Answer
Verified
398.4k+ views
Hint: In this question, we will proceed by finding out how many days it will take to complete the work by a single woman and a single man. Then we will find how many days it takes to complete the work by one man and one woman. So, use this concept to reach the solution of the given problem.
Complete step-by-step answer:
Given that work done by four men and three women in six days is equal to the work done by five men and seven women in four days.
Let us consider that one man takes \[x\] days and one woman takes \[y\] days to complete the work. So, work done per day by a man is \[\dfrac{1}{x}\] and the work done per day by a woman is \[\dfrac{1}{y}\].
Since 4 men and 3 women can done a work in 6 days, the work done by them per day is given by
\[ \Rightarrow \dfrac{4}{x} + \dfrac{3}{y} = \dfrac{1}{6}...........................\left( 1 \right)\]
Since 5 men and 7 women can done a work in 4 days, the work done by them per day is given by
\[ \Rightarrow \dfrac{5}{x} + \dfrac{7}{y} = \dfrac{1}{4}...........................\left( 2 \right)\]
Now, multiplying equation (1) with 5 we have
\[ \Rightarrow \dfrac{{20}}{x} + \dfrac{{15}}{y} = \dfrac{5}{6}...........................\left( 3 \right)\]
And, multiplying equation (2) with 4 we have
\[ \Rightarrow \dfrac{{20}}{x} + \dfrac{{28}}{y} = 1...........................\left( 4 \right)\]
Subtracting equation (3) from (4), we have
\[
\Rightarrow \left( {\dfrac{{20}}{x} + \dfrac{{28}}{y}} \right) - \left( {\dfrac{{20}}{x} + \dfrac{{15}}{y}} \right) = 1 - \dfrac{5}{6} \\
\Rightarrow \dfrac{{20}}{x} - \dfrac{{20}}{x} + \dfrac{{28}}{y} - \dfrac{{15}}{y} = \dfrac{{6 - 5}}{6} \\
\Rightarrow \dfrac{{28 - 15}}{y} = \dfrac{1}{6} \\
\Rightarrow \dfrac{{13}}{y} = \dfrac{1}{6} \\
\therefore y = 13 \times 6 = 78 \\
\]
Substituting \[y = 78\] in equation (1), we have
\[
\Rightarrow \dfrac{4}{x} + \dfrac{3}{{78}} = \dfrac{1}{6} \\
\Rightarrow \dfrac{4}{x} = \dfrac{1}{6} - \dfrac{3}{{78}} \\
\Rightarrow \dfrac{4}{x} = \dfrac{{13 - 3}}{{78}} \\
\Rightarrow \dfrac{4}{x} = \dfrac{{10}}{{78}} \\
\therefore x = \dfrac{{78 \times 4}}{{10}} = \dfrac{{156}}{5} \\
\]
Now, time taken to finish the work by 1 man and 1 woman \[ = \dfrac{1}{x} + \dfrac{1}{y}\]
\[
= \dfrac{1}{{\dfrac{{156}}{5}}} + \dfrac{1}{{78}} \\
= \dfrac{5}{{156}} + \dfrac{1}{{78}} \\
= \dfrac{{5 + 2}}{{156}} \\
= \dfrac{7}{{156}}{\text{ }} \\
\]
Therefore, the number of days taken to complete the work by one man and one woman is equal to \[\dfrac{{156}}{7} = 22\dfrac{2}{7}{\text{ days}}\].
Thus, the correct option is A. \[22\dfrac{2}{7}\].
Note: In questions like these, work done is calculated in fraction, this fraction is actually the reciprocal of the number of days in which work is done. So, the denominator of the fraction stands for the number of days. Here the obtained number of days in which only one man and one woman can finish the work must be less than the number of days taken to finish the work by four men and three women and also the number of days taken to finish the work by five men and seven women. Otherwise our obtained answer is wrong.
Complete step-by-step answer:
Given that work done by four men and three women in six days is equal to the work done by five men and seven women in four days.
Let us consider that one man takes \[x\] days and one woman takes \[y\] days to complete the work. So, work done per day by a man is \[\dfrac{1}{x}\] and the work done per day by a woman is \[\dfrac{1}{y}\].
Since 4 men and 3 women can done a work in 6 days, the work done by them per day is given by
\[ \Rightarrow \dfrac{4}{x} + \dfrac{3}{y} = \dfrac{1}{6}...........................\left( 1 \right)\]
Since 5 men and 7 women can done a work in 4 days, the work done by them per day is given by
\[ \Rightarrow \dfrac{5}{x} + \dfrac{7}{y} = \dfrac{1}{4}...........................\left( 2 \right)\]
Now, multiplying equation (1) with 5 we have
\[ \Rightarrow \dfrac{{20}}{x} + \dfrac{{15}}{y} = \dfrac{5}{6}...........................\left( 3 \right)\]
And, multiplying equation (2) with 4 we have
\[ \Rightarrow \dfrac{{20}}{x} + \dfrac{{28}}{y} = 1...........................\left( 4 \right)\]
Subtracting equation (3) from (4), we have
\[
\Rightarrow \left( {\dfrac{{20}}{x} + \dfrac{{28}}{y}} \right) - \left( {\dfrac{{20}}{x} + \dfrac{{15}}{y}} \right) = 1 - \dfrac{5}{6} \\
\Rightarrow \dfrac{{20}}{x} - \dfrac{{20}}{x} + \dfrac{{28}}{y} - \dfrac{{15}}{y} = \dfrac{{6 - 5}}{6} \\
\Rightarrow \dfrac{{28 - 15}}{y} = \dfrac{1}{6} \\
\Rightarrow \dfrac{{13}}{y} = \dfrac{1}{6} \\
\therefore y = 13 \times 6 = 78 \\
\]
Substituting \[y = 78\] in equation (1), we have
\[
\Rightarrow \dfrac{4}{x} + \dfrac{3}{{78}} = \dfrac{1}{6} \\
\Rightarrow \dfrac{4}{x} = \dfrac{1}{6} - \dfrac{3}{{78}} \\
\Rightarrow \dfrac{4}{x} = \dfrac{{13 - 3}}{{78}} \\
\Rightarrow \dfrac{4}{x} = \dfrac{{10}}{{78}} \\
\therefore x = \dfrac{{78 \times 4}}{{10}} = \dfrac{{156}}{5} \\
\]
Now, time taken to finish the work by 1 man and 1 woman \[ = \dfrac{1}{x} + \dfrac{1}{y}\]
\[
= \dfrac{1}{{\dfrac{{156}}{5}}} + \dfrac{1}{{78}} \\
= \dfrac{5}{{156}} + \dfrac{1}{{78}} \\
= \dfrac{{5 + 2}}{{156}} \\
= \dfrac{7}{{156}}{\text{ }} \\
\]
Therefore, the number of days taken to complete the work by one man and one woman is equal to \[\dfrac{{156}}{7} = 22\dfrac{2}{7}{\text{ days}}\].
Thus, the correct option is A. \[22\dfrac{2}{7}\].
Note: In questions like these, work done is calculated in fraction, this fraction is actually the reciprocal of the number of days in which work is done. So, the denominator of the fraction stands for the number of days. Here the obtained number of days in which only one man and one woman can finish the work must be less than the number of days taken to finish the work by four men and three women and also the number of days taken to finish the work by five men and seven women. Otherwise our obtained answer is wrong.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What organs are located on the left side of your body class 11 biology CBSE
Write an application to the principal requesting five class 10 english CBSE
What is the type of food and mode of feeding of the class 11 biology CBSE
Name 10 Living and Non living things class 9 biology CBSE