Name: Equations Reducible to a Pair of Linear Equations
Uploaded: 2019-07-19
Duration: 4 min 9 s
Description: Some equations may be in a form that can be reduced to a linear equation through substitution. Do you want to explore how this can be done? Watch the video t...

Question

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

women alone to finish the work, and also that taken by

1

man alone.

Accepted Answer

Let $5$ woman takes x days to finish the work.

Work finished by one woman in one day $= \frac{1}{x}$

Similarly,

Let $1$ man takes y days to finish the work.

Work finished by one man in one day $= \frac{1}{y}$

Given,

$2$ women and $5$ men complete the work in $4$ days

So, work finished by $2$ women and $5$ men in $1$ day $= \frac{1}{4}$

$∴$ $2$ $\times$ (work finished by one woman in one day) $+$ $5$ $\times$ ( work finished by one man in one day) $= \frac{1}{4}$

Therefore, $\frac{2}{x} + \frac{5}{y} = \frac{1}{4} . . . (i)$

Also, $3$ women and $3$ men complete the work in $3$ days

So, work finished by $3$ women and $6$ men in $1$ day $= \frac{1}{3}$

$3$ $\times$ (work finished by one woman in one day) $+$ $6$ $\times$ (work finished by one man in one day) $= \frac{1}{3}$

Therefore, $\frac{3}{x} + \frac{6}{y} = \frac{1}{3} . . . (ii)$

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

Then, $2 u + 5 v = \frac{1}{4} 8 u + 20 v = 1 . . (iii) Also, 3 u + 6 v = \frac{1}{3} 9 u + 18 v = 1 . . (iv)$

From (iii), we get

$8 u + 20 v = 1 u = \frac{1 - 20 v}{8}$

putting the value of u in (iv), we get

$9 u + 18 v = 1 9 (\frac{1 - 20 v}{8}) + 18 v = 1 9 - 180 v + 144 v = 8 - 36 v = - 1 v = \frac{1}{36}$

Putting the value of v in (iii), we get

$8 u + 20 v = 1 8 u + 20 (\frac{1}{36}) = 1 8 u = 1 - \frac{5}{9} u = \frac{1}{18}$

Now, putting u and v value in x an y, we get

$u = \frac{1}{x} and v = \frac{1}{y} \frac{1}{18} = \frac{1}{x} and \frac{1}{36} = \frac{1}{y} x = 18 and y = 36$

Therefore, time taken by one woman to finish the work alone $= x = 18 days$

And, time taken by one man to finish the work alone $= y = 36 days$

$2$ 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 $1$ women alone to finish the work, and also that taken by $1$ man alone.

Important Questions on Pair of Linear Equations in Two Variables

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