Solving Ratio Problems Involving Broken Eggs: A Chicken-Duck Egg Case Study

Mathematical problems involving ratios and proportions are quite common, especially when dealing with daily situations like managing and calculating quantities of different types of items. One such engaging problem is the scenario involving the ratio of chicken eggs to duck eggs, where the ratio changes due to some eggs being broken. Let's explore this problem in detail and understand the step-by-step solution.

Problem Statement

Li had a certain ratio of chicken eggs to duck eggs, which was 3:4. While putting the eggs into a carton, 12 chicken eggs were accidentally broken, and the new ratio of the number of chicken eggs to the number of duck eggs became 1:2. The challenge is to determine how many chicken eggs Li had initially.

Step-by-Step Solution

To solve this problem, we can begin by representing the initial quantities of eggs using algebraic expressions.

Expression Representation

We will denote the initial number of chicken eggs as 3x and the initial number of duck eggs as 4x, where x is a positive integer. The reason for using x as a common factor is to simplify our calculations.

Problem Setting

After breaking 12 chicken eggs, the new ratio of the number of chicken eggs to the number of duck eggs is given as 1:2. We can set up the ratio equation based on this information:

(3x - 12) : 4x 1 : 2

From the equation (3x - 12) : 4x 1 : 2, we can write the following proportion:

4x 2 * (3x - 12)

Let's solve this equation step-by-step:

Multiply both sides of the equation by 2: 4x 6x - 24 Subtract 6x from both sides: -2x -24 Divide both sides by -2: x 12

Now that we have the value of x, we can determine the initial number of chicken eggs:

3x 3 * 12 36

Conclusion

Hence, the initial number of chicken eggs Li had was 36. This solution emphasizes the importance of using algebraic methods to represent and solve real-world problems involving ratios and proportions.

Revisiting the Solution

For a deeper understanding, let's solve the problem using a different approach.

Alternative Solution

Initially, let's establish the ratio of chicken eggs to duck eggs as 3:4. This means that for every 3 chicken eggs, there are 4 duck eggs. If we represent the number of chicken eggs as 3x and duck eggs as 4x, where x is a common factor.

After breaking 12 chicken eggs, the new ratio becomes 1:2. This implies that for every chicken egg, there are 2 duck eggs. We can set up the following equation based on the new ratio:

(3x - 12) : 4x 1 : 2

From the equation (3x - 12) : 4x 1 : 2, we can write the following proportion:

4x 2 * (3x - 12)

By solving the equation, we will arrive at the same result that the initial number of chicken eggs is 36.

Conclusion

Understanding how to solve ratio problems involving changes can help students and professionals in various fields, including business, logistics, and agriculture. The key to solving such problems lies in systematically setting up and solving equations based on the given ratios and conditions.