Question

$2$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

Find the time taken by$1$ woman alone to finish the work, and also that taken by$1$ man alone.

Let the number of days taken by a woman and a man be$\mathrm{x}$ and$\mathrm{y}$.

$\begin{array}{rcl}4\left(\frac{2}{\mathrm{x}}+\frac{5}{\mathrm{y}}\right)& =& 1\\ \frac{2}{\mathrm{x}}+\frac{5}{\mathrm{y}}& =& \frac{1}{4}.......\left(1\right)\end{array}$

$\begin{array}{rcl}3\left(\frac{3}{\mathrm{x}}+\frac{6}{\mathrm{y}}\right)& =& 1\\ \frac{3}{\mathrm{x}}+\frac{6}{\mathrm{y}}& =& \frac{1}{3}.......\left(2\right)\end{array}$

Putting $\frac{1}{\mathrm{x}}=\mathrm{p}$ and$\frac{1}{\mathrm{y}}=\mathrm{q}$ in these equations,

we get,

$$\begin{gathered} 2\mathrm{p}+5\mathrm{q}=\frac{1}{4}\left[\mathrm{From}\left(1\right)\right] \\ 8\mathrm{p}+20\mathrm{q}=1......\left(3\right) \end{gathered}$$

and

$\begin{array}{rcl}3\mathrm{p}+6\mathrm{q}& =& \frac{1}{3}\left[\mathrm{From}\left(2\right)\right]\\ 9\mathrm{p}+18\mathrm{q}& =& 1........\left(4\right)\end{array}$

Multiply equation$\left(3\right)$ by$9$ and$\left(4\right)$ by$8$, we get

$$\begin{gathered} 72\mathrm{p}+180\mathrm{q}=9......\left(5\right) \\ 72\mathrm{p}+144\mathrm{q}=8......\left(6\right) \end{gathered}$$

Subtract equation$\left(6\right)$ from$\left(5\right)$,

$$\begin{gathered} 36\mathrm{q}=1 \\ \therefore \mathrm{q}=\frac{1}{36} \end{gathered}$$

Then put the value of$\mathrm{q}$ in equation$\left(4\right)$,

$\begin{array}{rcl}9\mathrm{p}+18\times \frac{1}{36}& =& 1\\ 9\mathrm{p}+\frac{1}{2}& =& 1\\ 9\mathrm{p}& =& 1-\frac{1}{2}\\ 9\mathrm{p}& =& \frac{1}{2}\\ \therefore \mathrm{p}& =& \frac{1}{18}\end{array}$

$$\begin{gathered} p=\frac{1}{x}=\frac{1}{18}andq=\frac{1}{y}=\frac{1}{36} \\ \therefore x=18andy=36 \end{gathered}$$

$\therefore$Number of days taken by a woman$=18$ and Number of days taken by a man$=36$