Solving Chicken and Rabbit Problems: A Systematic Approach

Introduction

Have you ever encountered a problem like this: 'I count 35 heads and 94 legs among the chickens and rabbits on a farm. How many rabbits and how many chickens do I have?' Such problems are common in basic algebra and can be solved using a systematic approach involving a system of equations. In this article, we will delve into the process and provide a step-by-step solution to this problem and several others.

Problem Statement

The problem at hand is:

I count 35 heads and 94 legs among the chickens and rabbits on a farm. How many rabbits and how many chickens do I have?

Solving the Problem

To solve this problem, let's define our variables:

Let c number of chickens Let r number of rabbits

We can set up the following two equations based on the information given:

Total number of heads: [ c r 35 ] Total number of legs: [ 2c 4r 94 ]

Step 1: Simplify the Second Equation

We can simplify the second equation by dividing it by 2:

Simplified total number of legs: [ c 2r 47 ]

Step 2: Set Up the System of Equations

Now, we have the following system of equations:

[ c r 35 ] [ c 2r 47 ]

Step 3: Subtract the First Equation from the Second

We subtract the first equation from the second to eliminate (c):

[ (c 2r) - (c r) 47 - 35 ] [ c 2r - c - r 12 ] [ r 12 ]

Step 4: Substitute (r 12) Back into the First Equation

Now, we substitute (r 12) back into the first equation:

[ c 12 35 ] [ c 35 - 12 ] [ c 23 ]

Conclusion

Thus, the number of chickens and rabbits on the farm is:

Chickens: 23 Rabbits: 12

We can validate our solution by checking the total heads and legs:

Heads: (23 12 35) Legs: (2 times 23 4 times 12 46 48 94)

Alternative Problems

Let's consider another problem with the same method:

Let (c) and (r) be chickens and rabbits respectively. Then:

[ c r 58 ] [ 2c 4r 192 ]

First, simplify the second equation:

[ 2c 4r 192 ] [ c 2r 96 ]

Set up the system of equations:

[ c r 58 ] [ c 2r 96 ]

Subtract the first equation from the second:

[ (c 2r) - (c r) 96 - 58 ] [ r 38 ]

Substitute (r 38) back into the first equation:

[ c 38 58 ] [ c 20 ]

Thus:

Chickens: 20 Rabbits: 38

General Solution for Similar Problems

Let's consider another example with a slightly different setup:

Let the number of chicken be (x)

Let the number of rabbit be (y)

So:

[ x y 72 ] [ 2x 4y 200 ]

Solving the pair of linear equations by substitution or elimination:

[ 2x 4y 200 ] [ x y 72 ]

Substitute (y 72 - x) into the second equation:

[ 2x 4(72 - x) 200 ] [ 2x 288 - 4x 200 ] [ -2x 200 - 288 ] [ -2x -88 ] [ x 44 ]

Substitute (x 44) back into the first equation:

[ 44 y 72 ] [ y 28 ]

Thus:

Chickens: 44 Rabbits: 28

Conclusion

These examples demonstrate the power of using a systematic approach to solve problems involving multiple variables. By setting up and solving systems of equations, we can find the solution to a variety of problems involving animals and numbers of different types.