frac{{20}}{{12}} \cr & = \frac{5}{3} \cr & = 1 + \frac{2}{3} \cr & \Rightarrow {\text{Efficiency of Y}} = \frac{2}{3} \cr} $$
Time taken by Y alone to complete the total work
$$\eqalign{ & = \frac{{24}}{{\frac{2}{3}}} \cr & = 36{\text{ days}} \cr} $$

Alternate :
$$\eqalign{ & {\text{X}} \times {\text{20}} = \left( {{\text{X}} + {\text{Y}}} \right) \times {\text{12}} \cr & \frac{{\text{X}}}{{{\text{X + Y}}}} \cr & = \frac{{12}}{{15}} \cr & = \frac{{3 \to {\text{Efficiency of X}}}}{{5 \to {\text{Efficiency of }}\left( {{\text{X}} + {\text{Y}}} \right)}} \cr & {\text{Efficiency of Y}} \cr & = 5 - 3 \cr & = 2{\text{units/day}} \cr & {\text{Total work}} \cr & = 24 \times 3 \cr & = 72{\text{units}} \cr & {\text{Total time taken by Y}} \cr & = \frac{{72}}{2} \cr & = {\text{36 days}} \cr} $$

Since we know efficiency and time are inversely proportion to each other.
$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{P}}:{\text{Q}} \cr & {\text{Efficiency}}\,\,\,3:4 \cr & {\text{Time}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4:3 \cr} $$

$$\eqalign{ & \left( {{\text{A}} + {\text{B}}} \right){\text{'s 2 days work}} \cr & = \left( {\frac{1}{{12}} + \frac{1}{{18}}} \right) \cr & = \frac{5}{{36}} \cr & {\text{Work done in 7 pairs of days}} \cr & = \left( {\frac{5}{{36}} \times 7} \right) \cr & = \frac{{35}}{{36}} \cr & {\text{Remaining work}} \cr & = \left( {1 - \frac{{35}}{{36}}} \right) \cr & = \frac{1}{{36}} \cr & {\text{On 15th day, it is A's turn}}{\text{.}} \cr & \frac{1}{{12}}{\text{ work is done by A in 1 day}}{\text{.}} \cr & \frac{1}{{36}}{\text{ work is done by A in}} \cr & = \left( {12 \times \frac{1}{{36}}} \right) \cr & = \frac{1}{3}{\text{ day}}{\text{.}} \cr & \therefore {\text{Total time taken}} = 14\frac{1}{3}{\text{ days}} \cr} $$

$$\eqalign{ & \left( {{\text{A}} + {\text{B}} + {\text{C}}} \right){\text{'s 3 days work}} \cr & = \frac{1}{{90}} + \frac{1}{{40}} + \frac{1}{{12}} \cr & = \frac{{43}}{{360}} \cr & \left( {{\text{A}} + {\text{B}} + {\text{C}}} \right){\text{'s 24 days work}} \cr & = \frac{{43}}{{360}} \times 8 \cr & = \frac{{344}}{{360}} \cr & {\text{Remaining work}} \cr & = \left( {1 - \frac{{344}}{{360}}} \right) \cr & = \frac{{16}}{{360}} \cr & = \frac{4}{{90}} \cr & {\text{On 25th day, it is A's turn}}{\text{.}} \cr & {\text{A's 1 day's work}} = \frac{1}{{90}} \cr & {\text{Remaining work}} \cr & = \left( {\frac{4}{{90}} - \frac{1}{{90}}} \right) \cr & = \frac{3}{{90}} \cr & = \frac{1}{{30}} \cr & {\text{On 26th day, it is B's turn}}{\text{.}} \cr & {\text{B's 1 day's work}} = \frac{1}{{40}} \cr & {\text{Remaining work}} \cr & = \left( {\frac{1}{{30}} - \frac{1}{{40}}} \right) \cr & = \frac{1}{{120}} \cr & {\text{On 27th day, it is C's turn}}{\text{.}} \cr & \frac{1}{{12}}{\text{ work is done by C in 1 day}}{\text{.}} \cr & \frac{1}{{120}}{\text{ work is done by C in }} \cr & = \left( {12 \times \frac{1}{{120}}} \right) \cr & = \frac{1}{{10}}{\text{ day}}{\text{.}} \cr} $$
Hence, the whole work is complete in $${\text{26}}\frac{1}{{10}}$$ days out of which A worked for 9 days,
B worked for 9 days and C worked for $${\text{8}}\frac{1}{{10}}$$ days.
Ratio of wages of A, B and C = Ratio of work done by A, B and C
$$ = \left( {\frac{1}{{90}} \times 9} \right):\left( {\frac{1}{{40}} \times 9} \right)$$     : $$\left( {\frac{1}{{12}} \times 8\frac{1}{{10}}} \right)$$
$$\eqalign{ & = \frac{1}{{10}}:\frac{9}{{40}}:\frac{{27}}{{40}} \cr & = 4:9:27 \cr & \therefore {\text{A's share}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{4}{{40}} \times 240} \right) \cr & = {\text{Rs}}{\text{.24}} \cr & {\text{B's share}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{9}{{40}} \times 240} \right) \cr & = {\text{Rs}}.54 \cr & {\text{C's share}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{27}}{{40}} \times 240} \right) \cr & = {\text{Rs}}{\text{.162}} \cr} $$

$$\eqalign{ & {\text{3 men's 1 day's work}} = \frac{1}{{20}} \cr & {\text{5 boy's 1 day's work}} = \frac{1}{{30}} \cr & \left( {{\text{3 men}} + {\text{5 boy}}} \right){\text{'s 1 day's work}} \cr & = \left( {\frac{1}{{20}} + \frac{1}{{30}}} \right) \cr & = \frac{5}{{60}} \cr & = \frac{1}{{12}} \cr} $$
∴ 3 men and 5 boys will complete the work in 12 days.

Let time taken by P = x days
Then, time taken by Q = 3x days
∴ 3x - x = 48
⇒ 2x = 48
⇒ x = 24
∴ (P + Q)'s 1 day's work
$$\eqalign{ & = \frac{1}{{24}} + \frac{2}{{72}} \cr & = \frac{{3 + 1}}{{72}} \cr & = \frac{1}{{18}} \cr} $$
∴ Required time = 18 days

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Alternate:
Ratio of times taken by P and Q = 1 : 3
The time difference is 3 - 1 = 2 days
while Q take 3 days and P takes 1 day
If difference of time is 2 days, Q takes 3 days
If difference of time is 48 days,
$$\eqalign{ & {\text{Q }}{\kern 1pt} {\text{takes}} = {\text{ }}{\kern 1pt} \left( {\frac{3}{2} \times 48} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 72\,{\kern 1pt} {\text{days}} \cr} $$
So, P takes 24 days to do the work.
$$\eqalign{ & {\text{P's 1 day's work}} = \frac{1}{{24}} \cr & {\text{Q's 1 day's work}} = \frac{1}{{72}} \cr & \left( {{\text{P + Q}}} \right){\text{'s 1 day's work}} \cr & {\text{ = }}\left( {\frac{{\text{1}}}{{24}}{\text{ + }}\frac{{\text{1}}}{{72}}} \right) = \frac{{\text{4}}}{{72}} = \frac{1}{{18}} \cr} $$
∴ A and B together can do the work in
$$\frac{{18}}{1} = {\text{18 }}{\kern 1pt} {\text{days}}$$

Let time taken by B = x
$${\text{Efficiency}} \propto \frac{1}{{{\text{Time}}\,{\text{taken}}}}$$
So, if B is 100% efficient. then A is 80% efficient
So,
$$\eqalign{ & \Rightarrow \frac{{80}}{{100}} = \frac{x}{{15/2}} \cr & \Rightarrow x = \frac{{80 \times 15}}{{100 \times 2}} \cr & \Rightarrow x = 6\,{\text{hours}} \cr} $$

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Alternate:
Number of hours taken by A to finish the work = $$7\frac{1}{2}$$ hours = $$\frac{15}{2}$$ hours
Work done by A in one hour : $$\frac{2}{15}$$
Let number of hours taken by B to finish the work : $$\frac{1}{{\text{x}}}$$
A can work 20% less than B that is $$\frac{20}{100}$$  = $$\frac{4}{5}$$ times of B’s work.
$$\eqalign{ & {\text{Here,}}\,\frac{4}{5}:1 = \frac{2}{{15}}:\frac{1}{{\text{x}}} \cr & \frac{4}{5} = \frac{{2{\text{x}}}}{{15}} \cr & {\text{x}} = \frac{{15 \times 4}}{{5 \times 2}} = 6\,{\text{hours}} \cr} $$