Time And Work - Aptitude MCQ Questions and Solutions with Explanations

According to the question,
Efficiency of a man, a woman and a child are
$$\eqalign{ & 5:4:2{\text{ units/day}} \cr & {\text{One day work of 2 men}} \cr & = 2 \times 5 \cr & = 10{\text{ units}} \cr & {\text{One day work of 3 women}} \cr & = 3 \times 4 \cr & = 12{\text{ units}} \cr & {\text{One day work of 4 children}} \cr & = 4 \times 2 \cr & = 8{\text{ units}} \cr} $$
Applying formula, let time taken is D days.
$$\frac{{\left( {10 + 12 + 8} \right) \times {{10}_{{\text{days}}}}}}{{{{10}_{{\text{hectare}}}}}} = $$     $$\left[ {\frac{{\left( {{6_{{\text{men}}}} \times 5} \right) + \left( {{4_{{\text{women}}}} \times 4} \right) + \left( {{7_{{\text{children}}}} \times 2} \right) \times {\text{D}}}}{{{{16}_{{\text{hectare}}}}}}} \right]$$
$$\eqalign{ & \Leftrightarrow \frac{{\left( {30} \right) \times 10}}{{10}} = \frac{{\left[ {60} \right] \times {\text{D}}}}{{16}} \cr & \Leftrightarrow {\text{D}} = 8{\text{ days}} \cr} $$

Applying formula
Let work done by 'n' men and women
$$\frac{{{{\text{p}}_{{\text{men}}}} \times {{\text{p}}_{{\text{hours}}}} \times {{\text{p}}_{{\text{days}}}}}}{{{{\text{p}}_{{\text{units}}}}}} = $$     $$\frac{{{{\text{n}}_{{\text{men}}}} \times {{\text{n}}_{{\text{hours}}}} \times {{\text{n}}_{{\text{days}}}}}}{{{{\text{W}}_{{\text{units}}}}}}$$
$$\eqalign{ & \Leftrightarrow {{\text{p}}^{\text{2}}} = \frac{{{{\text{n}}^{\text{3}}}}}{{\text{w}}} \cr & \Leftrightarrow {\text{W}} = \frac{{{{\text{n}}^{\text{3}}}}}{{{{\text{p}}^{\text{2}}}}} \cr} $$

$$\frac{{{{\text{2}}_{{\text{person}}}} \times {{\text{2}}_{{\text{days}}}}}}{{{{\text{2}}_{{\text{jobs}}}}}} = \frac{{{\text{100}} \times {\text{D}}}}{{{\text{10}}{{\text{0}}_{{\text{jobs}}}}}}$$      (D is number of days required)
$$\eqalign{ & {\text{After solving}} \cr & {\text{D}} = 2{\text{ days}} \cr} $$

(A's 1 day's work) : (B's 1 day's work)
$$\eqalign{ & = 2:1 \cr & \left( {{\text{A}} + {\text{B}}} \right){\text{'s 1 day's work}} = \frac{1}{{18}} \cr & \therefore {\text{A's 1 day's work}} \cr & = \left( {\frac{1}{{18}} \times \frac{2}{3}} \right) \cr & = \frac{1}{{27}} \cr} $$
Hence, A alone can finish the work in 27 days.

Ratio of times taken by A and B = 1 : 3
If difference of time is 2 days, B takes 3 days
If difference of time is 60 days, B takes
$$\eqalign{ & = \left( {\frac{3}{2} \times 60} \right) \cr & = {\text{90 days}} \cr} $$
So, A takes 30 days to do the work
$$\eqalign{ & {\text{A's 1 day's work}} = \frac{1}{{30}} \cr & {\text{B's 1 day's work}} = \frac{1}{{90}} \cr & \left( {{\text{A}} + {\text{B}}} \right){\text{'s 1 day's work}} \cr & = \left( {\frac{1}{{30}} + \frac{1}{{90}}} \right) \cr & = \frac{4}{{90}} \cr & = \frac{2}{{45}} \cr} $$
∴ A and B together can do the job in
$$\eqalign{ & = \frac{{45}}{2} \cr & = 22\frac{1}{2}\,{\text{days}} \cr} $$

(A's 1 day's work) : (B's 1 day's work)
$$\eqalign{ & = \frac{7}{4}:1 \cr & = 7:4 \cr} $$
Let A's and B's 1 day's work be 7x and 4x respectively
Then,
$$\eqalign{ & 7x + 4x = \frac{1}{7} \cr & \Rightarrow 11x = \frac{1}{7} \cr & \Rightarrow x = \frac{1}{{77}} \cr & \therefore {\text{A's 1 day's work}} \cr & = \left( {\frac{1}{{77}} \times 7} \right) \cr & = \frac{1}{{11}} \cr} $$
Hence, A alone can do the job in 11 days.

Let 'n' more number of man are required to complete the job in 20 day.
$$\frac{{{{80}_{{\text{days}}}} \times {\text{28}}{{\text{0}}_{{\text{worker}}}}}}{{{\text{3}}{\text{.5 km}}}} = $$     $$\frac{{{{\left( {{\text{280}} + {\text{n}}} \right)}_{{\text{worker}}}} \times {\text{2}}{{\text{0}}_{{\text{days}}}}}}{{{\text{1}}{\text{.5 km}}}}$$
$$\eqalign{ & {\text{After solving:}} \cr & \Rightarrow 480 = 280 + {\text{n}} \cr & \Rightarrow {\text{n}} = 200 \cr} $$

Put values in formula
$$\frac{{{{\left( {x - 1} \right)}_{{\text{men}}}} \times {{\left( {x + 1} \right)}_{{\text{days}}}}}}{{{{\text{9}}_{{\text{work}}}}}} = $$       $$\frac{{{{\left( {x + 2} \right)}_{{\text{men}}}} \times {{\left( {x - 1} \right)}_{{\text{days}}}}}}{{{{10}_{{\text{work}}}}}}$$
$$\eqalign{ & \Rightarrow \frac{{x + 1}}{9} = \frac{{x + 2}}{{10}} \cr & \Rightarrow 10x + 10 = 9x + 18 \cr & \Rightarrow x = 8 \cr} $$

	A	:	B
Efficiency →	100%	:	140%
	5	:	7
		⤩
Time	7	:	5
	×10↓		↓×10
Actual Time	70 days		50 days

Time taken by B to complete the work
$$\eqalign{ & = 12 \times \frac{{100}}{{160}} \cr & = \frac{{15}}{2}\,{\text{days}} \cr} $$
∴ (A + B)'s 1 day's work
$$\eqalign{ & = \frac{1}{{12}} + \frac{2}{{15}} \cr & = \frac{{5 + 8}}{{60}} \cr & = \frac{{13}}{{60}} \cr} $$
Hence, the work will be completed in $$\frac{{60}}{{13}}$$ days

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Alternate:
Ratio of time, taken by A and B = 160 : 100 = 8 : 5
If A takes 8 days, B takes 5 days.
If A takes 12 days, B takes = $$\frac{5}{8} \times 12 = \frac{{ 15}}{2}$$
Time take by A = 12 days
Time take by B = 7.5 days
L.C.M. of Total Work = 60
Ratio of time, taken by A and B = 8 : 5
Time taken by A and B together to complete the task
$$ = \frac{{60}}{{13}}{\text{days}}$$