Question

Formulate the following problems as a pair of equations, and hence find their solutions:

$2$ women and $5$ men can together finish an embroidery work 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 work, and also that taken by $1$ man alone.

Answer 1

Step 1: Formulate the equations

Let us assume that

Number of days taken by women to finish the work is $x$.

Number of days taken by men to finish the work is $y$.

Work done by women in one day $=\frac{1}{x}$

Work done by men in one day $=\frac{1}{y}$

As per the given data

$4\left(\frac{2}{x}+\frac{5}{y}\right)=1$

$\Rightarrow$$\left(\frac{2}{x}+\frac{5}{y}\right)=\frac{1}{4}$

And, $3\left(\frac{3}{x}+\frac{6}{y}\right)=1$

$\Rightarrow$ $\left(\frac{3}{x}+\frac{6}{y}\right)=\frac{1}{3}$

Now, let us substitute

$m=\frac{1}{x}$

$n=\frac{1}{y}$, we get,

$2m+5n=\frac{1}{4}$

$\Rightarrow$ $8m+20n=1\dots \dots \dots \dots \dots \dots \dots \left(1\right)$

$3m+6n=\frac{1}{3}$

$\Rightarrow$$9m+18n=1\dots \dots \dots \dots \dots \dots \dots .\left(2\right)$

Step 2: Find their solution

Upon multiplying $\left(1\right)$ with $9$ and equation $\left(2\right)$ with $8$, and the subtracting we get,

$72m+180n-72m-144m=9-8$

$\Rightarrow$ $36n=1$

$\Rightarrow$ $n=\frac{1}{36}$

Substituting this in equation $\left(1\right)$ we get,

$8m+\left(20\times \frac{1}{36}\right)=1$

$\Rightarrow$ $72m+5=9$

$\Rightarrow$ $72m=4$

$\Rightarrow$ $m=\frac{1}{18}$

Therefore,

Work done by women in one day

$m=\frac{1}{x}=18$

Work done by men in one day

$n=\frac{1}{y}=36$

Hence, the number of days taken by women is $18$ and the number of days taken by men is $36$.