Question

8 women and 12 men can together

finish a work in 10 days, while 6

women and 8 men can finish it in 14

days. Find the time taken by 1

woman alone to finish the work, and

also that taken by 1 man alone.​

Answer 1

Let 1 woman finish the work in x days and 1 man finish the work in y days.

work done by 1 woman in 1 day = 1/x

work done by 1 man in 1 day = 1/y

ATQ

Case 1:

8 women and 12 men finish work in 10 days

1 day’s work of 8 women and 12 men= 1/10 part of  work.

8/x + 12/y = 1/10

4(2/x + 3/y) = 1/10

2/x + 3/y = 1/40……….(1)

Case 2.

6 women and 8 men finish work in 14 days

1 day’s work of 6 women and 8 men= 1/14 part of  work.

6/x + 8/y = 1/14

2(3/x + 4/y) = 1/14

3/x + 4/y = 1/28……….(2)

Putting 1/x = p and 1/y = q in equations,1 & 2 ,

2p + 3q = 1/40………….(3)

3p + 4q = 1/28………….(4)

Multiply equation 3 by 4 and equation 4 by 3,

8p + 12q = 4/40

8p +12q = 1/10…………..(5)

9p + 12q = 3/28………….(6)

On subtracting equation 5 and 6,

8p +12q = 1/10

9p + 12q = 3/28

(-)   (-)      (-)

-----------------

- p = 1/10-3/28

-p = (14 - 15)/140

-p = -1/140

p = 1/140

On substituting p= 1/140 in equation 5,

8p +12q = 1/10

8(1/140) +12q = 1/10

8/140 + 12q = 1/10

12q = 1/10 - 2/35

12q = (7 - 4)/70

12q = 3/70

q= 3/(70×12)

q= 1/(70×4)

q= 1/280

Now p= 1/140= 1/x

x = 140

q= 1/280= 1/y

y = 280

Hence, the  time  taken  by one  woman alone to finish the work = 140 days and  one man alone to finish the work = 280 days.

HOPE THIS WILL HELP YOU.

Answer 2

Step-by-step explanation:

Given that:8 women and 12 men can together finish a work in 10 days, while 6 women and 8 men can finish it in 14 days.

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

Solution:

In such type of questions,one should always find the work done by men and women in 1 day.

Let

one woman alone can do the work in x days

So, work done by a woman in 1 day = (1/x)th part

one man alone can do the work in y days

So, work done by a man in 1 day = (1/y)th part

Case1: 8 women and 12 men can together finish a work in 10 days

$\frac{8}{x} + \frac{12}{y} = \frac{1}{10} \: \: \: \: ...eq1$

Case2: 6 women and 8 men can together finish a work in 14 days

$\frac{6}{x} + \frac{8}{y} = \frac{1}{14} \: \: \: ...eq2$

Now,convert these equations in linear equations ,by substitution

$let \\ \\ \frac{1}{x} = u \: \: and \: \frac{1}{y} = v$

So, equations becomes

$8u + 12v = \frac{1}{10} \\ \\ 6u + 8v = \frac{1}{14}$

after cross multiplication

$80u + 120v = 1 \\ 84u + 112v = 1$

For elimination method,equate the coefficient of u

$84(80u + 120v = 1) \\80( 84u + 112v = 1) \\ \\ 6720u + 10080v = 84 \\ 6720u + 8960v = 80 \\ ( - ) \: \: \: \: \: ( - ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: ( - ) \\ - - - - - - - - \\ 1120v = 4 \\ \\ v = \frac{4}{1120} \\ \\\bold{ v = \frac{1}{280} }$

Now put the value of v in any of the two equations

$80u + 120v = 1 \\ \\ 80u + 120 \times \frac{1}{280} = 1 \\ \\ 80u = 1 - \frac{120}{280} \\ \\ 80u = 1 - \frac{3}{7} \\ \\ 80u = \frac{4}{7} \\ \\ u = \frac{4}{7 \times 80} \\ \\ \bold{u = \frac{1}{140} }$

Thus

$\frac{1}{x} = \frac{1}{140} \\ \\\bold{ x = 140} \\ \\ and \\ \\ \frac{1}{y} = \frac{1}{280} \\ \\\bold{ y = 280}$

One man alone can finish the work in 280 days and one woman alone can finish the work in 140 days.

Hope it helps you.