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

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.

Answer 2

Answer:

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