Question

8 men and 12 women can finish a piece of work in 10days. While 6 men and 8 women can
finish it in 14 days. Find the time taken by I man alone and that by 1 woman alone to finish the
work​

Answer 1

${ \huge{ \underline{ \underline{ \texttt{ \blue{AnSwer:- }}}}}}$

Let us assume the time taken by man be x days and time taken by woman be y days respectively.

So,

☢ Work done in 1 day :

• 1 Man = 1/x
• 1 Woman = 1/y

The eqn becomes,

$\longrightarrow \sf \: \: \dfrac{8}{x} + \dfrac{12}{y} = \dfrac{1}{10} - - - - (1)$

And,

$\longrightarrow \sf \: \: \dfrac{6}{x} + \dfrac{8}{y} = \dfrac{1}{14} - - - - (2)$

Again, Let us consider u=1/x & v=1/y

$\longrightarrow \sf \: \: 8u + 12v = \dfrac{1}{10} - - - - (3)$

$\longrightarrow \sf \: \: 6u + 8v = \dfrac{1}{14} - - - - (4)$

Now,

Multiply :

• eqn(3) by (2)
• eqn(4) by (3)

$\longrightarrow \sf \: \: 16u + 24v = \dfrac{2}{10} - - - - (5)$

$\longrightarrow \sf \: \: 18u + 24v = \dfrac{3}{14} - - - - (6)$

Also, subtract eqn(5) as well as eqn(6),

$\longrightarrow \sf \: \: - 2u = - \dfrac{3}{14} + \dfrac{2}{10} \\ \\ \\ \longrightarrow \sf \: \: - 2u = \dfrac{ - 3 \times 10 + 2 \times 14 \: }{140} \\ \\ \\ \longrightarrow \sf \: \: - 2u = \dfrac{ - 30 + 28}{140} \\ \\ \\ \longrightarrow \sf \: \: - 2u = \dfrac{ - 2}{140} \\ \\ \\ \longrightarrow \sf \: \:u = \dfrac{ \cancel- 2}{140} \times \dfrac{1}{ \cancel- 2} \\ \\ \\ \longrightarrow \sf \: \:u = \dfrac{1}{140}$

Substitute the value of u in eqn(5)

$\longrightarrow \sf \: \:16u + 24v = \dfrac{2}{10} \\ \\ \\ \longrightarrow \sf \: \:16 \times \dfrac{1}{140} + 24v = \dfrac{2}{10} \\ \\ \\ \longrightarrow \sf \: \: \dfrac{4}{35} + 24v = \dfrac{1}{5} \\ \\ \\ \longrightarrow \sf \: \:24v = \dfrac{1}{5} - \dfrac{4}{35} \\ \\ \\ \longrightarrow \sf \: \:24v = \dfrac{1 \times 7 - 4}{35} \\ \\ \\ \longrightarrow \sf \: \:24v = \dfrac{3}{35} \\ \\ \\ \longrightarrow \sf \: \:v = \dfrac{3}{35} \times \frac{1}{24} \\ \\ \\ \longrightarrow \sf \: \:v = \dfrac{1}{280}$

As we have considered :

• u = 1/x

$\longrightarrow \sf \: \: \dfrac{1}{140} = \dfrac{1}{x} \\ \\ \\ \longrightarrow \sf \: \:{ \underline{ \boxed{ \texttt{ \red{x = 140}}}}} \: \bigstar$

• v = 1/y

$\longrightarrow \sf \: \: \dfrac{1}{280} = \dfrac{1}{y} \\ \\ \\ \longrightarrow \sf \: \:{ \underline{ \boxed{ \texttt{ \pink{y = 280}}}}} \: \bigstar$

Therefore, a man complete the work in 140 days while a woman can complete the work in 280 days.

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