On a certain pasture the grass grow at an even rate. It is known that 40 cows can graze on it for 40 days before the grass is exhausted, but 30 cows can graze there for as long as 60 days. How many days would the pasture last if 20 cows were to graze on it ?

Options

A. 90 days
B. 80 days
C. 100 days
D. 120 days

Answer

To solve the problem step by step, we will use the information given about the grazing capacity of cows and the growth of grass on the pasture.

### Step 1: Define Variables
Let:
- \( x \) = initial quantity of grass (in units)
- \( y \) = daily growth rate of grass (in units per day)

### Step 2: Set Up the First Equation
From the first condition, we know that 40 cows can graze for 40 days. The total amount of grass consumed by the cows over this period can be expressed as:
- Total grass consumed = Number of cows × Number of days = \( 40 \times 40 = 1600 \)

The equation representing the total grass consumed is:
\[ x + 40y = 1600 \] (Equation 1)

### Step 3: Set Up the Second Equation
From the second condition, we know that 30 cows can graze for 60 days. The total amount of grass consumed by these cows is:
- Total grass consumed = Number of cows × Number of days = \( 30 \times 60 = 1800 \)

The equation representing the total grass consumed is:
\[ x + 60y = 1800 \] (Equation 2)

### Step 4: Solve the Equations
Now we have two equations:
1. \( x + 40y = 1600 \)
2. \( x + 60y = 1800 \)

We can subtract Equation 1 from Equation 2 to eliminate \( x \):
\[ (x + 60y) - (x + 40y) = 1800 - 1600 \]
This simplifies to:
\[ 20y = 200 \]
Thus, we find:
\[ y = 10 \]

### Step 5: Substitute \( y \) Back to Find \( x \)
Now that we have \( y \), we can substitute it back into Equation 1 to find \( x \):
\[ x + 40(10) = 1600 \]
\[ x + 400 = 1600 \]
\[ x = 1600 - 400 = 1200 \]

### Step 6: Calculate the Duration for 20 Cows
Now we need to find out how long the pasture will last if 20 cows graze on it. The total grass available is \( x + \text{(growth over days)} \):
- Total grass available = \( 1200 + 10D \) (where \( D \) is the number of days)

The total grass consumed by 20 cows grazing for \( D \) days is:
- Total grass consumed = \( 20D \)

Setting the total grass available equal to the total grass consumed gives us:
\[ 1200 + 10D = 20D \]

### Step 7: Solve for \( D \)
Rearranging the equation:
\[ 1200 = 20D - 10D \]
\[ 1200 = 10D \]
\[ D = \frac{1200}{10} = 120 \]

### Conclusion
The pasture would last for **120 days** if 20 cows were to graze on it.