A box weighs 8.5 kg when it is completely filled with sand and 5.5 kg when it is half filled with sand. What is the weight, in kilograms, of the empty box alone?

Introduction / Context:
This is a classic linear equation word problem involving total weight, where the weight of the box and the weight of the sand combine to give the observed measurements. You are given the weight when the box is full of sand and when it is half full, and you must deduce the weight of the empty box. Problems like this test your ability to model real-life situations using simple algebra.

Given Data / Assumptions:

• Weight of box completely full of sand = 8.5 kg.
• Weight of box half full of sand = 5.5 kg.
• Let B be the weight of the empty box in kilograms.
• Let S be the weight of the sand when the box is completely full.

Concept / Approach:
We express the two given situations as linear equations in terms of B and S. For the full box, the total weight is B + S. For the half full box, the total weight is B + S/2 because only half the sand is present. These two equations form a small system that can be solved easily by subtraction. Once S is found, B is obtained by substituting back into either equation. This approach illustrates a simple but powerful application of algebraic thinking.

Step-by-Step Solution:
Let B be the weight of the empty box and S be the weight of the sand when the box is full.From the full box condition: B + S = 8.5.From the half full box condition: B + S/2 = 5.5.Subtract the second equation from the first: (B + S) − (B + S/2) = 8.5 − 5.5.This simplifies to S/2 = 3, so S = 6 kg.Substitute S = 6 into B + S = 8.5 to find B: B + 6 = 8.5, so B = 8.5 − 6 = 2.5 kg.

Verification / Alternative check:
Check both conditions using B = 2.5 kg and S = 6 kg. For the full box, total weight = 2.5 + 6 = 8.5 kg, which matches the given value. For the half full box, sand weight is S/2 = 3 kg, so total = 2.5 + 3 = 5.5 kg, which also matches the given information. Since both conditions are satisfied, B = 2.5 kg is correct.

Why Other Options Are Wrong:

• If the empty box weighed 5 kg or 6 kg, then the sand would need to have inconsistent weights in the two scenarios, contradicting the problem statement.
• 4.5 kg or 3 kg for the box leaves too little or too much sand weight when compared with the given total weights.
• Only 2.5 kg fits both the full and half full conditions simultaneously.

Common Pitfalls:

• Treating the difference in total weights (3 kg) as the weight of the empty box instead of half the sand.
• Forgetting that half filled means half the sand weight, not half the total weight.
• Setting up equations incorrectly, such as writing S/2 + S instead of B + S/2.

Final Answer:
2.5 kg