Required Mathematical Intelligence

Chapter 87 The Famous Problem of Cow Eating Grass

In Newton's famous book "General Arithmetic", there is also a well-known topic, that is, the topic of cattle grazing on the pasture. Later, people called this kind of application problem Newton's problem.

"There is a piece of pasture grass, if you graze 27 cows, you can eat up the grass in 6 weeks; if you graze 23 cows, you can eat up the grass in 9 weeks; if you graze 21 cows, how many weeks can you eat it?" Eat up the grass?"

To solve this problem, we assume that the pasture is equally dense everywhere, grows at the same rate, and that each cow eats the same amount of grass each week.

Can you solve this problem?
[Answer: Analysis and Solution When grazing cattle on a pasture, the cattle not only eat the original grass on the pasture, but also eat the newly grown grass on the pasture.Therefore, the key to answering this question is to know the original amount of grass on the pasture and the amount of grass growth every week.

Assume that the amount of grass that each cow eats per week is 1.
27头牛6个星期的吃草量为27×6=162，这既包括牧场上原有的草，也包括6个星期长的草。

23头牛9个星期的吃草量为23×9=207，这既包括牧场上原有的草，也包括9个星期长的草。

因为牧场上原有的草量一定，所以上面两式的差207-162=45正好是9个星期生长的草量与6个星期生长的草量的差。由此可以求出每星期草的生长量是45÷（9-6）=15.
牧场上原有的草量是162-15×6=72，或207-15×9=72.
前面已假定每头牛每星期的吃草量为1，而每星期新长的草量为15，因此新长出的草可供15头牛吃。今要放牧21头牛，还余下21-5=6头牛要吃牧场上原有的草，这牧场上原有的草量够6头牛吃几个星期，就是21头牛吃完牧场上草的时间。72÷6=12（星期）。

That is to say, grazing 21 cows can eat up the grass on the pasture in 12 weeks. ]
(End of this chapter)