Time And Work - Aptitude MCQ Questions and Solutions with Explanations

1^st method:
A and B complete a work in = 15 days
One day's work of (A + B) = $$\frac{1}{{15}}$$
B complete the work in = 20 days;
One day's work of B = $$\frac{1}{{20}}$$
Then, A's one day's work
$$\eqalign{ & = \frac{1}{{15}} - \frac{1}{{20}} \cr & = \frac{{4 - 3}}{6} \cr & = \frac{1}{{60}} \cr} $$
Thus, A can complete the work in = 60 days.

2^nd method:
(A + B)'s one day's % work = $$\frac{{100}}{{15}}$$ = 6.66%
B's one day's % work = $$\frac{{100}}{{20}}$$ = 5%
A's one day's % work = 6.66 - 5 = 1.66%
Thus, A need = $$\frac{{100}}{{1.66}}$$ = 60 days to complete the work.

$$\eqalign{ & {\text{One}}\,{\text{day's}}\,{\text{work}}\,{\text{of}} \cr & \left( {A + B} \right) = \frac{1}{{18}}\,.......(1) \cr & {\text{One}}\,{\text{day's}}\,{\text{work}}\,{\text{of}} \cr & \left( {A + C} \right) = \frac{1}{{12}}\,.......\left( 2 \right) \cr & {\text{One}}\,{\text{day's}}\,{\text{work}}\,{\text{of}} \cr & \left( {B + C} \right) = \frac{1}{9}\,.......\left( 3 \right) \cr & {\text{Adding}}\,\left( {\text{1}} \right){\text{,}}\,\left( {\text{2}} \right)\,{\text{and}}\,\left( {\text{3}} \right) \cr & 2 \times \left( {A + B + C} \right) = { {\frac{1}{{18}}} + {\frac{1}{{12}}} + {\frac{1}{9}} } \cr & 2(A + B + C) = \frac{1}{4} \cr & {\text{One day's work of}} \cr & {A + B + C} = \frac{1}{8} \cr & B = {\frac{1}{8}} - \left( {A + C} \right) \cr & B = {\frac{1}{8}} - {\frac{1}{{12}}} \cr & {\text{One day's work of}} \cr & B = \frac{{ {3 - 2} }}{{24}} = \frac{1}{{24}} \cr & B\,{\text{need}}\,24\,{\text{days}} \cr} $$

1^st Method:
(A+B)'s one day's work = $$\frac{1}{3}$$ part
(A+B) works 2 days together = $$\frac{2}{3}$$ part
Remaining work = $$1 - \frac{2}{3}$$  = $$\frac{1}{3}$$ part
$$\frac{1}{3}$$ part of work is completed by A in two days
Hence, one day's work of A = $$\frac{1}{6}$$
Then, one day's work of B = $$\frac{1}{3} - \frac{1}{6}$$  = $$\frac{1}{6}$$
So, B alone can complete the whole work in 6 days.

2^nd Method:
(A+B)'s one day's % work = $$\frac{{100}}{3}$$ = 33.3%
Work completed in 2 days = 66.6%
Remaining work = 33.4%
One day's % work of A = $$\frac{{33.4}}{2}$$ = 16.7%
One day's work of B = 33.4 - 16.7 = 16.7%
B alone can complete the work in,
= $$\frac{{100}}{{16.7}}$$
= 5.98 days
≈ 6 days.

1^st Method:
$$\eqalign{ & \left( {B + C} \right)\,2\,{\text{days}}\,{\text{work}} \cr & = 2 \times \left( {\frac{1}{{20}} + \frac{1}{{30}}} \right) \cr & = 2 \times {\frac{{3 + 2}}{{60}}} \cr & = \frac{1}{6}{\text{part}} \cr & {\text{Remaining}}\,{\text{work}} \cr & = 1 - \frac{1}{6} \cr & = \frac{5}{6}\text{part} \cr & {\text{A's}}\,{\text{one}}\,{\text{day's}}\,{\text{work}} \cr & = \frac{1}{{18}}{\text{part}} \cr & {\text{Time taken to complete the work}} \cr & = \frac{{ {\frac{5}{6}} }}{{ {\frac{1}{{18}}} }}\,{\text{days}} \cr & {\text{Hence,}} \cr & {\text{Time taken to complete the work}} \cr & = {\frac{5}{6}} \times 18 \cr & = 15\,{\text{days}} \cr} $$

2^nd Method:
% of work B completes in one day = $$\frac{{100}}{{20}}$$ = 5%;
% of work C completes in one day = $$\frac{{100}}{{30}}$$ = 3.33%;
% of work (A + B) completes together in one day = 5 + 3.33 = 8.33%;
% work (A + B) completes together in 2 days = 8.66 × 2 = 16.66%;
Remaining work = 100 - 16.66 = 83.34%;
% of work A completes in 1 day = $$\frac{{100}}{{18}}$$ = 5.55%
Time taken to complete the remaining work by A
= $$\frac{{83.34}}{{5.55}}$$
= 15 days

1^st Method:
Working 5 hours a day, A can complete the work in 8 days i.e.
= 5 × 8 = 40 hours
Working 6 hours a day, B can complete the work in 10 days i.e.
= 6 × 10 = 60 hours
(A + B)'s 1 hour's work,
$$\eqalign{ & = \frac{1}{{40}} + \frac{1}{{60}} \cr & = \frac{{3 + 2}}{{120}} \cr & = \frac{5}{{120}} \cr & = \frac{1}{{24}} \cr} $$
Hence, A and B can complete the work in 24 hours i.e. they require 3 days to complete the work.

2^nd Method:
% 1 hour's work of A = $$\frac{{100}}{{40}}$$ = 2.5%
% 1 hour's work of B = $$\frac{{100}}{{60}}$$ = 1.66%
(A + B) one hour's % work,
= (2.5 + 1.66) = 4.16%
Time to complete the work,
= $$\frac{{100}}{{4.16}}$$ = 24 hours
Then, $$\frac{{24}}{8}$$ = 3 days
They need 3 days, working 8 hours a day to complete the work.

According to question,
Ganga begins at 9 am and she does 3 units/hours
Saraswati begins at 10 am and she does 2 units/hours
So by 11 am they complete 5 units
Time $$ = \frac{{{\text{T}}{\text{.W}}.}}{{3 + 2}} = \frac{{24}}{5}$$
(4 cycle of 2 hrs each + 4 units left)
And now ganga will complete 3 unit out of 4 units in 1 hr
Now, rest 1 unit work done by = $$\frac{1}{2}$$ hr
Total time = 8 + 1 + $$\frac{1}{2}$$ = 9$$\frac{1}{2}$$ hr
Hence, Work finished at
= 9 am + 9$$\frac{1}{2}$$ hr
= 6:30 PM

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Alternate Solution:
Work done by Ganga in 1 hour = $$\frac{1}{8}$$
Work done by Saraswati in 1 hour = $$\frac{1}{12}$$
They are working alternatively with Ganga beginning the job.
Work done in every two hours = $$\frac{1}{8}$$ + $$\frac{1}{12}$$ = $$\frac{5}{24}$$
Work done in 4 × 2 = 8 hours = $$\frac{5\times4}{24}$$ = $$\frac{5}{6}$$
Remaining work = 1 - $$\frac{5}{6}$$ = $$\frac{1}{6}$$
In 9th hour, Ganga starts the work and does $$\frac{1}{8}$$ of the work
Work remaining = $$\frac{1}{6}$$ - $$\frac{1}{8}$$ = $$\frac{1}{24}$$
In 10th hour, Saraswati starts the work
Time needed to finish the remaining work
$$\eqalign{ & = \frac{{\frac{1}{{24}}}}{{\frac{1}{{12}}}} \cr & = \frac{1}{{24}} \times 12 \cr} $$
  $$=$$ 0.5 hours
  $$=$$ 30 minutes
i.e., work will be completed in 9 hour 30 minutes, after 9 AM
i.e., at 6:30 PM

Here, we use work equivalence method;
10 × 12 = 12 × x
Or, x = 10 days

To understand the work equivalence method, we use a graphic as follows:

Men   Days
10 ↓    12
12     ↑ x (let)
Here, the two arrows, downward (↓) and upward (↑) show variation between men and days.
[If downward arrows show decrements then upward arrows show increments and vice-verse.]
Thus,
$$\eqalign{ & \frac{{10}}{{12}} = \frac{{\text{x}}}{{12}} \cr & {\text{or, x}} = \frac{{10 \times 12}}{{12}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 10\,{\text{days}} \cr} $$

$$\eqalign{ & {\text{We have}} \cr & {\text{B}} = \frac{3}{2} \times {\text{A}} \cr & \to {\text{A}} = \frac{2}{3} \times {\text{B}} \cr & {\text{One day's work,}} \cr & \Rightarrow {\text{A}} + {\text{B}} = \frac{1}{{18}} \cr & \Rightarrow \frac{2}{3} \times {\text{B}} + {\text{B}} = \frac{1}{{18}} \cr & \Rightarrow \frac{5}{3} \times {\text{B}} = \frac{1}{{18}} \cr & {\text{One}}\,{\text{day's}}\,{\text{work}}\,{\text{of}}\,{\text{B}} \cr & = \frac{3}{{90}} \cr} $$
B alone can complete the work in
$$\eqalign{ & = \frac{{90}}{3} \cr & = 30\,{\text{days}} \cr} $$

10 men = 20 boys
→1 men = 2 boys
8 men = 2 × 8 boys = 16 boys
Then,
(16 boys + 4 boys) = 20 boys can make 260 mats in 20 days
Now,
It can be calculated by work equivalence method:
20 × 260 × 20 = x × 20 × 20
x = 260 mats

$$\frac{7}{{10}}$$ part of work has been completed by A in 15 days. Then,
Rest work = 1 - $$\frac{7}{{10}}$$ = $$\frac{3}{{10}}$$ part
Given, That $$\frac{3}{{10}}$$ part of the work is completed by A and B together in 4 days. Means,
(A + B) completed the $$\frac{3}{{10}}$$ of work in 4 days
So, (A + B)'s 1 day's work = $$\frac{3}{{10 \times 4}}$$  = $$\frac{3}{{40}}$$
Hence,
(A + B) can complete the work in $$\frac{{40}}{3}$$ = $$13\frac{1}{3}$$ days