Time and Work - 01

Explanation:-

According to the question,
As they complete the same work in same time
6 men + 5 women
= 6 men + 3 men
= 9 men
If, 3 men does a work in 12 days
1 men does a work in 12 × 3
9 men does a work in $$\frac{{{\text{12}} \times {\text{3}}}}{9}$$   = 4 days

Alternate :
$$\eqalign{ & 3{\text{ men}} \times {\text{12}} = {\text{5 women}} \times {\text{12}} \cr & \frac{{{\text{men}}}}{{{\text{women}}}} = \frac{5}{3} \cr & {\text{Total work}} \cr & = 3 \times 5 \times 12 \cr & = 15 \times 12 \cr & = 180{\text{ units}} \cr & {\text{Efficiency of 6 men}} + {\text{5 women}} \cr & = 6 \times 5 + 5 \times 3 \cr & = 45 \cr & {\text{Time taken by 6 men}} + {\text{5 women}} \cr & = \frac{{180}}{{45}} \cr & = 4{\text{ days}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{According to the question,}} \cr & {\text{15 men}} = {\text{20 days}} \cr & {\text{300 men}} = 1{\text{ days}}.....{\text{(i)}} \cr & {\text{24 women}} = {\text{20 days}} \cr & {\text{480 men}} = 1{\text{ days}}......{\text{(ii)}} \cr & {\text{Compare equation (i) and (ii)}} \cr & {\text{300 men}} = 480{\text{ women}} \cr & {\text{5 men}} = 8{\text{ women}}.....{\text{(iii)}} \cr & {\text{10 men}} + 8{\text{ women}} = ? \cr & {\text{10 men}} + {\text{5 men}} = ? \cr & 15\,{\text{men}} = ? \cr}$$
$${\text{15 men}} \times {\text{20 days}}$$     = $${\text{15 men}}$$   $$\times$$ $$x{\text{ days}}$$
Alternate
$$\eqalign{ & {\text{15m}} \times {\text{20 days}} = 24{\text{w}} \times 20{\text{ days}} \cr & \frac{{\text{m}}}{{\text{w}}} = \frac{8}{5} \cr}$$
So, 1 man work 8 units work in one day
and 1 woman work 5 units work in one day
Total work = 15 × 8 × 20
Hence, (10 men + 8 women) work whole in D days
$$\eqalign{ & \left( {{\text{10m}} + {\text{8w}}} \right) \times {\text{D}} = 15 \times 8 \times 20 \cr & \left( {{\text{10}} \times {\text{8}} + {\text{8}} \times {\text{5}}} \right) \times {\text{D}} = 15 \times 8 \times 20 \cr & \left( {{\text{10}} + 40} \right) \times {\text{D}} = 15 \times 8 \times 20 \cr & {\text{D}} = 20{\text{ days}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Let}}\,{\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = x\,{\text{and}} \cr & {\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = y \cr & {\text{Then}},\,x + y = \frac{1}{{30}} \cr & 16x + 44y = 1 \cr & {\text{Solving}}\,{\text{these}}\,{\text{two}}\,{\text{equations,}}\,{\text{we}}\,{\text{get}} \cr & x = \frac{1}{{60}}\,{\text{and}}\,y = \frac{1}{{60}} \cr & \therefore {\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{60}} \cr & {\text{Hence,}}\,{\text{B}}\,{\text{alone}}\,{\text{shall}}\,{\text{finish}}\,{\text{the}} \cr & {\text{whole}}\,{\text{work}}\,{\text{in}}\,{\text{60}}\,{\text{days}} \cr}$$

Explanation:-

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

Explanation:-

$$\eqalign{ & {\text{Work}}\,{\text{done}}\,{\text{by}}\,{\text{X}}\,{\text{in}}\,{\text{8}}\,{\text{days}} \cr & = \left( {\frac{1}{{40}} \times 8} \right) = \frac{1}{5} \cr & {\text{Remaining}}\,{\text{work}} \cr & = \left( {1 - \frac{1}{5}} \right) = \frac{4}{5} \cr & {\text{Now}},\,\frac{4}{5}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{Y}}\,{\text{in}}\,{\text{16}}\,{\text{days}} \cr & {\text{Whole}}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{Y}}\,{\text{in}}\, \cr & \left( {16 \times \frac{5}{4}} \right) = 20\,{\text{days}} \cr & \therefore {\text{X's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{40}} \cr & \therefore {\text{Y's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{20}} \cr & \left( {{\text{X + Y}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \left( {\frac{1}{{40}} + \frac{1}{{20}}} \right) = \frac{3}{{40}} \cr & {\text{Hence,}}\,{\text{X}}\,{\text{and}}\,{\text{Y}}\,{\text{will}}\,{\text{together}}\,{\text{complete}}\,{\text{the}}\,{\text{work}}\,{\text{in}} \cr & \left( {\frac{{40}}{3}} \right) = 13\frac{1}{3}\,{\text{days}} \cr}$$

Explanation:-

$$\eqalign{ & \left( {{\text{B + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \left( {\frac{1}{9} + \frac{1}{{12}}} \right) = \frac{7}{{36}} \cr & {\text{Work}}\,{\text{done}}\,{\text{by}}\,{\text{B}}\,{\text{and}}\,{\text{C}}\,{\text{in}}\,{\text{3}}\,{\text{days}} \cr & = \left( {\frac{7}{{36}} \times 3} \right) = \frac{7}{{12}} \cr & {\text{Remaining}}\,{\text{work}} \cr & = \left( {1 - \frac{7}{{12}}} \right) = \frac{5}{{12}} \cr & {\text{Now}},\,\frac{1}{{24}}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & {\text{So}},\,\frac{5}{{12}}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}} \cr & \left( {24 \times \frac{5}{{12}}} \right) = 10\,{\text{days}} \cr}$$

Explanation:-

$$\eqalign{ & \left( {20 \times 16} \right){\text{women}}\,{\text{can}}\,{\text{complete}}\,{\text{the}}\,{\text{work}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & \therefore {\text{1}}\,{\text{woman's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{320}} \cr & \left( {16 \times 15} \right)\,{\text{men}}\,{\text{can}}\,{\text{complete}}\,{\text{the}}\,{\text{work}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & \therefore {\text{1}}\,{\text{man's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{240}} \cr & {\text{So,}}\,{\text{required}}\,{\text{ratio}} \cr & = \frac{1}{{240}}:\frac{1}{{320}} \cr & = \frac{1}{3}:\frac{1}{4} \cr & = 4:3\,\left( {{\text{cross}}\,{\text{multiplied}}} \right) \cr}$$

Explanation:-

$$\eqalign{ & {\text{Ration}}\,{\text{of}}\,{\text{rates}}\,{\text{of}}\,{\text{working}}\,{\text{of}}\,{\text{A}}\,{\text{and}}\,{\text{B}} \cr & = 2:1 \cr & {\text{So,}}\,{\text{ratio}}\,{\text{of}}\,{\text{times}}\,{\text{taken}} = 1:2 \cr & {\text{B's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{12}} \cr & \therefore {\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \frac{1}{3};\,({\text{2times}}\,{\text{of}}\,{\text{B's}}\,{\text{work}}) \cr & \left( {{\text{A + B}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \left( {\frac{1}{6} + \frac{1}{{12}}} \right) = \frac{3}{{12}} = \frac{1}{4} \cr & {\text{So,}}\,{\text{A}}\,{\text{and}}\,{\text{B}}\,{\text{together}}\,{\text{can}}\,{\text{finish}}\,{\text{the}} \cr & {\text{work}}\,{\text{in}}\,{\text{4}}\,{\text{days}}{\text{.}}\, \cr}$$

Explanation:-

$$\eqalign{ & {\text{2(A + B + C)'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \left( {\frac{1}{{30}} + \frac{1}{{24}} + \frac{1}{{20}}} \right) \cr & = \frac{{15}}{{120}} = \frac{1}{8} \cr & \therefore \left( {{\text{A + B + C}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \frac{1}{{2 \times 8}} = \frac{1}{{16}} \cr & {\text{Work}}\,{\text{done}}\,{\text{by}}\,{\text{A,}}\,{\text{B,}}\,{\text{C}}\,{\text{in}}\,{\text{10}}\,{\text{days}} \cr & = \frac{{10}}{{16}} = \frac{5}{8} \cr & {\text{Remaining}}\,{\text{work}} \cr & = \left( {1 - \frac{5}{8}} \right) = \frac{3}{8} \cr & {\text{A's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \left( {\frac{1}{{16}} - \frac{1}{{24}}} \right) = \frac{1}{{48}} \cr & {\text{Now}},\,\frac{1}{{48}}\,{\text{work}}\,{\text{isdone}}\,{\text{by}}\,{\text{A}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & {\text{So}},\,\frac{3}{8}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}} \cr & \left( {48 \times \frac{3}{8}} \right) = 18\,{\text{days}} \cr}$$

Explanation:-

$$\eqalign{ & \left( {{\text{A + B}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \left( {\frac{1}{{15}} + \frac{1}{{10}}} \right) = \frac{1}{6} \cr & {\text{Work}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{and}}\,{\text{B}}\,{\text{in}}\,{\text{2}}\,{\text{days}} \cr & = \left( {\frac{1}{6} \times 2} \right) = \frac{1}{3} \cr & {\text{Remaining}}\,{\text{work}} \cr & = \left( {1 - \frac{1}{3}} \right) = \frac{2}{3} \cr & {\text{Now}},\,\frac{1}{{15}}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{A}}\,{\text{in}}\,{\text{1}}\,{\text{day}} \cr & \therefore \frac{2}{3}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{a}}\,{\text{in}} \cr & \left( {15 \times \frac{2}{3}} \right) = 10\,{\text{days}} \cr & {\text{Hence,}}\,{\text{the}}\,{\text{total}}\,{\text{time}}\,{\text{taken}} \cr & = \left( {10 + 2} \right) = 12\,{\text{days}} \cr}$$