Solving the Rabbit and Pigeon Ratio Problem with Detailed Steps and Explanation

Imagine a charming park where rabbits and pigeons coexist in a specific ratio. The problem is to determine the number of rabbits if the total number of legs in the park is 300, and the ratio of rabbits to pigeons is 4:7. This puzzle involves a step-by-step approach to solving a ratio and proportion problem. Let's walk through it together.

Understanding the Problem

The puzzle provides two key pieces of information:

The ratio of rabbits to pigeons is 4:7. The total number of legs in the park is 300.

A rabbit has 4 legs, while a pigeon has 2. Using this information, we can create an equation to solve for the number of rabbits and pigeons in the park.

Step-by-Step Solution

Step 1: Define the Variables

Let x represent the number of groups of rabbits, and y represent the number of groups of pigeons. Given the ratio, we have:

x : y 4 : 7

This can be written as:

7x 4y

Solving for y in terms of x:

y 7x/4

Step 2: Write the Equation for Legs

We know that rabbits have 4 legs each, and pigeons have 2 legs each. The total number of legs is given by:

4x 2y 300

Substituting y 7x/4 into the equation:

4x 2(7x/4) 300

Simplifying the equation:

4x 14x/4 300

4x 3.5x 300

7.5x 300

x 300/7.5 40

Step 3: Calculate the Number of Rabbits

Since x represents the number of groups of rabbits:

Number of groups of rabbits x 40

Therefore, the number of rabbits is:

4x 4(40) 160/4 40

Conclusion

The solution to the puzzle is that there are 40 rabbits in the park. This detailed solution not only provides the answer but also illustrates the step-by-step process of solving a ratio and proportion problem using algebra.

Additional Insights

It might seem unusual to find so many rabbits and pigeons in one park, but such a scenario is possible in a wildlife sanctuary or a large, well-managed park where such animals are allowed to roam freely.

Cross-Verification

To cross-verify the solution:

Number of pigeons 7x 7(40) 280/2 70

Total legs 4x 2(7x/4) 160 140 300

This confirms our solution is correct.