Question

2 women and 5 men can finish a piece of embroidery in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the embroidery and that taken by 1 man alone?

Answer 1

Hint: Start by assuming the number of days taken by men and women for the work as some variable, Then compute the work done by men , women in one day , this gives us their efficiency. Form the equations using them as per the statement given in the question and solve for the values of variables.

Complete step-by-step answer:
Let us consider , 1 woman can finish the embroidery work in ‘x’ days and 1 man can finish it in ‘y’ days.
So , 1 woman’s one day work $={\displaystyle \frac{1}{x}}$
Similarly , 1 man’s one day work $={\displaystyle \frac{1}{y}}$
According to the statement in the question,
One day work of 2 women and 5 men $={\displaystyle \frac{2}{x}}+{\displaystyle \frac{5}{y}}={\displaystyle \frac{1}{4}}$
One day work of 3 women and 6 men $={\displaystyle \frac{3}{x}}+{\displaystyle \frac{6}{y}}={\displaystyle \frac{1}{3}}$
Let ${\displaystyle \frac{1}{x}}=a$ and ${\displaystyle \frac{1}{y}}=b$
Now, $2a+5b={\displaystyle \frac{1}{4}}\to eqn.1$
$3a+6b={\displaystyle \frac{1}{3}}\to eqn.2$
Multiply eqn.1 by 3 and eqn.2 by 2, Then subtract them eqn.1 – eqn.2 , we get
$3b={\displaystyle \frac{1}{12}}\Rightarrow b={\displaystyle \frac{1}{36}}$
Putting $b={\displaystyle \frac{1}{36}}$ in eqn.1 , we get
$22a+5\left({\displaystyle \frac{1}{36}}\right)={\displaystyle \frac{1}{4}}$
Multiply by 36 both side
$72a+5=9$
Taking 5 to right side
$$\begin{gathered} 72a=9-5 \\ \Rightarrow a={\displaystyle \frac{1}{18}} \end{gathered}$$
Now ,Substituting the values to get x and y . We get,
$a={\displaystyle \frac{1}{x}}={\displaystyle \frac{1}{18}},b={\displaystyle \frac{1}{y}}={\displaystyle \frac{1}{36}}$
So, One woman can finish the embroidery work in 18days and one man can finish the work in 36 days.

Note: Similar questions can be asked with a slight twist by adding or removing people midway of the project. In that case form equations till the time of adding or removing keeping in mind the amount of work completed or needs to be completed, In this way we can work with complex problems also.