Question

2 women, five men
can together finish.
an embroydery work!
in 4days while 3 women
6 men can finish it
in 3 days find
time taken by one
Alone women alone and
one men alone to finish
the work​

Answer 1

Question -

2 women, 5 men can together finish an embroydery work in 4 days while 3 women 6 men can finish it in 3 days. Find the time taken by one women alone and one man alone to finish the work.

Step-by-step solution -

Let time taken by one woman to complete the work be x and time taken by one man to complete the work by y.

Given conditions are :

$\sf{\dfrac{2}{x}\:+\:\dfrac{5}{y}\:=\:\dfrac{1}{4}}$ ...(1)

$\sf{\dfrac{3}{x}\:+\:\dfrac{6}{y}\:=\:\dfrac{1}{3}}$ ...(2)

$\rule{150}2$

2 women, 5 men can together finish a piece of work in 4 days.

$\implies\:\sf{\dfrac{2}{x}\:+\:\dfrac{5}{y}\:=\:\dfrac{1}{4}}$

$\implies\:\sf{\dfrac{2y\:+\:5x}{xy}\:=\:\dfrac{1}{4}}$

$\implies\:\sf{4(2y\:+\:5x)\:=\:xy}$

$\implies\:\sf{8y\:+\:20x\:=\:xy}$ ...(3)

3 women 6 men complete a piece of work in 3 days.

$\implies\:\sf{\dfrac{3}{x}\:+\:\dfrac{6}{y}\:=\:\dfrac{1}{3}}$

$\implies\:\sf{\dfrac{3y\:+\:6x}{xy}\:=\:\dfrac{1}{3}}$

$\implies\:\sf{3(3y\:+\:6x)\:=\:xy}$

$\implies\:\sf{9y\:+\:18x\:=\:xy}$ ...(4)

Now,

On multiplying equation (3) with 9 and equation (4) with 8 we get,

$\implies\:\sf{72y\:+\:180x\:=\:9xy}$

$\implies\:\sf{72y\:=\:9xy\:-\:189x}$ ...(5)

$\implies\:\sf{72y\:+\:144x\:=\:8xy}$

$\implies\:\sf{72y\:=\:8xy\:-\:144x}$ ...(6)

On comparing equation (5) & (6) we get,

$\implies\:\sf{9xy\:-\:189x\:=\:8xy\:-\:144x}$

$\implies\:\sf{9xy\:-\:8xy\:=\:36x}$

$\implies\:\sf{xy\:=\:36x}$

$\implies\:\sf{\bold{y\:=\:36}}$

Substitute value of x in equation (4)

$\implies\:\sf{9(36)\:+\:18x\:=\:x(36)}$

$\implies\:\sf{324\:+\:18x\:=\:36x}$

$\implies\:\sf{324\:=\:36x\:-\:18x}$

$\implies\:\sf{324\:=\:18x}$

$\implies\:\sf{\frac{324}{18}\:=\:x}$

$\implies\:\sf{\bold{x\:=\:18}}$

•°• Time taken by one woman to complete the work is 18 days and on man is 36 days.

Answer 2

Solution

Case I

Let the time taken by 1 woman to finish the embroidary work be x days

So, Work done by 1 women in 1 day = $\dfrac{1}{x}$

Work done by 2 women in 1 day = $2 \times \dfrac{1}{x} = \dfrac{2}{x}$

Let the time taken by 1 man to finish the embroidary work be y days

So, Work done by 1 man in 1 day = $\dfrac{1}{y}$

Work done by 5 man in 1 day = $5 \times \dfrac{1}{y} = \dfrac{5}{y}$

Total time taken to finish the work by 2 women and 5 men to finish the work = 4 days

Work done by 2 women and 5 men in 1 days = $\dfrac{1}{4}$

$\implies \dfrac{2}{x} + \dfrac{5}{y} = \dfrac{1}{4}$

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

$\implies \dfrac{8}{x} + \dfrac{20}{y}= 1 \longrightarrow(1)$

Case II

Work done by 3 women in 1 day = $3 \times \dfrac{1}{x} = \dfrac{3}{x}$

Work done by 6 men in 1 day = $6 \times \dfrac{1}{y} = \dfrac{6}{y}$

Total time taken to finish the work by 3 women and 6 men to finish the work = 3 days

Work done by 3 women and 6 men in 1 days = $\dfrac{1}{3}$

$\implies \dfrac{3}{x} + \dfrac{6}{y} = \dfrac{1}{3}$

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

$\implies \dfrac{9}{x} + \dfrac{18}{y} = 1 \longrightarrow(2)$

Now, substituting a = 1/x and b = 1/y in (1) & (2)

→ 8a + 20b = 1 ----- (3)

→ 9a + 18b = 1 ------ (4)

LCM of 8 and 9 is 72. Multiply (3) with 9 and (4) with 8. Equations will be :

i.e ( 8a + 20b = 1 ) * 9

→ 72a + 180b = 9 ---- (5)

i.e ( 9a + 18b = 1 ) * 8

→ 72a + 144b = 8 --- (6)

Subtracting (6) from (5)

→ 72a + 180b - ( 72a + 144b ) = 9 - 8

→ 72a + 180b - 72a - 144b = 1

→ 36b = 1

→ b = 1/36

But b = 1/y

→ 1/36 = 1/y

→ y = 36

Substituting b = 1/36 in (4)

$\implies 9a + 18b =1$

$\implies 9a + 18 \bigg( \dfrac{1}{36} \bigg) =1$

$\implies 9a + \dfrac{1}{2} =1$

$\implies 9a = 1 - \dfrac{1}{2}$

$\implies 9a = \dfrac{1}{2}$

$\implies a = \dfrac{1}{18}$

But a = 1/x

→ 1/18 = 1/x

→ x = 18

Hence, time taken by 1 women alone is 18 days and time taken by 1 men alone is 36 days.