Medium 6

Let $a$ and $b$ be the number of days on which Eric writes $2$ and $3$ poems, respectively.

Let $c$ and $d$ be the number of days on which Marina writes $2$ and $3$ poems, respectively.

Then we have 3 equations to work with, namely

(i) $2a + 3b = 43$ , (ii) $2c + 3d = 61$ and (iii) $a + b = c + d$ .

Subtracting (i) from (ii) gives us equation (iv), $2(c - a) + 3(d - b) = 18$ .

Rearranging (iii), we have that $c - a = -(d - b)$ .

Plugging this into (iv) gives us that $d - b = 18$ .

Now from (i) and (ii) it is clear that $b \ge 1$ and $d \le 19$ . Given these conditions, and that $d - b = 18$ , the only possible solutions are $b = 1$ and $d = 19$ .

With $b = 1$ we have that $a = 20$ , giving us a final solution of $a + b = \boxed{21}$ .

From the question itself we can know that they wrote with different tenacity. we can tell Eric wrote 2 per day .... and Marina wrote 3 per day ,since in same time period Marina wrote more. Then Eric would take 20 days and one extra day to write 3 poems..... Marina would take 19 days and 2 extra days to write 2 poems a day... So we get as 21 .