Question

Yogesh requires 3 days more than Vivek to do a work completely. If both
of them work together, the work can be completed in 2 days. Find the
number of days required for each of them to do the work completely.

Answer 1

Suppose, Vivek completes a work in x days.
Yogesh completes the same work in ( x + 3) days.
∴ Work done by Vivek in one day  = `1/x`
and work done by Yogesh in one day ` = 1/(x+3)`

Work done by both of them together in one day = `1/2`
from the given condition,

`1/x +1/(x+3) = 1/2`

` ∴ (x +3 +x )/(x(x+3))= 1/2`

` ∴(2x +3 )/(x^2 +3x ) = 1/2`

∴ x² + 3x = 2(2x + 3)           
∴ x² + 3x = 4x + 6
∴ x² + 3x - 4x -6 = 0
∴ x² - x - 6 = 0
∴ x² -3x +2x - 6 =0
∴ x (x-3) + 2 (x -3) =0
∴ (x -3) (x +2) = 0
∴ x - 3 = 0 or x + 2 = 0
∴   x = 3   or   x = -2

or, a = 1, b = -1, c = -6
∴ x =`(-b+-sqrt(b^2 - 4ac))/(2a)`

      =`(1+-sqrt((-1)^2-4(1)(-6)))/2`

      `=(1+-sqrt(25))/2`

∴ x = `(1+5)/2 = 3  or x =( 1-5)/2 =-2`

but the number of days is not negative
∴ x = 3                   ∴ x + 3 = 3 +3 = 6
∴ Vivek completes the work in 3 days and Yogesh in 6 days.