Time and Work - 02

Explanation:-

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

Explanation:-

$$\eqalign{ & {\text{A's}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} = \frac{1}{4}; \cr & {\text{(B + C)'s}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} = \frac{1}{3}; \cr & {\text{(A + C)'s}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} = \frac{1}{2}. \cr & {\text{(A + B + C)'s}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} \cr & = \left( {\frac{1}{4} + \frac{1}{3}} \right) = \frac{7}{{12}} \cr & {\text{B's}}\,{\text{1}}\,{\text{hour's}}\,{\text{work}} \cr & = \left( {\frac{7}{{12}} - \frac{1}{2}} \right) = \frac{1}{{12}} \cr & \therefore {\text{B}}\,{\text{along}}\,{\text{will}}\,{\text{take}}\,{\text{12}}\,{\text{hours}}\,{\text{to}}\,{\text{do}}\,{\text{the}}\,{\text{work}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{C's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \frac{1}{3} - \left( {\frac{1}{6} + \frac{1}{8}} \right) = \frac{1}{3} - \frac{7}{{24}} = \frac{1}{{24}} \cr & {\text{A's}}\,{\text{wages:B's}}\,{\text{wages:C's}}\,{\text{wages}} \cr & = \frac{1}{6}:\frac{1}{8}:\frac{1}{{24}} \cr & = 4:3:1 \cr & \therefore {\text{C's}}\,{\text{share}}\,{\text{(for}}\,{\text{3}}\,{\text{days}}\,{\text{)}} \cr & = Rs.\,\left( {3 \times \frac{1}{{24}} \times 3200} \right) \cr & = Rs.\,400 \cr}$$

Explanation:-

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

Explanation:-

$$\eqalign{ & {\text{A's}}\,{\text{2}}\,{\text{day's}}\,{\text{work}} = \left( {\frac{1}{{20}} \times 2} \right) = \frac{1}{{10}} \cr & \left( {{\text{A + B + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \left( {\frac{1}{{20}} + \frac{1}{{30}} + \frac{1}{{60}}} \right) = \frac{6}{{60}} = \frac{1}{{10}} \cr & {\text{Work}}\,{\text{done}}\,{\text{in}}\,{\text{3}}\,{\text{days}} = \left( {\frac{1}{{10}} + \frac{1}{{10}}} \right) = \frac{1}{5} \cr & {\text{Now}},\,\frac{1}{5}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{in}}\,{\text{3}}\,{\text{dys}} \cr & \therefore {\text{Whole}}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{in}}\, \cr & \left( {3 \times 5} \right) = 15\,{\text{days}} \cr}$$

Explanation:-

$$\eqalign{ & \left( {{\text{A + B + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{4} \cr & {\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{16}} \cr & {\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{12}} \cr & \therefore {\text{C's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \frac{1}{4} - \left( {\frac{1}{{16}} + \frac{1}{{12}}} \right) = \left( {\frac{1}{4} - \frac{7}{{48}}} \right) = \frac{5}{{48}} \cr & {\text{So,}}\,{\text{C}}\,\,{\text{alone}}\,{\text{can}}\,{\text{do}}\,{\text{the}}\,{\text{work}}\,{\text{in}} \cr & \frac{{48}}{5} = 9\frac{3}{5}\,{\text{days}} \cr}$$

Explanation:-

Pipe A can fill the empty tank in = 0.9 hours
So work rate of the Pipe A = $$\frac{{100}}{{0.9}}$$ % per hour.
Pipe B can empty the tank in = 0.7 hours
Negative Work rate of B = $$\frac{{100}}{{0.7}}$$ % per hour. (B is removing water, so, taken as negative work.)
Tank fill per hour = $$\frac{{100}}{{0.7}} - \frac{{100}}{{0.9}}$$   = 31.75% per hour.
Time Taken to empty the tank = $$\frac{{100}}{{31.75}}$$   = 3.14 hours.
So, Capacity of the tank = 3.14 × 180 = 567 liters.

Explanation:-

Given,
A is 60% more efficient of B Means,
A = B + 60% of B.
A = $$\frac{{8{\text{B}}}}{5}$$
A can complete whole work in 15 days. So,
One day work of A = $$\frac{1}{{15}}$$
One day work of $$\frac{{8{\text{B}}}}{5}$$ = $$\frac{1}{{15}}$$
One day work of B = $$\frac{5}{{120}}$$  = $$\frac{1}{{24}}$$
One day work, (A + B),
$$= \frac{1}{{15}} + \frac{1}{{24}}$$
One day work, (A + B) = $$\frac{{24 + 15}}{{360}}$$   = $$\frac{{39}}{{360}}$$
So, Time taken to finish the work by A and B together = $$\frac{{360}}{{39}}$$ = $$\frac{{120}}{{13}}$$

Alternatively
We can solve it through percentage method
Given,
A = $$\frac{{8{\text{B}}}}{5}$$
A can complete whole work in 15 days
Work rate of A = $$\frac{{100}}{{15}}$$ = 6.66% per day
Work rate of $$\frac{{8{\text{B}}}}{5}$$ = 6.66%
Work rate of B = 4.16% per day
Work rate of (A + B) = 6.66 + 4.16 = 10.82% per day
So, A and B can complete 100% work

Explanation:-

His Per day pay = Rs. 26
Total pay employee got = Rs. 829
Total pay he gets if he did not remain idle a single day,
= 26 × 56 = Rs.1456
He Forfeits or fined = 1456 - 829 = Rs. 627
Per day he Forfeits Rs. 7 Means per idle day he loses = 26 + 7 = Rs. 33
So, Total idle days = $$\frac{{627}}{{33}}$$ = 19 days

Alternatively,
Let he works for X days and remain idle for (56 - X)
Now, According to question,
X × 26 - 7 × (56 - X) = 829
26X + 7X - 392 = 829
33X = 829 + 392 = 1221
X = 37 days
No of days remain idle = 56 -37 =19 days

Explanation:-

To do half of the work in one sixth of the time means A is 3 times as efficient as B. To understand this point just think if same work would have been done in one sixth of the time then A would have been six times as efficient as B.
Now they together complete the work in 10 days. B is working with someone who is 3 times as efficient as himself. That means 4 people of B's efficiency finished the work in 10 days. So B alone would have done it in 40 days. A alone would have taken one third of this time.

Alternatively,
Given,
$${\text{A}} \times \frac{1}{6} = {\text{B}} \times \frac{1}{2}$$
So, A = 3B
Now,
Given they together complete the work in 10 days
So, One Day''s work of,
(A + B) = $$\frac{{100}}{{10}}$$ = 10%
(3B + B) = 10%
4B = 10%
So, one day work of B = $$\frac{{10}}{4}$$ = 2.5%
So, B can comp