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 woman alone to finish the work, and also that taken by 1 man alone.solve it by elimination method.​

Answer 1

$\huge{\blue{\bold{\underline{\underline{Answer :}}}}}$

$\:\:$

$\large{\green{\underline \bold{\tt{Given :-}}}}$

$\:\:$

• 2 women and 5 men can together finish an embroidery work in 4 days

$\:\:$

• 3 women and 6 men can finish it in 3 days

$\:\:$

$\large{\red{\underline \bold{\tt{To \: Find :-}}}}$

$\:\:$

• Time taken by 1 woman alone to finish the work, and also that taken by 1 man alone

$\:\:$

$\large{\orange{\underline{\tt{Solution :-}}}}$

$\:\:$

• Let 1 woman can finish the work alone in "x" days

• Let 1 man can finish the work alone in "y" days

$\:\:$

$\underline{\bold{\texttt{One woman's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 1 } { x }$

$\:\:$

$\underline{\bold{\texttt{One man's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 1 } { y }$

$\:\:$

$\underline{\bold{\texttt{2 women's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 2 } { x }$

$\:\:$

$\underline{\bold{\texttt{5 men's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 5 } { y }$

$\:\:$

$\underline{\bold{\texttt{3 women's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 3 } { x }$

$\:\:$

$\underline{\bold{\texttt{6 men's one day work :}}}$

$\:\:$

➠ $\sf \dfrac { 6 } { y }$

$\:\:$

$\underline{\bold{\texttt{2 women's and 5 men's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 2 } { x } + \dfrac { 5 } { y }$

$\:\:$

$\underline{\bold{\texttt{2 women's and 5 men's 4 day work :}}}$

$\:\:$

➠ $\sf (\dfrac { 2 } { x } + \dfrac { 5 } { y }) \times 4$

$\:\:$

$\underline{\bold{\texttt{3 women's and 6 men's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 3 } { x } + \dfrac { 6 } { y }$

$\:\:$

$\underline{\bold{\texttt{3 women's and 6 men's 3 day work :}}}$

$\:\:$

➠ $\sf (\dfrac { 3 } { x } + \dfrac { 6} { y }) \times 3$

$\:\:$

$\purple{\underline \bold{According \: to \: the \ question :}}$

$\:\:$

2 women and 5 men can together finish an embroidery work in 4 days

$\:\:$

So,

$\:\:$

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

$\:\:$

• Let $\sf \dfrac { 1 } { x } = a$

$\:\:$

• Let $\sf \dfrac { 1 } { y} = b$

$\:\:$

So the above equation will be

$\:\:$

➜ $\sf (2a + 5b) \times 4 = 1$

$\:\:$

➠ 8a + 20b = 1 --------- (1)

$\:\:$

Also given that , 3 women and 6 men can finish it in 3 days

$\:\:$

So,

$\:\:$

➜ $\sf (\dfrac { 3 } { x } + \dfrac { 6} { y }) \times 3 = 1$

$\:\:$

• Let $\sf \dfrac { 1 } { x } = a$

$\:\:$

• Let $\sf \dfrac { 1 } { y} = b$

$\:\:$

So the above equation will be

$\:\:$

➜ $\sf (3a + 6b) \times 3 = 1$

$\:\:$

➠ 9a + 18b = 1 --------- (2)

$\:\:$

$\underline{\bold{\texttt{Multiplying (1) by 9 }}}$

$\:\:$

➜ 8a + 20b = 1

$\:\:$

➠ 72a + 180b = 9 ------ (3)

$\:\:$

$\underline{\bold{\texttt{Multiplying (2) by 8 }}}$

$\:\:$

➜ 9a + 18b = 1

$\:\:$

➠ 72a + 144b = 8 ----- (4)

$\:\:$

$\underline{\bold{\texttt{Subtracting (4) from (3) }}}$

$\:\:$

➜ 72a + 180b - 72a - 144b = 9 - 8

$\:\:$

➜ 36b = 1

$\:\:$

➨ $\sf b = \dfrac { 1 } { 36 }$ ------ (5)

$\:\:$

$\underline{\bold{\texttt{Putting b = \dfrac { 1 } { 36 } from (5) to (2) }}}$

$\:\:$

➠ 9a + 18b = 1

$\:\:$

➜ $\sf 9a + 18( \dfrac { 1 } { 36 }) = 1$

$\:\:$

➜ $\sf 9a + \dfrac { 1 } { 2} = 1$

$\:\:$

➜ $\sf \dfrac { 18a + 1 } { 2 } = 1$

$\:\:$

➜ 18a = 2 - 1

$\:\:$

➜ 18a = 1

$\:\:$

➨ $\sf a = \dfrac { 1 } { 18 }$ ------- (6)

$\:\:$

$\underline{\bold{\texttt{Putting the real values of a \& b :}}}$

$\:\:$

➠ $\sf b = \dfrac { 1 } { 36 } = \dfrac { 1 } { y }$

$\:\:$

➨ y = 36

$\:\:$

• Hence, 1 man can finish the work alone in 36 days

$\:\:$

➠ $\sf a = \dfrac { 1 } { 18 } = \dfrac { 1 } { x }$

$\:\:$

➨ x = 18

$\:\:$

• Hence, 1 woman can finish the work alone in 18 days

Answer 2

$\huge{\blue{\bold{\underline{\underline{Answer :}}}}}$

$\:\:$

$\large{\green{\underline \bold{\tt{Given :-}}}}$

$\:\:$

• 2 women and 5 men can together finish an embroidery work in 4 days

$\:\:$

• 3 women and 6 men can finish it in 3 days

$\:\:$

$\large{\red{\underline \bold{\tt{To \: Find :-}}}}$

$\:\:$

• Time taken by 1 woman alone to finish the work, and also that taken by 1 man alone

$\:\:$

$\large{\orange{\underline{\tt{Solution :-}}}}$

$\:\:$

• Let 1 woman can finish the work alone in "x" days

• Let 1 man can finish the work alone in "y" days

$\:\:$

$\underline{\bold{\texttt{One woman's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 1 } { x }$

$\:\:$

$\underline{\bold{\texttt{One man's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 1 } { y }$

$\:\:$

$\underline{\bold{\texttt{2 women's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 2 } { x }$

$\:\:$

$\underline{\bold{\texttt{5 men's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 5 } { y }$

$\:\:$

$\underline{\bold{\texttt{3 women's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 3 } { x }$

$\:\:$

$\underline{\bold{\texttt{6 men's one day work :}}}$

$\:\:$

➠ $\sf \dfrac { 6 } { y }$

$\:\:$

$\underline{\bold{\texttt{2 women's and 5 men's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 2 } { x } + \dfrac { 5 } { y }$

$\:\:$

$\underline{\bold{\texttt{2 women's and 5 men's 4 day work :}}}$

$\:\:$

➠ $\sf (\dfrac { 2 } { x } + \dfrac { 5 } { y }) \times 4$

$\:\:$

$\underline{\bold{\texttt{3 women's and 6 men's 1 day work :}}}$

$\:\:$

➠ $\sf \dfrac { 3 } { x } + \dfrac { 6 } { y }$

$\:\:$

$\underline{\bold{\texttt{3 women's and 6 men's 3 day work :}}}$

$\:\:$

➠ $\sf (\dfrac { 3 } { x } + \dfrac { 6} { y }) \times 3$

$\:\:$

$\purple{\underline \bold{According \: to \: the \ question :}}$

$\:\:$

2 women and 5 men can together finish an embroidery work in 4 days

$\:\:$

So,

$\:\:$

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

$\:\:$

Let $\sf \dfrac { 1 } { x } = a$

$\:\:$

Let $\sf \dfrac { 1 } { y} = b$

$\:\:$

So the above equation will be

$\:\:$

➜ $\sf (2a + 5b) \times 4 = 1$

$\:\:$

➠ 8a + 20b = 1 --------- (1)

$\:\:$

Also given that , 3 women and 6 men can finish it in 3 days

$\:\:$

So,

$\:\:$

➜ $\sf (\dfrac { 3 } { x } + \dfrac { 6} { y }) \times 3 = 1$

$\:\:$

Let $\sf \dfrac { 1 } { x } = a$

$\:\:$

Let $\sf \dfrac { 1 } { y} = b$

$\:\:$

So the above equation will be

$\:\:$

➜ $\sf (3a + 6b) \times 3 = 1$

$\:\:$

➠ 9a + 18b = 1 --------- (2)

$\:\:$

$\underline{\bold{\texttt{Multiplying (1) by 9 }}}$

$\:\:$

➜ 8a + 20b = 1

$\:\:$

➠ 72a + 180b = 9 ------ (3)

$\:\:$

$\underline{\bold{\texttt{Multiplying (2) by 8 }}}$

$\:\:$

➜ 9a + 18b = 1

$\:\:$

➠ 72a + 144b = 8 ----- (4)

$\:\:$

$\underline{\bold{\texttt{Subtracting (4) from (3) }}}$

$\:\:$

➜ 72a + 180b - 72a - 144b = 9 - 8

$\:\:$

➜ 36b = 1

$\:\:$

➨ $\sf b = \dfrac { 1 } { 36 }$ ------ (5)

$\:\:$

$\underline{\bold{\texttt{Putting b = \dfrac { 1 } { 36 } from (5) to (2) }}}$

$\:\:$

➠ 9a + 18b = 1

$\:\:$

➜ $\sf 9a + 18( \dfrac { 1 } { 36 }) = 1$

$\:\:$

➜ $\sf 9a + \dfrac { 1 } { 2} = 1$

$\:\:$

➜ $\sf \dfrac { 18a + 1 } { 2 } = 1$

$\:\:$

➜ 18a = 2 - 1

$\:\:$

➜ 18a = 1

$\:\:$

➨ $\sf a = \dfrac { 1 } { 18 }$ ------- (6)

$\:\:$

$\underline{\bold{\texttt{Putting the real values of a \& b :}}}$

$\:\:$

➠ $\sf b = \dfrac { 1 } { 36 } = \dfrac { 1 } { y }$

$\:\:$

➨ y = 36

$\:\:$

• Hence, 1 man can finish the work alone in 36 days

$\:\:$

➠ $\sf a = \dfrac { 1 } { 18 } = \dfrac { 1 } { x }$

$\:\:$

➨ x = 18

$\:\:$

• Hence, 1 woman can finish the work alone in 18 days