Decision and conditions
Separate total volume, dissolved substance and transferred amounts. Mixtures are uniform before removal; evaporation removes water only when explicitly stated.
Worked example
A tank contains 12 liters of pure solution A. A quarter of the liquid is removed and replaced with water, mixed uniformly, then the same removal and replacement is repeated. What fraction of the final liquid is A? After the first replacement the A fraction is 3/4. The next replacement retains 3/4 of that remaining A. Final fraction is (3/4)² = 9/16. Method choice: Each replacement retains three quarters of the A currently present.
Contrast: change the reasoning
15 liters of a 20% salt solution loses only water through evaporation until its concentration is 25%. How many liters evaporated? The salt amount is 3 liters-equivalent. Final volume is 3/0.25=12. Water lost is 15−12=3. Method choice: Salt amount remains fixed while total volume decreases.
Transfer with solution
Container A holds 15 liters and B holds 6 liters. How much must be transferred from A to B so that A holds twice as much as B? 15−t=2(6+t). Collecting gives 3t=3. Transfer 1 liters. Method choice: The transfer decreases one container and increases the other by the same amount.
Boundary check
12 liters of a 25% solution is mixed with 3 liters of a 50% solution. How many liters of pure water must be added to obtain a 20% solution? Total solute is 3+3/2=9/2. At 20%, total volume must be 45/2. Subtract current volume 15; add 15/2 liters. Method choice: Find total solute from both sources before changing total volume.
Misconception to reject
Removing a fraction of a mixture removes that same fraction of its solute. Subtracts two quarters of the original A, ignoring dilution before the second removal.
A tank contains 12 liters of pure solution A. A quarter of the liquid is removed and replaced with water, mixed uniformly, then the same removal and replacement is repeated. What fraction of the final liquid is A?
- After the first replacement the A fraction is 3/4.
- The next replacement retains 3/4 of that remaining A.
- Final fraction is (3/4)² = 9/16.
Compare the other answer choices
Subtracts two quarters of the original A, ignoring dilution before the second removal.
Performs only one replacement.
Multiplies the removed fractions rather than the retained fractions.
Reports the water fraction instead of A.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
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