Time And Work - Aptitude MCQ Questions and Solutions with Explanations

	A	:	B
Efficiency →	100%	:	125%
	4	:	5
Time	5	:	4
	×1.2↓		↓×1.2
Actual time	6 days		$${\text{4}}\frac{4}{5}$$ days

Let time taken by B and C = x days
∴ Time taken by A = 3x days
∴ Part of work done by A, B and C in 1 day
$$\eqalign{ & = \frac{1}{{\text{x}}} + \frac{1}{{3{\text{x}}}} = \frac{{3 + 1}}{{3{\text{x}}}} = \frac{4}{{3{\text{x}}}} \cr & \therefore \frac{4}{{3{\text{x}}}} = \frac{1}{{24}} \cr & \Rightarrow 3{\text{x}} = 4 \times 24 \cr & \Rightarrow {\text{x}} = \frac{{4 \times 24}}{3} = 32\,{\text{days}} \cr} $$
∴ Time taken by A = 32 × 3 = 96 days.

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Alternet :
Work done by all of them together in $$\frac{1}{{24}}$$
Efficiency of A : Efficiency of (B + C) = 1 : 3
Work done by A in 1 day = $$\frac{1}{{24}}$$ × $$\frac{1}{{4}}$$ = $$\frac{1}{{96}}$$
i.e., A alone can finish the job in 96 days.

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Alternet :
Suppose A can complete the job in 3x days and (B + C) can complete the job in x days.
$$\eqalign{ & \frac{{3{\text{x}} \times {\text{x}}}}{{3{\text{x}} + {\text{x}}}} = 24 \cr & \Rightarrow \frac{{3{\text{x}}}}{4} = 24 \cr & \Rightarrow {\text{x}} = 32 \cr & 3{\text{x}} = 96\,{\text{days.}} \cr} $$

5 men's 15 day's work = 5 men's 10 day's work + 10 women's 5 day's work
⇒ 5 men's 5 day's work = 5 women's 5 day's work
⇒ 10 women's 5 day's work
$$\eqalign{ & = \left( {\frac{1}{{15}} \times 5} \right) \cr & = \frac{1}{3} \cr} $$
⇒ 10 women's 1 day's work = $$\frac{1}{{15}}$$
∴ 10 women can complete the work in 15 days.

100 men's 40 day's work + 100 men's 5 day's work = 1
⇒ 100 men's 45 day's work = 1
So, if the contractor had not engaged additional men, 100 men would have finished the work in 45 days.
Difference in time = (45 - 40) = 5 days

(1 man + 1 woman)'s 1 day's work
$$\eqalign{ & = \frac{1}{6} + \frac{1}{9} \cr & = \frac{5}{{18}} \cr & {\text{Remaining work}} \cr & = \left( {1 - \frac{5}{{18}}} \right) \cr & = \frac{{13}}{{18}} \cr} $$
Work done by 1 boy in 1 day = $$\frac{1}{{18}}$$
$$\eqalign{ & \therefore {\text{Number of boys required}} \cr & = \left( {\frac{{13}}{{18}} \times 18} \right) \cr & = 13 \cr} $$

	Man	:	Woman	:	Boy
Efficiency	4	:	2	:	1

Total work = Time × (Efficiency of man + woman + boy)
⇒ 7 days × (4 + 2 + 1) = 49 units
∴ Boy can do this work in = $$\frac{{49}}{1}$$ = 49 days

$$\eqalign{ & {\text{According to the question,}} \cr & \Rightarrow 2{\text{A}} = {\text{3B}} \cr & \Rightarrow \frac{{\text{A}}}{{\text{B}}} = \frac{3}{2} \cr & \Rightarrow {\text{Then efficiency ratio }} \cr & {\text{A}}:{\text{B}} = 3:2 \cr} $$
⇒ We know that time is inversely proportional to efficiency
⇒ Then time taken by them in ratio
$$\eqalign{ & {\text{A}}:{\text{B}} = \mathop {\mathop {{\text{ }}2}\limits_{4 \times \downarrow } {\text{ }}}\limits_{8{\text{days}}} :\mathop {\mathop 3\limits_{{\text{ }} \downarrow \times 4} }\limits_{12{\text{days}}} \cr & \because {\text{A can do the work in 8 days}} \cr & \Rightarrow {\text{i}}{\text{.e, 2 units}} \to {\text{8}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{1 unit}} \to {\text{4}} \cr & \Rightarrow {\text{Time taken by B}} \to {\text{3 units}} \cr & = 3 \times 4 \cr & = 12{\text{ days}} \cr} $$

A + B + C = 6 days [6 days total work]
$$\eqalign{ & {\text{According to the question,}} \cr & {\text{Ratio of their efficiencies,}} \cr & {\text{A}}:{\text{B}}:{\text{C}} \cr & 6{\text{ }}:3{\text{ }}:2 \cr & {\text{Total efficiencies}} \cr & \left( {6 + 3 + 2} \right){\text{units}} = 11{\text{ units}} \cr & {\text{Total work}} = 11 \times 6 = 66{\text{ units}} \cr} $$
Therefore, time taken by C to complete the work
$$\frac{{{\text{Total work}}}}{{{\text{Efficiencies}}}} = \frac{{66}}{2} = 33{\text{ days}}$$

A : B = 1 : 2
B : C = 1 : 2 (Multiply by 2)
B : C = 2 : 4
A : B : C = 1 : 2 : 4

$$\eqalign{ & {\text{1 men's 1 day's work}} \cr & = \frac{1}{{21 \times 3}} \cr & = \frac{1}{{63}} \cr & {\text{1 boy's 1 day's work}} \cr & = \frac{1}{{21 \times 9}} \cr & = \frac{1}{{189}} \cr & \left( {{\text{5 men}} + {\text{6 boy}}} \right){\text{'s 1 day's work}} \cr & = \frac{5}{{63}} + \frac{6}{{189}} \cr & = \frac{5}{{63}} + \frac{2}{{63}} \cr & = \frac{7}{{63}} \cr & = \frac{1}{9} \cr} $$
Hence, 5 men's and 6 boy's together can do the work in 9 days.