Introduction to the Problem

Imagine the scenario: Peter has 48 apples. He decides to give away a portion of his apples to his son and his daughter, and the remaining apples are shared in a specific manner between himself and his wife. The question is, how many apples do Peter and his wife receive? Let's solve this problem step-by-step.

Step 1: Distributing to the Son

Peter first gives 1/4 of his apples to his son. We calculate this as follows:

(frac{1}{4} times 48 12)

Son receives 12 apples. Now, let's find out how many apples are left:

(48 - 12 36)

Step 2: Distributing to the Daughter

Next, Peter gives 1/3 of the remaining apples to his daughter:

(frac{1}{3} times 36 12)

Daughter receives 12 apples. Now, let's find out how many apples are left:

(36 - 12 24)

Step 3: Sharing the Remaining Apples

The problem states that Peter's wife gets 4 more apples than he does. Let’s denote the number of apples Peter receives as (x). Therefore, his wife receives (x 4) apples. We know that the total remaining apples are 24:

(x (x 4) 24)

Simplifying the equation:

(2x 4 24)

(2x 20)

(x 10)

Peter receives 10 apples, and his wife receives:

(10 4 14)

Conclusion and Verification

To verify, let's check if the distribution meets the initial conditions:

- Son receives 12 apples.

- Daughter receives 12 apples.

- Peter receives 10 apples.

- Wife receives 14 apples.

Total: (12 12 10 14 48) apples.

The distribution indeed totals 48 apples, confirming our solution.

Additional Insights

This problem demonstrates the practical application of basic algebra and fractional operations in real-life scenarios. It also highlights the importance of step-by-step calculations and equation formulation in solving complex problems.

For further exploration, you might consider other such problems involving fractional distributions or algebraic equations. This type of problem-solving skill is highly valued in many fields, including mathematics, finance, and engineering.