Question

to complete a work ajay needs 5 days more than vijay. after 4days vijay left job then ajay complete the remaining job within 5 days then how much does each one take to complete the works?
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Answer 1

let bijay takes x days.

so Ajay takes (x + 5) days.

so bijays 1 day work = 1/x

Ajay's 1 day work = 1/(x + 5)

their 1 day work =

$\frac{1}{x} + \frac{1}{x + 5} \\ \frac{x + 5 + x}{x(x + 5)} \\ \frac{2x + 5}{x(x + 5)}$

so their 4 days work

$4( \frac{2x + 5}{x(x + 5)} ) \\ \frac{8x + 20}{ x(x + 5) }$

remaining work =

$1 - \frac{8x + 20}{x (x + 5}) \\ \frac{ {x}^{2} + 5x - (8x + 20) }{x(x + 5)} \\ \frac{ {x}^{2} + 5x - 8x - 20}{x(x + 5)} \\ \frac{ {x}^{2} - 3x - 20}{x(x + 5)}$

Ajay's 5 days work =

$5 \times \frac{1}{x + 5} \\ \frac{5}{x + 5}$

since Ajay completes the remaining work in 5 days so,

$\frac{ {x}^{2} - 3x - 20 }{x(x + 5)} = \frac{5}{x + 5} \\ {x}^{2} - 3x - 20 = 5x \\ {x}^{2} - 3x - 5x - 20 = 0 \\ {x}^{2} - 8x - 20 = 0 \\ {x}^{2} - 10x + 2x - 20 = 0 \\ x(x - 10) + 2(x - 10) = 0 \\ (x - 10)(x + 2) = 0 \\ if \: x - 10 = 0 \: then \: x = 10 \\ if \: x + 2 = 0 \: then \: x = - 2$

so x = 10 is correct.

therefore bijay takes 10 days.

Ajay takes 15 days.