How to solve the cow grazing problem in mathematics and find the method.

Hello, I found a little information that I once collected and sent it to you, I hope it will help you a little.

Newton, the great British scientist, once wrote a book on mathematics. There is a very famous topic in the book about cattle grazing on pastures, which was later called the "Newtonian problem".

The "Newtonian problem" goes like this: "There was a pasture with 27 cattle that were known to be raised, and the grass was eaten up in six days; Raise 23 cows and eat all the grass in 9 days. If you raise 21 cows, how many days can you eat all the grass on the pasture? And the grass on the pasture is constantly growing. ”

The general solution to this type of problem is: consider the grass that a cow eats in a day as 1, then there is:

1) The amount of grass eaten by 27 cows for 6 days was: 27 6 162

These 162 include the original grass of the pasture and the new grass of 6 days. ）

2) The amount of grass that 23 cows ate for 9 days was: 23 9 207

These 207 include the pasture's original grass and the 9-day-old grass that grows newly. ）

3) The new grass grows at 1 day: (207 162) (9 6) 15

4) The original grass on the pasture is: 27 6 15 6 72

5) The newly grown grass is enough for 15 cows to eat every day, 21 cows minus 15, and the remaining 6 cows eat the grass of the original pasture:

72 (21 15) 72 6 12 (days).

So it takes 21 cows to eat up the grass on the pasture in 12 days.

Do the math, please.

There is a pasture, if you raise 25 sheep, you can eat all the grass in 8 days; Raise 21 sheep and eat all the grass in 12 days. What if you raise 15 sheep and eat up all the grass growing on the pasture in a few days?

All the grass that the cow eats during that time - (existing grass + grass that the cow eats at the time it grows).

The amount of grass eaten by the cow - the amount of grass grown = the amount of grass consumed.

There are four basic formulas commonly used to solve the problem of cattle grazing, which are:

1) The growth rate of grass = (the corresponding number of cow heads, the number of days that eat more, the corresponding number of cow heads, the number of days that eat less) (the number of days that eat more, the number of days that eat less);

2) The amount of original grass = the number of cattle head The number of days eaten The growth rate of the grass The number of days eaten; `

3) The number of days eaten = the amount of grass (the number of cows, the growth rate of grass);

4) The number of cattle heads = the original amount of grass and the number of days eaten + the growth rate of the grass.

These four formulas are the basis for solving Newton's problems.

1) The growth rate of grass = (the corresponding number of cow heads, the number of days that eat more, the corresponding number of cow heads, the number of days that eat less) (the number of days that eat more, the number of days that eat less);

2) The amount of original grass = the number of cattle head The number of days eaten The growth rate of the grass The number of days eaten; 3) The number of days eaten = the amount of grass (the number of cows, the growth rate of grass);

4) The number of cattle heads = the original amount of grass and the number of days eaten + the growth rate of the grass.

If the amount of grass that a cow eats in a day is considered to be 1, then there is:

The 27 cows ate the following grass for 6 days: 27 6 162 These 162 include the grass that was originally used in the pasture and the grass that grows new in 6 days. The 23 cows ate the grass for 9 days

23 9 207 (These 207 include the grass that was originally in the pasture and the grass that grew new in 9 days.) The grass that grows new on 1 day is: (207 162) (9 6) 15 The original grass on the pasture is:

27 6 15 6 72 or 23 9-15 9 72

The new grass grows enough to feed 15 cows every day, and 21 cows minus 15, leaving 6 cows to eat the grass from the original pasture

72 (21 15) 72 6 12 (days) So raise 21 cows, and it takes 12 days to eat all the grass on the pasture (if you don't understand, please ask!).

The original amount of grass in the pasture is 1, and the new grass grows x every week;

1 (1-5x)/20*5

1-6x)/16*6

2 (1-5x)/20*5

1-yx)/11y

From (1) we get x = 1 30, and substituting (2) gives y = 8 (days).

The problem of cattle grazing, also known as the problem of growth and decline or Newton's pasture, was proposed by the great British scientist Newton in the 17th century. The condition of a typical cow grazing problem is to find how many cows can eat the same grass by assuming that the growth rate of grass is fixed, and the number of days it takes for different numbers of cattle to eat the same grass is different. Because the number of days eaten is different, and the grass grows every day, the stock of grass constantly changes with the number of days the cow eats.

There are four basic formulas commonly used to solve the problem of cattle grazing, which are:

1) The growth rate of grass = (the corresponding number of cow heads, the number of days that eat more, the corresponding number of cow heads, the number of days that eat less) (the number of days that eat more, the number of days that eat less);

2) The amount of original grass = the number of cattle head The number of days eaten The growth rate of the grass The number of days eaten; `

3) The number of days eaten = the amount of grass (the number of cows, the growth rate of grass);

4) The number of cattle heads = the original amount of grass and the number of days eaten + the growth rate of the grass.

These four formulas are the basis for solving the problem of growth and decline.

Since the grass is constantly growing in the process of grazing in the process of cattle grazing, the key to solving the problem of growth and decline is to find ways to find invariants from change. The original grass on the pasture is unchanged, and although the new grass is changing, the amount of new grass growing every day should be the same because it grows at a uniform rate. It is because of this invariant that the above four basic formulas can be derived.

The problem of cattle grazing is often given that different numbers of cattle eat the same grass, and the field has both the original grass and the new grass that grows every day. Due to the different number of cows that eat grass, find out how many days the grass in the field can be eaten by several cows.

The key to solving the problem is to figure out the known conditions, conduct comparative analysis, and then find the number of new grass growing every day, and then find the amount of original grass in the grassland, and then solve the problem that the question is always asked.

The basic quantitative relationship for this type of problem is:

1.(Number of cows, days of grazing more - number of cows, number of days of grazing) (number of days of eating more - days of eating less) = the amount of new grass growing in the grass per day.

2.Number of cows Grass days - new growth per day Number of grass days = original grass in the meadow.