Use a before-and-after ledger. Keep the percentage basis, the initial solution quantity, and the final solution quantity visible, then calculate the component amount on both sides.
Start with a conserved quantity
OpenStax derives the dilution relation from unchanged solute amount and requires compatible units. Its molarity treatment uses C1 × V1 = C2 × V2; extending the calculation requires checking what each concentration measures. OpenStax dilution derivation.
For this original paper exercise, use a fictional nonreacting component S and a declared mass/volume basis. No actual mixing, measurement, or preparation procedure is supplied.
| Before-and-after field | Supplied or expected value |
|---|---|
| Initial concentration C1 | 8g S per 100mL solution, or 8% m/v |
| Initial solution volume V1 | 25mL |
| Final solution volume V2 | 200mL |
| Expected final concentration C2 | 1g S per 100mL solution, or 1% m/v |
The initial component mass is (8g / 100mL) × 25mL = 2g. Put that same 2g over the supplied final 200mL: (2g / 200mL) × 100mL = 1g per 100mL.
Check the result backward: (1g / 100mL) × 200mL = 2g. The before-and-after equality is the central result. The final-to-initial solution-volume ratio is 200 / 25 = 8, so the concentration numeral decreases by that factor within this same basis.
Final volume is not an amount to add
V2 labels the whole final solution. It does not label the solvent added. Using 200mL as an added quantity would change the meaning of the supplied record.
The difference between the two stated solution volumes is 175mL. This arithmetic exercise does not establish that adding 175mL of a solvent physically produces the final solution volume. Leave any preparation procedure outside the packet. The ledger checks a supplied final record; it does not prescribe how to make it.
Give mass percentages their own ledger
A separate fictional record uses 8g S per 100g solution, an initial solution mass of 25g, and a final solution mass of 200g. Its expected answer is 1% m/m.
The initial component mass is (8g S / 100g solution) × 25g solution = 2g S. The final concentration is 2g S in 200g solution, or 1g S per 100g solution. The component mass again remains 2g.
Mass percentage uses solution mass, while the earlier m/v record uses solution volume. OpenStax percentage concentration definitions.
Do not plug milliliters into the mass-percent ledger just because a familiar formula contains V. The two records are independent, and their shared numeral 1 does not identify one physical solution. No density is supplied to convert between them.
Identify impossible and unchanged targets
For the declared volume-mode exercise, require positive initial and final solution volumes, with final volume at least as large as initial volume. Initial concentration may be zero but must be finite and nonnegative; the basis and units must be known.
A target of 16% on the same basis exceeds the 8% starting concentration. Solvent-only dilution cannot reach that target under this unchanged-component contract. Hold it as outside the dilution task rather than returning an unlabeled smaller final volume.
A positive target equal to the positive starting concentration is an identity case: the required solution quantity is unchanged, so no dilution is described. It is not a calculation error. The zero-to-zero case needs the separate treatment below.
Zero requires separate treatment. A zero initial concentration gives zero final concentration for any valid supplied positive final quantity. In an inverse calculation, zero stock cannot produce a positive target. If both stock and target are zero, the equation does not determine a unique final quantity; do not manufacture one through division by zero.
Blank fields, zero denominators, mismatched bases, and nonfinite values should clear any old accepted answer and show the unresolved condition. These guardrails define this teaching exercise. Solute loss, reactions, or combining stocks fall outside its model.
Design a helper around the conservation check
The local HTML example shows the starting record, final record, component amount before and after, and basis together. Keep a native worksheet and checked expected-results matrix. Leave actual-result cells blank until the helper is implemented and tested.
Editing inputs or changing the basis must invalidate the old answer. Reset should restore the supplied example and clear active calculation feedback. Controls should do those specific jobs rather than suggest an actual solution has been prepared.
NOS can save and render a prepared HTML version of this conservation view. That can make the two component-amount calculations easier to compare; the native worksheet remains the source, and runtime behavior still needs testing. Explore NOS.
Copyable conservation-packet prompt
Prepare an English fictional percent-dilution worksheet, checked expected-results matrix, and specification for an original helper. Do not implement, execute, mix materials, or provide a preparation recipe.
Use nonreacting component S with unchanged component amount. First record: 8g S per 100mL solution, initial solution volume 25mL, supplied final solution volume 200mL. Expected initial and final component mass is 2g, final concentration is 1g per 100mL or 1% m/v, and final/initial solution-volume ratio is 8. Show unit cancellation and check the final result backward.
V2 means final solution volume, not solvent added. Do not turn the numerical 175mL volume difference into an instruction to add solvent or assume volumes add.
Second independent record: 8g S per 100g solution, initial solution mass 25g, final solution mass 200g. Component mass remains 2g; final answer is 1% m/m. Use solution mass, not volume, for this ledger. Do not infer density or treat the two 1% answers as one physical solution.
Volume mode requires finite positive initial/final solution volumes, final at least initial, finite nonnegative initial concentration and explicit matching basis/units. Hold a 16% target from 8% stock as unreachable by solvent-only dilution. Equal positive stock and target concentrations give an identity/no-dilution case. Zero stock gives zero final concentration for a supplied positive final quantity; inverse positive-target solving from zero stock is impossible, while zero stock and zero target leave the final quantity underdetermined.
Hold blanks, mismatched bases, zero denominators and nonfinite values; remove old accepted answers. Input/basis edits invalidate prior results. Reset restores the fixture and clears active feedback. Provide expected and blank actual-result columns, keyboard/focus behavior, and pending runtime/narrow-screen checks. Native worksheet/key remain primary. Any NOS connection is conditional saving/rendering of prepared HTML only. Cover no solute loss, reaction, stock mixing, density defaults, dosage or reagent instructions.
Accept the dilution result only when the labeled before-and-after records account for the same amount of S.