Question 9
A Mongolian yak herdsman manages two different grazing areas: Pasture A and Pasture B. In each pasture, the grass grows at a constant daily rate, although the growth rates may differ between the two pastures. Each yak consumes grass at the same constant rate.
Through years of experience, the herdsman has made the following observations:
For Pasture A:
- $29$ yaks can completely consume all the grass in exactly $7$ days,
- $17$ yaks can completely consume all the grass in exactly $28$ days.
For Pasture B:
- $41$ yaks can completely consume all the grass in exactly $6$ days,
- $21$ yaks can completely consume all the grass in exactly $26$ days.
The herdsman currently owns $32$ yaks, all of which must graze every day. On any given day, he may divide the yaks between the two pastures in any way he chooses, and the allocation of yaks may vary from day to day.
What is the maximum number of days the $32$ yaks can continue grazing before the grass in both pastures is completely consumed?
Through years of experience, the herdsman has made the following observations:
For Pasture A:
- $29$ yaks can completely consume all the grass in exactly $7$ days,
- $17$ yaks can completely consume all the grass in exactly $28$ days.
For Pasture B:
- $41$ yaks can completely consume all the grass in exactly $6$ days,
- $21$ yaks can completely consume all the grass in exactly $26$ days.
The herdsman currently owns $32$ yaks, all of which must graze every day. On any given day, he may divide the yaks between the two pastures in any way he chooses, and the allocation of yaks may vary from day to day.
What is the maximum number of days the $32$ yaks can continue grazing before the grass in both pastures is completely consumed?