Solution Y is 30% liquid X and 70% water. If 2 kilograms of water evaporate from 8 kilograms of solutions...

1. Translate the problem requirements: Solution Y has \(30\%\) liquid X and \(70\%\) water. We start with \(8\) kg of solution Y, then \(2\) kg of water evaporates (leaving \(6\) kg), then \(2\) kg of fresh solution Y is added (making \(8\) kg total). We need to find the percentage of liquid X in this final \(8\) kg mixture.
2. Track the liquid X content throughout the process: Calculate how much liquid X is present initially, after evaporation, and after adding more solution Y, since liquid X doesn't evaporate.
3. Track the total mixture weight at each stage: Follow the weight changes as water evaporates and new solution is added to determine the final total weight.
4. Calculate the final percentage: Use the final amount of liquid X divided by the final total weight to get the percentage.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English first. We start with a solution that's like a mixture of juice and water - \(30\%\) is the "good stuff" (liquid X) and \(70\%\) is just water. We have \(8\) kilograms of this mixture initially.

Then two things happen: some water evaporates (disappears), and we add more of the original mixture. We need to figure out what percentage of our final mixture is the "good stuff" (liquid X).

More specifically:

• Solution Y contains \(30\%\) liquid X and \(70\%\) water
• We start with \(8\) kg of solution Y
• \(2\) kg of water evaporates, leaving us with \(6\) kg
• We then add \(2\) kg of fresh solution Y, giving us \(8\) kg total again
• We need to find what percent of this final \(8\) kg is liquid X

Process Skill: TRANSLATE - Converting the problem description into clear, trackable quantities

2. Track the liquid X content throughout the process

The key insight is that liquid X doesn't evaporate - only water does. So we need to track how much liquid X we have at each stage.

Initially (\(8\) kg of solution Y):
Liquid X = \(30\%\) of \(8\) kg = \(0.30 \times 8 = 2.4\) kg
Water = \(70\%\) of \(8\) kg = \(0.70 \times 8 = 5.6\) kg

After \(2\) kg of water evaporates:
Liquid X = \(2.4\) kg (unchanged - liquid X doesn't evaporate)
Water = \(5.6 - 2 = 3.6\) kg
Total = \(2.4 + 3.6 = 6\) kg ✓

After adding \(2\) kg of fresh solution Y:
The \(2\) kg of fresh solution Y contains:

• Liquid X: \(30\%\) of \(2\) kg = \(0.6\) kg
• Water: \(70\%\) of \(2\) kg = \(1.4\) kg

So our final mixture has:
Liquid X = \(2.4 + 0.6 = 3.0\) kg
Water = \(3.6 + 1.4 = 5.0\) kg

3. Track the total mixture weight at each stage

Let's verify our weight tracking:

• Initially: \(8\) kg
• After evaporation: \(8 - 2 = 6\) kg
• After adding fresh solution: \(6 + 2 = 8\) kg

Our final mixture weighs \(8\) kg total, with \(3.0\) kg of liquid X and \(5.0\) kg of water.
Check: \(3.0 + 5.0 = 8.0\) kg ✓

4. Calculate the final percentage

Now we can find what percentage of our final mixture is liquid X:

Percentage of liquid X = \(\frac{\text{Amount of liquid X}}{\text{Total weight}} \times 100\%\)
Percentage of liquid X = \(\frac{3.0 \text{ kg}}{8.0 \text{ kg}} \times 100\%\)
Percentage of liquid X = \(\frac{3}{8} \times 100\%\)
Percentage of liquid X = \(0.375 \times 100\% = 37.5\%\)

Since 37.5% = 37½%, our answer is 37½%.

Final Answer

The final liquid solution contains 37½% liquid X.

This matches answer choice C: 37½%.

We can verify this makes sense: we started with 30% liquid X, then removed only water (concentrating the liquid X), then added back some original solution. The final percentage should be higher than 30% but not too much higher, and 37½% fits this expectation perfectly.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding what evaporates
Students often assume that when "\(2\) kg of water evaporate," it means \(2\) kg of the total solution evaporates, maintaining the same 30%-70% ratio. They fail to recognize that ONLY pure water evaporates, not the liquid X component. This leads them to incorrectly calculate that \(0.6\) kg of liquid X and \(1.4\) kg of water evaporate, rather than understanding that all \(2\) kg that evaporates is pure water.

Faltering Point 2: Confusion about the sequence of operations
Students may get confused about the order of events and try to work backwards or mix up the steps. The problem involves: (1) starting with \(8\) kg, (2) losing \(2\) kg through evaporation to get \(6\) kg, then (3) adding \(2\) kg of fresh solution to get back to \(8\) kg. Some students might try to solve this by thinking about the net effect rather than tracking each step systematically.

Faltering Point 3: Misinterpreting "\(2\) kg of solution Y are added"
Students might think that \(2\) kg of pure liquid X is added rather than \(2\) kg of the original solution Y (which is 30% liquid X and 70% water). This misinterpretation would lead them to add \(2\) kg of liquid X directly instead of adding \(0.6\) kg liquid X and \(1.4\) kg water.

Errors while executing the approach

Faltering Point 1: Arithmetic errors in percentage calculations
Even with the correct approach, students frequently make calculation errors when computing 30% of various quantities. For example, miscalculating 30% of \(8\) kg as \(2.8\) kg instead of \(2.4\) kg, or 30% of \(2\) kg as \(0.8\) kg instead of \(0.6\) kg. These small arithmetic errors compound and lead to incorrect final answers.

Faltering Point 2: Losing track of quantities after evaporation
Students correctly identify that \(2.4\) kg of liquid X exists initially, but then incorrectly assume some of it evaporates or gets lost in the process. They might recalculate the liquid X content after evaporation instead of recognizing that the \(2.4\) kg of liquid X remains constant throughout the evaporation step.

Faltering Point 3: Incorrect addition when combining solutions
When adding the \(2\) kg of fresh solution Y to the remaining \(6\) kg, students may forget to break down the fresh solution into its components (\(0.6\) kg liquid X + \(1.4\) kg water) and instead just add \(2\) kg to their previous totals without proper accounting for each component.

Faltering Point 4: Converting between decimal and fractional forms
Students correctly calculate 37.5% but fail to recognize this equals 37½%. They might select 37% (if available) or get confused between the decimal representation (37.5%) and the mixed number representation (37½%) shown in the answer choices.