Question

6men and 8 boys can finish a work in 14days while 8 men and 12boys finish same work in 10 days find time taken by one man alone and that of one boy alone to finish same work

Answer 1

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$\mathbb{\underline{\blue{SOLUTION:}}}$

Let time taken by one man to complete the work be x &
time taken by one boy to complete he work be y

In one day,
$A\: man\: can\: complete\: \frac{1}{x}\: part\: of\: the\: work$
$A\: boy\: can\: complete\: \frac{1}{y}\: part\: of\: the\: work$

Given that
• 6men and 8 boys can finish a work in 14days

=> $\frac{6}{x} + \frac{8}{y} = \frac{1}{14}$

• 8 men and 12boys finish same work in 10 days

=> $\frac{8}{x} + \frac{12}{y} = \frac{1}{10}$

Since, the above equations are non linear
Consider
$\frac{1}{x} = a$
$\frac{1}{y} = b$

Now, we have
=>$6a + 8b = \frac{1}{14}$ ----Eq.(1)

=>$8a + 12b = \frac{1}{10}$ ----Eq.(2)

Do 4×Eq.(1) & 3×Eq.(2):

=> $24a + 32b = \frac{4}{14}$ ----Eq.(3)

=>$24a + 36b = \frac{3}{10}$ ----Eq.(4)

Do Eq.(4) - Eq.(3):

=> $24a + 36b - 24a - 32b = \frac{3}{10} - \frac{2}{7}$

=> $4b = \frac{1}{70}$

=> $b = \frac{1}{280}$

Substitute in Eq.(2) to get value of a:
=> $8a + 12(\frac{1}{280}) = \frac{1}{10}$

=> $8a + \frac{3}{70} = \frac{1}{10}$

=> $8a = \frac{1}{10} - \frac{3}{70}$

=> $8a = \frac{70-30}{70}$

=> $8a = \frac{40}{70}$

=> $2a = \frac{1}{70}$

=> $a = \frac{1}{140}$

So,
$x = \frac{1}{a} = 140$
$y = \frac{1}{b} = 280$

Hence,
One man can finish the work in 140 days &
One boy can finish the work in 280 days

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$\mathcal{\huge{\orange{Hope\: it\: Helps}}}$