Laxmi and Kartik together complete a work in 4 days. If they work seperately, then the time taken by Kartik would be 6 days more than the time taken by Laxmi. Calculate the time taken by Laxmi alone to complete the work.

Explanation

Let Laxmi alone complete the work in x days.
`:.` Kartik alone can complete the work in (x+6) days.
They together complete `((1)/(x)+(1)/(x+6))` part of the work in 1 day, i.e., they complete `(x+6+x)/(x(x+6))"part"=(2x+6)/(x^(2)+6x)` parts of the work in 1 day.
`:.` They together complete the work in `(x^(2)+6x)/(2x+6)` days.`(x^(2)+6x)/(2x+6)` days.
As per question, `(x^(2)+6x)/(2x+6)=4`
or, `x^(2)+6x=8x+24" "or,x^(2)-2x-24=0`
or, `x^(2)-(6-4)x-2=0`
or, `x^(2)-6x+4x-24=0`
or, `x(x-6)+4(x-6)=0`
or, `(x-6)(x+4)=0`
`:.` either `x-6=0" "or,x+4=0`
`impliesx=6" "or,x=-4`
But the number of days can not be negative, `:.xne-4,:.x=6`
Hence, Laxmi alone can complete the work in 6 days.