牛吃草问题

Cow Eating Grass Problem (Classic Rate Problem)

The “cow eating grass” problem that deals with **changing quantities** — specifically, grass that continues to grow while being eaten.

Typical Problem Statement

There is a pasture where grass grows at a constant rate.

- If 10 cows graze on it, the grass will be completely eaten in 20 days.

- If 15 cows graze on it, the grass will be completely eaten in 10 days.

**Question:** How many cows are needed to eat all the grass in 5 days?

(Or other variations: How many days will it take for a certain number of cows? What is the daily growth rate of the grass? etc.)

### Core Idea

The total amount of grass consists of two parts:

1. The grass that is already present at the beginning.

2. The new grass that grows every day.

While the cows are eating, the grass keeps growing. Therefore, the problem cannot be solved by simple division — it requires setting up equations that account for both the **consumption rate** of the cows and the **growth rate** of the grass.

General Approach

Let:

- \( x \) = initial amount of grass

- \( y \) = daily growth of grass

- \( c \) = amount one cow eats per day

Then for different numbers of cows and days, you can form equations such as:

\[

x + 20y = 10 \times 20 \times c

\]

\[

x + 10y = 15 \times 10 \times c

\]

Solving the system gives the relationship between growth and consumption, which can then be used to answer further questions.

This type of problem trains students to model real-world situations where quantities are changing simultaneously, and is a classic example of **linear equations with rates**.