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Chandan and Falguni work on alternet days. Chandan working on 1st  day, Falguni on 2nd , then Chandan again by 3rd , followed by Falguni on 4th  and so on. They can finish the work in $25$ days. The work done by chandan varies everyday. On a perticular day $d$ work done by Chandan $d$ units. Falguni works at a constant rate.The ratio of work done by Chandan on 1st day to done by Falguni on 2nd day$=1:4.$ The time require Falguni alone finish the work ____________ days

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 Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Person ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ ${\color{Blue} F}$ ${\color{Red} C}$ Work done (in units) 1 4 3 4 5 4 7 4 9 4 11 4 13 4 15 4 17 4 19 4 21 4 23 4 25

The ratio of work done by $Chandan$ on $1^{st}$ day to done by $Falguni$ on $2^{nd}$ day=$1:4$

Let $Chandan$(${\color{Red} C}$) does $1$ unit of work on $1^{st}$ day and $Falguni$(${\color{Blue} F}$) does $4$ units of work on $2^{nd}$ day

$\because$ $Falguni$ does a constant amount of work everyday so she will work $4$ units on every alternate day

$\Rightarrow$ Total work done by $Falguni$ = $4+4+4...12$ times $= 4*12 = 48$ units

$\because$ $Chandan$ does $d$ amount of work on $d$ day so he will do different units of work every alternate day

$\Rightarrow$ Total work done by $Chandan$ = $1+3+5+7...25$ = $\frac{13}{2}(1+25) = 13 *13 = 169$ ( $\because$ sum of a.p. = $\frac{n}{2}(a+l)$)

So total work done by $Falguni$ and $Chandan$ in $25$ days = $48+169$ =$217$ units.

Question is asking that in how many days will $Falguni$ be able to do $217$ units of work if she does $4$ units of work everyday

$\therefore$ Days required by $Falguni$ to complete the work alone = $\frac{217}{4}$ = $54\tfrac{1}{4}$ = 54.25 days

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