Solving the Zoo Animal Legs Puzzle

This problem involves a classic type of word problem that can be solved using a system of linear equations. We are given the total number of heads and the total number of legs for two types of animals: deers and peacocks.

Defining the Variables

Let's define our variables based on the animals:

• Let p represent the number of peacocks.
• Let d represent the number of deers.

Formulating the Equations

We can create two equations based on the information given:

1. Equation based on Heads: Each animal (deer or peacock) has exactly one head. The total number of heads is 80.
   The equation is: \( d + p = 80 \)
2. Equation based on Legs: Deers have 4 legs, and peacocks have 2 legs. The total number of legs is 200.
   The equation is: \( 4d + 2p = 200 \)

Solving the System of Equations

Now we have a system of two linear equations with two variables:

\( d + p = 80 \quad \text{(Equation 1)} \)

\( 4d + 2p = 200 \quad \text{(Equation 2)} \)

We can solve this system using several methods, such as substitution or elimination. Let's use the elimination method, as it can be quite straightforward here.

Using the Elimination Method

Our goal is to eliminate one variable (either \(d\) or \(p\)) by manipulating the equations.

1. Multiply Equation 1 by 4 to make the coefficient of \(d\) the same as in Equation 2:
   \( 4 \times (d + p) = 4 \times 80 \)
   \( 4d + 4p = 320 \quad \text{(New Equation 3)} \)
2. Now subtract Equation 2 (\( 4d + 2p = 200 \)) from New Equation 3 (\( 4d + 4p = 320 \)):
   \( (4d + 4p) - (4d + 2p) = 320 - 200 \)
   \( 4d - 4d + 4p - 2p = 120 \)
   \( 2p = 120 \)
3. Solve for \(p\):
   \( p = \frac{120}{2} \)
   \( p = 60 \)

So, there are 60 peacocks.

Finding the Number of Deers (Optional Check)

We can substitute the value of \(p=60\) back into Equation 1 (\( d + p = 80 \)) to find the number of deers:

\( d + 60 = 80 \)

\( d = 80 - 60 \)

\( d = 20 \)

There are 20 deers.

Verification

Let's check if these numbers satisfy the conditions:

• Total heads: \( d + p = 20 + 60 = 80 \). Correct.
• Total legs: \( 4d + 2p = 4(20) + 2(60) = 80 + 120 = 200 \). Correct.

The numbers match the problem statement.

Final Answer: Number of Peacocks

The problem asked for the number of peacocks.

We found that \(p = 60\).

Revision Table: Zoo Animals Equations

Additional Information: Solving Systems of Equations

Solving systems of linear equations is a fundamental skill in algebra. Besides the elimination method used above, another common method is the substitution method.

Substitution Method Explained

Here's how you could solve the same problem using the substitution method:

1. Start with the system:
   \( d + p = 80 \quad \text{(Equation 1)} \)
   \( 4d + 2p = 200 \quad \text{(Equation 2)} \)
2. Solve one equation for one variable. From Equation 1, it's easy to isolate \(d\) (or \(p\)):
   \( d = 80 - p \quad \text{(Equation 3)} \)
3. Substitute this expression for \(d\) into Equation 2:
   \( 4(80 - p) + 2p = 200 \)
4. Now, solve the resulting equation for \(p\):
   \( 320 - 4p + 2p = 200 \)
   \( 320 - 2p = 200 \)
   Subtract 320 from both sides:
   \( -2p = 200 - 320 \)
   \( -2p = -120 \)
   Divide by -2:
   \( p = \frac{-120}{-2} \)
   \( p = 60 \)
5. Once you have \(p=60\), substitute this value back into Equation 3 (\( d = 80 - p \)) to find \(d\):
   \( d = 80 - 60 \)
   \( d = 20 \)

Both methods yield the same result, confirming the number of peacocks is 60 and deers is 20.