Question

Question 2 (ii)
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

Let the number of days taken by a woman and a man be x and y respectively.
Therefore, work done by a woman in 1 day = $\frac{1}{x}$
According to the question,
$4(\frac{2}{x}+\frac{5}{y})=1$
$\frac{2}{x}+\frac{5}{y}=\frac{1}{4}$
$3(\frac{3}{x}+\frac{6}{y})=1$
$\frac{3}{x}+\frac{6}{y}=\frac{1}{3}$
$Putting\frac{1}{x}=pand\frac{1}{y}=q\text{in these equations, we get}$
$2p+5q=\frac{1}{4}$
$8p+20q=1$ -----(i)
$3p+6q=\frac{1}{3}$
$9p+18q=1$ -----(ii)
$\text{By cross multiplication, we get}$
$\frac{p}{-20-(-18)}=\frac{q}{-9-(-8)}=\frac{1}{144-180}$
$\frac{p}{-2}=\frac{q}{-1}=\frac{1}{-36}$
$\frac{p}{-2}=-\frac{1}{36}and\frac{q}{-1}=\frac{1}{-36}$
$p=\frac{1}{18}andq=\frac{1}{36}$
$p=\frac{1}{x}=\frac{1}{18}andq=\frac{1}{y}=\frac{1}{36}$
$\text{x = 18 and y = 36}$
Number of days taken by a woman = 18
Number of days taken by a man = 36