Solving the Puzzle: Determining the Weight of an Empty Crate

Recently, a mathematical puzzle has been circulating the web, involving a crate filled with apples. The puzzle states that a crate full of apples weighs 15 kg, and when it is two-thirds full of apples, it weighs 11 kg. The challenge is to determine the weight of the empty crate. Let's break down the problem step-by-step to find the solution.

Problem Description and Variables

Let's denote the weight of the empty crate as C and the weight of the apples when the crate is full as A. From the problem, we have the following two equations:

C A 15 kg (Equation 1) C (frac{2}{3})A 11 kg (Equation 2)

By solving these equations, we can determine the unknowns and find the weight of the empty crate.

Solving the Equations

Step 1: Solve Equation 1 for A

From Equation 1:

A 15 - C (Equation 3)

Step 2: Substitute Equation 3 into Equation 2

Substituting A from Equation 3 into Equation 2:

C (frac{2}{3})(15 - C) 11

Step 3: Simplify the equation

Expanding the equation:

C 10 - (frac{2}{3})C 11

Combining like terms:

C - (frac{2}{3})C 1

This simplifies to:

(frac{1}{3})C 1

Step 4: Solve for C

Subtract 10 from both sides:

(frac{1}{3})C 1

Multiply both sides by 3:

C 3

Conclusion: The weight of the empty crate is 3 kg.

Alternative Method:

Let's confirm the solution using another method. If only 1/3 full of the crate weighs only 1/3 of the crate's apples, then the weight difference when the crate is 1/3 full compared to being empty is:

4 kg 15 kg - 11 kg

Therefore, the full weight of the apples when the crate is full is:

4 kg × 3 12 kg

Thus, the weight of the empty crate is:

15 kg - 12 kg 3 kg

Verification: By substituting back into the original equations:

15 kg 3 kg 12 kg 11 kg 3 kg (2/3)×12 kg 3 kg 8 kg 11 kg

The solution satisfies both equations in the system.

In conclusion, the weight of the empty crate is clearly determined to be 3 kg through both mathematical methods and checks. Such puzzles help improve logical thinking and algebraic skills.