### Barbara and David can finish a piece of work in $10$ and $20$ days respectively. If this work is done by both of them on alternate days (starting with Barbara), how many days will it take for them to finish the work$?$

$13$ days

Step by Step Explanation:
1. 10 = rem
m = 10
n = 5
2. Barbara takes $10$ days to finish the work. So in $1$ day, she finishes $\dfrac { 1 }{ 10 }$ of the work.
3. David takes $20$ days to finish the work. So in $1$ day, he finishes $\dfrac { 1 }{ 20 }$ of the work.
4. Since the work is done on alternate days, so the work done in two days $= \dfrac{ 1 }{ 10 } + \dfrac{ 1 } { 20 } = \dfrac{ 2 + 1 }{ 20 } = \dfrac{ 3 }{ 20 }$
5. Multiply $\dfrac{ 3 }{ 20 }$ by number such that it is less than or equal to $1.$
The work done in $12$ days $= 6 \times \dfrac{ 3 }{ 20 } = \dfrac { 9 }{ 10 }$
6. Remaining work $= 1 - \dfrac { 9 }{ 10 } = \dfrac { 1 }{ 10 }$
7. Since, they work in alternate days, on $13^{th}$ day remaining work $\dfrac { 1 }{ 10 }$ is done by Barbara.
8. Thus, they will take $13$ days to finish the work.