Question

a can do a work in 5 days less than the time taken by b to do it. if both of them take 11 1/9 days then time taken by b alone to do it is

Answer 1

Let b takes x days to complete the work.
Then, a takes x-5 days to complete
Together they complete in
$\frac{x(x - 5)}{x + (x - 5)} = \frac{ {x}^{2} - 5x }{2x - 5}$
Given,
$\frac{ {x}^{2} - 5x }{2x - 5} = 11 \frac{1}{9} = \frac{100}{9} \\ 9 {x}^{2} - 45x = 200x - 500 \\ 9 {x}^{2} - 245x + 500 = 0 \\ x = \frac{245 + \: or \: - \sqrt{ {( - 245)}^{2} - 4 \times 9 \times 500) } }{2 \times 9} \\ = \frac{245 + \: or \: - 205}{18} \\ = 25 \: \: or \: \: \frac{20}{9} \\ \\ x \: \: cannot \: \: be \: \: 25 \: \: since \: \: combined \: \: \\ working \: \: day \: \: is \: \: only \: \: \frac{100}{9} \\ \\ so \: \: b \: \: alone \: \: can \: \: do \: \: the \: \: work \\ in \: \: \frac{20}{9} \: \: days$