Question

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

Answer 1

Let time taken by 1 women alone to finish work = x days

And by 1 man = y days

So, 1 woman does $\dfrac{1}{x}$ part of total work in one day.

And 1 man does $\dfrac{1}{y}$ part of total work in one day.

2 women do $\dfrac{2}{x}$ work in 1 day

5 men do $\dfrac{5}{y}$ work in 1 day

Therefore, 2 women and 5 men do $\dfrac{2}{x} + \dfrac{5}{y}$ work in one day (Equation 1)

Now,

Given that 2 women and 5 men complete work in 4 days

So in 1 day they do $\dfrac{1}{4th}$ part of work (Equation 2)

Clearly, we can see that Equation 1 = Equation 2

$\implies \dfrac {2}{x} + \dfrac{5}{y} = \dfrac {1}{4}$ (Equation 3)

Similarly

3 women and 6 men complete work $\dfrac{1}{3}$ part of work in work in one day

$\implies \dfrac{3}{x} + \dfrac{6}{y}= \dfrac{1}{3}$ (Equation 4)

Let $\dfrac{1}{x} = p\; and\; \dfrac{1}{y} = q$

Putting this in Equation 3 and Equation 4

2p + 5p = $\dfrac{1}{4}$

3p + 6q = $\dfrac{1}{3}$

8p + 20q = 1 (Equation 5)

9p + 18q = 1 (Equation 6)

Multiplying Equation 5 by 9 and Equation 6 by 8

72p + 180q = 9 (Equation 7)

72p + 144q = 8 (Equation 8)

Subtracting Equation 8 from Equation 7

36q = 1

q = $\dfrac{1}{36}$

Putting value in Equation 6

9p + 18 $\dfrac{1}{36}$= 1

9p = $\dfrac{1}{2}$

p = $\dfrac{1}{18}$

Putting values of 'p' and 'q' in $\dfrac{1}{x}$ and $\dfrac{1}{y}$ respectively

x = 18

y = 36

So,

1 woman completes work in 18 days alone

1 man completes work in 36 days alone