Cows Eat Grass How to expand into 300 words

When I was in the Olympiad class today, the teacher told us a very interesting Olympiad problem and shared it with you.

The question is as follows: There is a pasture with 27 cows, and the grass is eaten in 6 days; There were 23 cows, and the grass was eaten in 9 days. If there are 21 cows, how many days can they eat all the grass?

The teacher said, "This problem is called the Newtonian problem, also known as the cow eating grass problem. The teacher wrote this question on the blackboard, and I read it three times without a clue.

Seeing that we were all frowning and thinking, the teacher gave us a suggestion: "The difficulty of this problem is that the total amount of forage is undetermined, and it should be analyzed as time increases. "Hey!

It's okay not to dial, but I'm even more confused when I dial it. The cow eats the grass and eats it, so how can it still grow with time? Just when I was puzzled, the teacher said:

The key is to analyze the relationship between the total amount of forage grass that is undetermined and the amount of grass that is unchanged. Oh! I had a little idea, and hurriedly raised my hand, and the teacher saw that I understood through his prompts, and called me.

I said, "Teacher, don't you look at the amount of grass that each cow eats every day?" That is, the amount of grass eaten by 27 cows in 6 days is 27 6 = 162;The amount of grass eaten by 23 cows for 9 days was 23 9 = 207.

Excellent! The teacher then asked, "If you find out the amount of grass that each cow eats every day, will you find out how much grass grows every day?" This is simple, its extra unit is 207-162 = 45, that is, 9-6 = 3 days more, that is an average of 45 more per day 3 = 15, that is, the new grass to be born in the pasture every day is enough for 15 cattle to eat for a day.

The teacher added that the original amount of grass was 162-15 6=72. Then, the teacher asked the students, "Think about it, what can you do according to the questions?" Suppose 15 cows eat the grass that grows every day, and the rest of the cattle eat the original grass, how many days can they eat?

The answer is already clear: 72 (21-15) = 12 days. The teacher saw that some students were still confused, so he asked everyone to discuss it together, and the teacher did not explain the next question until the students knew it.

It's so much fun to do math Olympiad problems, so try it too!

Industrious cows eat delicious hay with relish.

3 and 1 3 hectares = 10 3 hectares.

Let's set the grazing allowance for each cow 1 per week.

Let's start with the first piece of land.

12 cows, grazing for 4 weeks: 12 4 = 48 servings.

Looking at the second piece of land, the area is the first one: 10 10 3 = 3 times.

Can be fed by 12 3 = 36 cows for 4 weeks.

Grass eaten in total: 36 4 = 144 servings.

Now it feeds 21 cows for 9 weeks.

Grass eaten in total: 21 9 = 189 servings.

Difference: 189-144 = 45 parts.

These 45 parts are the grass that grows in the second plot of land in 9-4 = 5 weeks: 45 5 = 9 parts.

It turned out that there was grass: 189-9 9 = 108 parts.

Looking at the third plot, the area is 24 10 = times that of the second plot.

It turned out that there was grass: 108 servings.

Grass per week: 9 servings.

After 18 weeks, a total of grass is eaten:

Serving an average of grazing per week:

648 18 = 36 servings.

So it can feed 36 cows for eighteen weeks.

Uniform Area:

The first plot: 10 3 hectares of pasture with 12 cows for 4 weeks, then the first 1 hectare of pasture can be raised for 12 (10 3) = cattle for 4 weeks.

24 hectares can raise cattle for 4 weeks.

The second block: 10 hectares of pasture, raising 21 cows, can be maintained for 9 weeks, then the first 1 hectare pasture, can be raised 21 10 = cattle for 9 weeks.

24 hectares can raise cattle for 9 weeks.

Set: A cow eats 1 portion of grass per week.

Servings) Weekly grass size:

Serving) original grass amount:

Serving) 18 circumference grass amount: servings).

There is a total of grass: portions).

To eat for 18 weeks, you need the number of heads of the cow:

648 18 = 36 (head).

It should be a drop in the bucket, or it may be a small test.

The formula used for this question is: Number of working members Working hours Productivity per unit of time of a single member = Total amount of work.

Column equations are required, let the unknowns:

1) The total amount of forage at the beginning of the pasture is: x (the unit can be taken in kilograms);

2) the growth rate of forage is: y (kilogram day);

3) The rate at which the cow grazes is: z (kilogram per day, meaning "each cow eats z kilograms of grass per day");

4) Cattle grazing at the beginning have w heads;

If this is the case, we have four unknowns, but according to the problem, there are only three conditions, and we can only list three equations, and we cannot solve all the unknowns. However, many of these quantities are not of concern to us, so we can reduce the number of unknowns by changing the units of unknowns.

We define a new unit for measuring the amount of forage: "cow day", which means "the amount of grass that each cow eats per day", for example, 10 cow days means enough grass for 10 cows to eat for 1 day, and also means the amount of grass that 5 cows eat for 2 days, which is obviously equal. But we don't care how many cows will eat for a few days.

Now we can say that the amount of grass eaten by 10 cows in 20 days must be: 10 20 = 200 cow days.

It's actually a unit that doesn't make any sense, and we don't know exactly how many kilograms of grass are in 1 ox day, but we don't need to care about that. The purpose of this is to eliminate an unknown – z, because the rate at which cattle are grazing is now a known amount: z = 1 cow day per day – each cow can of course only eat 1 cow day of grass per day.

Now we have unknowns:

1) X: Unit: Niu Tian;

2) y: unit: cow day day;

3) W: Unit: Head.

Equations according to the question:

Feeds 17 cows for 30 days: x + 30 * y = 17 * 30 * 1;(Multiplying by 1 makes sense).

It can also be fed by 19 cows for 24 days: x + 24 * y = 19 * 24 * 1;

w*6 + w - 4)*2 = x + 6 + 2)*y；

According to , you can get:

x = 240；The implication is that the original amount of pasture in the pasture is 240 cow days, which is enough for 240 cattle to eat for 1 day;

y = 9；The meaning is that the pasture grows 9 cow days of pasture every day, enough for 9 cows to eat for 1 day; That is, if there are only 9 cows on the pasture, then the amount of grass in the pasture can remain the same and be in a state of equilibrium;

Substituting the above results into , we get:

w*6 + w - 4)*2 = 240+ (6 + 2)*9

solution, yield: w = 40

That is, 40 cows were grazing at the beginning.

The great scientist Isaac Newton 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 is a pasture with 27 cattle that are known to be raised, and the grass is eaten in 6 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 kind of problem is: If the grass eaten by a cow in one day is considered as 1, then there is: (1) The grass eaten by 27 cows in 6 days is:

27 6 162 (These 162 include the grass that was originally in the pasture and the grass that grew new in 6 days.) 2) The grass that 23 cows ate for 9 days is: 23 9 207 (These 207 include the original grass of the pasture and the new grass that grows in 9 days.

3) The new grass on 1 day is: (207 162) (9 6) 15 (4) The original grass on the pasture is: 27 6 15 6 72 (5) The new grass grows every day is enough for 15 cows, 21 cattle minus 15, and the remaining 6 eat the grass of the original pasture:

72 (21 15) 72 6 12 (days) So with 21 cows, it takes 12 days to eat up all the grass on the pasture.

Solution: How many kilograms of grass does a cow eat every day, how many kilograms of grass does a sheep eat every day, 3x+8y=48

5x+15y=85

The solution is x=8y=38+3=11

Answer: A cow and a sheep eat a total of 11 kilograms of grass per day

Solution: Let a cow eat grass x kilograms every day, and a sheep eat grass y kilograms every day to list the equations.

3x+8y=48

5x+15y=85

The solution is x=8y=38+3=11

Answer: A cow and a sheep eat a total of 11 kilograms of grass per day

【Answer】4 cows less to eat for 2 days, which is equivalent to 8 days less 4 2 8 1 cows to eat. Just ask for a few cows for 8 days, and then add 1 cow.

Each cow should eat 1 serving of grass per day.

17 cows ate 17 30 510 servings for 30 days, and 19 cows ate 19 24 456 servings for 24 days.

30 24 6 days grew 510 456 54 parts of grass, 54 6 9 parts of new grass grew every day, the original grass had (19 9) 24 240 parts, 8 days to eat 240 8 30 cattle to eat the original grass.

There were 30 9 1 40 cows.

Is that okay?