How many cows are there? > In a field, the grass increases in a constant rate. $17$ cows can eat all the grass in the field in $30$ days. $19$ cows can eat all the grass in $24$ days. Suppose, a group of cows started eating grass for $6$ days. Then $4$ cows are sold. The rest cows took $2$ more days to eat the remaining grass. What is the number of cows in the group? ## My trying: A cow eats a certain amount of grass in one day, call it $c$. The field grows by a certain amount each day, call it $g$. The field has some initial amount of grass: $i$ \begin{equation} \begin{aligned} i + 30g - 17\cdot30c &= 0\\\ i+24g-19\cdot24c&=0 \end{aligned} \end{equation} Solving these two equations we get , $g = 9c$ . That means It takes 9 cows to eat one day's growth in one day.

$$\begin{aligned} V+30x&=30\times 17\times y\\\ V+24x&=24\times 19\times y\\\ V+8x&=6\times k\times y+2(k-4)\times y\\\ V&=510y-30x\\\ x&=9y\\\ 510y-22x&=(8k-8)y\\\ 510y-198y&=(8k-8)y\\\ k&=40 \end{aligned}$$ where $V$ - value of grass on the field, $x$ - speed of growth grass for a day, $y$ - speed of 1 cow for a day, $k$ - number of cows.