When a liquid freezes into its solid state, its volume increases by 4%. By what percentage does the volume decrease when this solid melts back into the liquid state?

Introduction / Context:
This question checks the understanding of reverse percentage change. An increase in one direction does not mean the same percentage decrease will bring you back to the original value. Here, the physical context is a liquid expanding slightly when it freezes and then contracting when it melts. The mathematics is the same as the standard successive percentage problems in aptitude tests.

Given Data / Assumptions:
- When the liquid freezes, its volume increases by 4 percent.
- When the solid melts back to liquid, the volume decreases from the larger value back to the original value.
- We are asked for the percentage decrease based on the larger (solid) volume.

Concept / Approach:
If a quantity increases by a certain percentage, say a%, the new value is original * (1 + a / 100). To reverse the change, we need to find a percentage decrease b% such that new value * (1 - b / 100) returns to the original. In general, these percentages a and b are not equal because the bases are different. The same idea appears in price rise and reduction problems.

Step-by-Step Solution:
Let the original volume of the liquid be V units. After freezing, there is a 4% increase, so solid volume = V * (1 + 4 / 100) = 1.04V. Now, when the solid melts, its volume goes from 1.04V back to V. Let the required percentage decrease be d%. Then V = 1.04V * (1 - d / 100). Divide both sides by V: 1 = 1.04 * (1 - d / 100). So, 1 / 1.04 = 1 - d / 100. Compute 1 / 1.04 = 100 / 104 = 25 / 26. Thus, 25 / 26 = 1 - d / 100. So, d / 100 = 1 - 25 / 26 = 1 / 26. Therefore, d = 100 / 26 = 50 / 13 = 3 11/13 percent (approximately 3.846 percent).

Verification / Alternative check:
Assume an original volume of 100 units. After a 4% increase, the volume becomes 104 units. To find the decrease needed to return to 100, compute the difference: 104 - 100 = 4 units. The percentage decrease relative to 104 is (4 / 104) * 100 = 400 / 104 = 100 / 26 = 3 11/13 percent. This numerical check confirms the algebraic result.

Why Other Options Are Wrong:
3 3/13% and 4 1/13%: These values are close but do not match the exact calculation of 100 / 26 percent.
4%: This assumes symmetry between increase and decrease, which is not correct when the bases differ.
5%: This is too large and would bring the volume below the original value.

Common Pitfalls:
Many learners think that a 4% increase followed by a 4% decrease restores the original quantity, but this is incorrect because the second percentage is taken on a different base (the increased value). Others may try to compute the decrease as 4 / 1.04 incorrectly or approximate too early. Always set up the equation with the new and original values and solve systematically to find the exact percentage change.

Final Answer:
When the solid melts back into liquid, the volume decreases by 3 11/13 percent of the solid volume.