Question

5 women and 2 men can together finish an embroidery work in 4 days, while 6 women and 3 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone to finish the work.

Answer 1

Hint: In this type of question where men and women together perform a work. The first thing is to assume the one day work of a woman and one day work of a man will be x and y respectively. After this form the two equations in two variables using the given data in question.

Complete step-by-step answer:
Let time taken by 1 woman alone to finish the work = x days
Let time taken by 1 man alone to finish the work = y days
So, 1 woman’s 1 day work = \[\left( {\dfrac{1}{x}} \right)\]th part of the work
And, 1 man’s 1 day work =\[\left( {\dfrac{1}{y}} \right)\]th part of the work
So, 5 women’s 1 day work =\[\left( {\dfrac{5}{x}} \right)\]th part of the work
And, 2 men’s 1 day work =\[\left( {\dfrac{2}{y}} \right)\]th part of the work
Therefore, 5 women and 2 men’s 1 day work =\[\left( {\left( {\dfrac{5}{x}} \right) + \left( {\dfrac{2}{y}} \right)} \right)\]th part of the work (1)
It is given that 5 women and 2 men complete work in = 4 days
It means that in 1 day, they will be completing \[\left( {\dfrac{1}{4}} \right)\]th part of the work. (2)
Clearly, we can see that (1) = (2)
\[ \Rightarrow \dfrac{5}{x} + \dfrac{2}{y} = \dfrac{1}{4}\;\;\;\;\;\;\;\;\;\] (3)
Similarly, 6 women’s 1 day work =\[\left( {\dfrac{6}{x}} \right)\]th part of the work
And, 3 men’s 1 day work =\[\left( {\dfrac{3}{y}} \right)\]th part of the work
Therefore, 6 women and 3 men’s 1 day work =\[\left( {\left( {\dfrac{6}{x}} \right) + \left( {\dfrac{3}{y}} \right)} \right)\]th part of the work (4)
It is given that 6 women and 3 men complete work in = 3 days
It means that in 1 day, they will be completing \[\left( {\dfrac{1}{3}} \right)\]rd part of the work. (5)
Clearly, we can see that (4) = (5)
 \[ \Rightarrow \dfrac{6}{x} + \dfrac{3}{y} = \dfrac{1}{3}\;\;\;\;\;\;\;\;\;\](6)
Let \[\dfrac{1}{x} = p\]and \[\dfrac{1}{y} = q\]
 Putting this in (3) and (6), we get
\[ \Rightarrow \dfrac{5}{x} + \dfrac{2}{y} = \dfrac{1}{4}\;\;\;\;\;\;\;\;\;\]and \[ \Rightarrow \dfrac{6}{x} + \dfrac{3}{y} = \dfrac{1}{3}\;\;\;\;\;\;\;\;\;\]
\[ \Rightarrow 5p + 2q = \dfrac{1}{4}\;\;\;\;\;\;\;\;\;\]and \[ \Rightarrow 6p + 3q = \dfrac{1}{3}\;\;\;\;\;\;\;\;\;\]
\[ \Rightarrow 20p + 8q = 1\;\;\;\;\;\;\;\;\](7) and \[ \Rightarrow 18p + 9q = 1\;\;\;\;\;\;\;\;\;\] (8)
Multiplying (7) by 9 and (8) by 8, we get
\[ \Rightarrow 180p + 72q = 9\;\;\;\;\;\;\;\;\] (9)
\[ \Rightarrow 144p + 72q = 8\;\;\;\;\;\;\;\;\] (10)
Subtracting (10) from (9), we get
\[ \Rightarrow 36p = 1 \Rightarrow p = \dfrac{1}{{36}}\]
Putting this in (8), we get
\[ \Rightarrow 18\left( {\dfrac{1}{{36}}} \right) + 9q = 1\]
\[ \Rightarrow 1 + 18q = 2\]
\[ \Rightarrow q = \dfrac{2}{{18}} = \dfrac{1}{9}\]
Putting values of p and q in (\[\dfrac{1}{x} = p\] and \[\dfrac{1}{y} = q\]), we get
\[x = 36\] and \[y = 9\]
Therefore, 1 woman completes work in = 36 days
And, 1 man completes work in = 9 days

Note: Linear equations in two variables are used to find when two unknown variables must be found. Linear equations in one variable are used to find when we have only one unknown variable.