Mixing Two Concentrations to Make a Target Solution

A concentration-mixing problem asks for volumes that produce a target strength. Keep units consistent and use a weighted average; simply averaging the two percentages works only when the volumes are equal.

Set up the amount-of-drug equation

Let x be the milliliters of the 70% solution. The remaining 100 − x mL comes from the 10% solution. The amount of active ingredient is 0.70x + 0.10(100 − x), and it must equal 0.30 × 100. Solving gives 0.70x + 10 − 0.10x = 30, so 0.60x = 20 and x = 33.3 mL.

Find the second volume

The total volume is 100 mL, so use 100 − 33.3 = 66.7 mL of the 10% solution. The calculation assumes volumes are additive and the ingredients are compatible; actual compounding must follow the prescription, formulation and pharmacy procedures.

Use alligation as a shortcut

Place the higher and lower strengths around the target. The differences are 70 − 30 = 40 parts of the lower solution and 30 − 10 = 20 parts of the higher solution. The ratio is 20 parts of 70% to 40 parts of 10%, or 1:2. For 100 mL, that is 33.3 mL and 66.7 mL.

Check the weighted concentration

Calculate the total active amount: 0.70 × 33.3 + 0.10 × 66.7 ≈ 23.31 + 6.67 = 29.98 percentage-milliliters, which is approximately 30% of 100 mL after rounding. In actual compounding, use required measurement precision and do not round until the final step.

Exam takeaway

Use a weighted equation or alligation to determine the ratio, keep volumes and concentration units consistent, and verify by calculating the final active amount.

Set up the mass-balance equation

Alligation is a shortcut for blending two concentrations to reach a target when ingredients are compatible and the final quantity is known. The underlying equation is a weighted average: strong concentration × strong volume plus weak concentration × weak volume equals target concentration × final volume. The shortcut only solves that equation; it cannot establish that the ingredients may safely be mixed.

The target must lie between the starting concentrations. If the requested target is outside that range, a positive-volume blend of only those two solutions cannot make it. This quick feasibility check catches sign errors before arithmetic begins.

Worked example with verification

Make 100 mL of a 30% solution from 70% and 10% solutions. Let x be the volume of 70% solution; remaining volume is 100 − x mL. Equation: 0.70x + 0.10(100 − x) = 0.30(100). Solving gives 0.60x = 20, so x = 33.3 mL; the weaker solution is 66.7 mL.

Verify: 0.70 × 33.3 = 23.31 units of solute and 0.10 × 66.7 = 6.67, totaling 29.98 units in 100 mL, or about 30%. The slight difference is rounding. If exact final volume matters, use the authorized procedure; do not assume component volumes are always additive.

Cross-check with alligation

The diagonal setup uses parts rather than solving for x. Subtract target from strong concentration to get parts of weak: 70 − 30 = 40. Subtract weak from target to get parts of strong: 30 − 10 = 20. The ratio is 20 parts of the 70% solution to 40 parts of the 10% solution, or 1:2. In 100 mL, one-third is strong and two-thirds weak, matching the equation.

The diagonal placement is easy to reverse. Differences are opposite the ingredient they represent: high minus target gives low-strength parts; target minus low gives high-strength parts. Check that the answer is closer to the concentration used in the larger amount.

Units, compatibility, and practice

Do not mix percent w/v, percent v/v, ratio strengths, mg/mL, and mass fractions as though they were automatically the same units. Convert to a common basis only when enough information is supplied; density can matter when converting mass to volume. Compatibility, stability, sterility, labeling, beyond-use dating, and compounding authorization are separate questions arithmetic cannot answer.

For PTCE problems, show units and round at the end unless told otherwise. In practice, use the pharmacist-approved formula and compounding record, verify ingredients and concentrations, and complete required independent checks. Stop if units differ, target is out of range, final-volume assumptions are unsupported, or ingredients are unclear.

Applying the concept in practice

Keep concentration units consistent throughout. For example, a 1:1000 ratio may need conversion to mass per volume before combining it with a percentage. If the conversion is not given and cannot be derived from the supplied information, stop rather than treating the numbers as comparable. Similar caution applies to percent w/w versus percent w/v: density may be needed to convert mass to volume.

A final verification checks three things separately: component volumes add to the requested quantity under the problem's assumptions; the weighted concentration equals the target; and the ingredients may be combined. In practice, the pharmacist-approved worksheet, measuring method, labeling, storage, and beyond-use date control. Alligation is a calculation aid, not authorization to compound.

Applying the concept in practice

Keep concentration units consistent throughout. For example, a 1:1000 ratio may need conversion to mass per volume before combining it with a percentage. If the conversion is not given and cannot be derived from the supplied information, stop rather than treating the numbers as comparable. Similar caution applies to percent w/w versus percent w/v: density may be needed to convert mass to volume.

A final verification checks three things separately: component volumes add to the requested quantity under the problem's assumptions; the weighted concentration equals the target; and the ingredients may be combined. In practice, the pharmacist-approved worksheet, measuring method, labeling, storage, and beyond-use date control. Alligation is a calculation aid, not authorization to compound.

A final practical check

Alligation can be checked with common sense before finalizing. A target closer to the strong stock should require more of the strong stock; a target closer to the weak stock should require more weak stock. If the calculated ratio runs the opposite way, revisit the diagonal subtraction or algebra. Keep the total amount distinct from the ratio: a 1:2 ratio means three total parts, not two. Convert parts to the requested quantity only after the ratio is correct, and preserve enough significant figures for the required measuring device.