Mixture Concentration: Weighted Average Problem
How much of each—20% lidocaine and 30% lidocaine—are needed to make 500 mL of a 28% concentration solution?
Already Got the Answer? Check It Here
- Setup: let
x= mL of 20% solution, so500 − x= mL of 30% solution. - Balance the pure lidocaine:
0.20x + 0.30(500 − x) = 0.28(500) - Result:
x = 100mL of 20% and400mL of 30%. - One-line check:
20 + 120 = 140mL of pure lidocaine, and140 ÷ 500 = 28%✓
If your answer matches, the sections on alligation, pitfalls, and the general formula below will help you tackle harder variants. If it doesn't, start with Method 1.
What You Will Learn
- The "amount of pure stuff" idea. Percentages can't be added directly, but the pure ingredient they describe can. Volume × concentration gives the quantity that actually adds up.
- How to use one variable for two unknowns. Because the two volumes must total 500 mL, naming one
xforces the other to be500 − x. - Why a mixture is a weighted average. The result lands between the two ingredient concentrations, closer to whichever you use more of.
- The alligation shortcut. A pencil-and-paper method that gives the ratio of the two ingredients without writing an equation. It works because of the weighted-average structure.
- How to estimate before calculating. A 28% target is much nearer 30% than 20%, so you can predict that the 30% solution dominates before doing any algebra.
Solution: Method 1 — The Pure-Lidocaine Balance Equation
The key move in every mixture problem is to stop thinking about the solutions and start thinking about the active ingredient. A 20% solution contains 0.20 mL of pure lidocaine in every mL of liquid. Pure lidocaine is conserved when you pour solutions together, so its total before mixing must equal its total after mixing.
Step 1 — Define the variable
We have two unknown volumes, but they are linked: together they must make 500 mL. So one variable is enough.
Step 2 — Find how much pure lidocaine the final mixture must contain
The target is 500 mL at 28% concentration, so the final mixture needs:
Step 3 — Write the balance equation
Pure lidocaine from the 20% solution plus pure lidocaine from the 30% solution must equal 140 mL:
Read it aloud: "20% of the first volume, plus 30% of the second volume, equals the 140 mL we need."
Step 4 — Solve
Distribute the 0.30, combine the x terms, and isolate x:
Step 5 — Find the second volume
The 30% solution is the remainder: 500 − 100 = 400 mL.
Solution: Method 2 — The Alligation Shortcut
Alligation skips the equation and works from distances. Each ingredient's concentration sits some distance from the target. The key idea is that the amount of each ingredient is proportional to the distance of the other ingredient from the target. Think of a seesaw: the heavier side sits closer to the pivot.
| Ingredient | Concentration | Distance from the target (28%) | Parts used |
|---|---|---|---|
| Weaker solution | 20% | 28 − 20 = 8 | 2 parts (the other distance) |
| Stronger solution | 30% | 30 − 28 = 2 | 8 parts (the other distance) |
The distances 8 and 2 are "crossed": the 20% solution gets the 2 (the stronger solution's distance), and the 30% solution gets the 8 (the weaker solution's distance). The ratio of 20% to 30% solution is therefore 2 : 8 = 1 : 4.
That is 5 parts in total, so each part is 500 ÷ 5 = 100 mL.
Why the crossing works: the pure lidocaine contributed by the 20% solution falls short of the target by 8 percentage points per mL, and the 30% solution overshoots by 2 percentage points per mL. For the shortfalls and overshoots to cancel, 8 × (volume of 20%) = 2 × (volume of 30%). That gives a ratio of volume of 20% : volume of 30% = 2 : 8.
Solution: Method 3 — Weighted Average with Fractions of the Batch
Instead of volumes, let p be the fraction of the final mixture that is the 20% solution. The 30% solution is then the fraction 1 − p. The final concentration is a weighted average:
So 20% of the batch is the weaker solution: 0.2 × 500 = 100 mL, and the other 0.8 × 500 = 400 mL is the stronger solution. The advantage here is that the batch size never enters the equation, so you can solve once and scale to any batch size.
The Answer
Mix 100 mL of the 20% lidocaine solution with 400 mL of the 30% lidocaine solution to make 500 mL of a 28% solution.
Verification
Check the volumes:100 + 400 = 500 mL ✓
Check the pure lidocaine:
From the 20% solution: 0.20 × 100 = 20 mL
From the 30% solution: 0.30 × 400 = 120 mL
Total: 20 + 120 = 140 mL
Check the concentration:140 ÷ 500 = 0.28 = 28% ✓
Independent check against the original equation:0.20(100) + 0.30(500 − 100) = 20 + 120 = 140 = 0.28(500) ✓
Sanity Check: Where Should the Answer Land?
You can often predict the shape of the answer before computing. The final concentration must fall between 20% and 30%, and 28% is 80% of the way from 20% to 30%. So about 80% of the mixture should be the stronger solution, which is exactly what 400 out of 500 mL is.
| Mix (20% / 30%) | Pure lidocaine | Concentration |
|---|---|---|
| 500 mL / 0 mL | 100 mL | 20% |
| 250 mL / 250 mL (equal parts) | 125 mL | 25% |
| 100 mL / 400 mL (our answer) | 140 mL | 28% |
| 0 mL / 500 mL | 150 mL | 30% |
The table also shows why any target outside 20%–30% would be impossible: no mixture of two solutions can be stronger than the strongest or weaker than the weakest.
What Trips People Up
Averaging the two concentrations assumes equal volumes, and the target of 28% is not 25%. The final concentration is a weighted average, and the weights are the volumes. A quick check shows the problem: 250 mL of each gives 25%, not 28%.
The right-hand side of the equation must be the amount of pure lidocaine in the final mixture (0.28 × 500 = 140), not the total volume. The left side is already measured in pure lidocaine, so the right side needs the same units. Also, don't mix forms, such as using 20x on one side and 0.28 on the other. Pick percentages or decimals and stay consistent.
The most common alligation error is to put each distance on the same row as the ingredient it came from. The distances must be crossed. Always apply the sense test: 28% is close to 30%, so the 30% solution must be the larger share. Your answer should have 400 mL of 30%, not 100 mL.
Using x for both volumes means you've told the equation they are equal, which would give 25%. The second volume must be written as 500 − x (or as a second variable y with x + y = 500).
The Pattern Behind Every Two-Solution Mixture
Whenever you blend two solutions with concentrations c₁ < c₂ into a total volume V at target concentration t (where c₁ ≤ t ≤ c₂), the same algebra gives a closed formula:
For our problem: 500 · (30 − 28)/(30 − 20) = 500 · 0.2 = 100 mL, and 500 · (28 − 20)/(30 − 20) = 500 · 0.8 = 400 mL. This is the alligation method written as a formula. The denominator c₂ − c₁ is the total width of the "seesaw," and the numerators are the two distances from the target.
Limits of the formula: it only applies to two ingredients and only when the target lies between the two concentrations. With three or more ingredients, the problem has infinitely many solutions unless extra conditions are added (see What-If #3). The structure also carries over to anything that blends by weighted average: alloys, test-score weighting, even the average price of a fuel blend.
Beyond the Textbook
- Pharmacy compounding. Pharmacists adjust strengths of creams, solutions, and IV preparations with exactly this calculation, though real-world work follows institutional protocols and validated sterile technique. This page is a math exercise, not a clinical procedure.
- Chemistry labs. Preparing a working solution from two stock solutions of different molarity is the same balance equation with different units.
- Metallurgy and fuel blending. A jeweler combining 14-karat and 18-karat gold, or a refinery blending two octane grades, is solving a weighted average problem.
A Brief History
Alligation is one of the oldest calculation techniques taught in commercial arithmetic. Medieval and Renaissance merchant textbooks used it to blend metals, wines, and spices to a target price or purity, and apothecaries used it to dilute drugs. It persisted in pharmacy training for centuries because it gives the correct ratio with only subtraction and no equation to solve. Today's algebra shows why it works, which is the real payoff of learning both.
Push Further
Try each problem on your own first, then press the button to compare. They get progressively harder.
Let x = mL of 20% solution. Then 500 − x = mL of 30% solution.
0.26 × 500 = 130 mL.
0.20x + 0.30(500 − x) = 130
0.20x + 150 − 0.30x = 130, so 150 − 0.10x = 130 and x = 200. The 30% solution is 500 − 200 = 300 mL. Answer: 200 mL of 20% and 300 mL of 30%.
0.20(200) + 0.30(300) = 40 + 90 = 130, and 130 ÷ 500 = 26% ✓. The target moved closer to 20%, so more of the 20% solution is used than before (200 mL instead of 100 mL), as expected.
150 + 350 = 500 mL.
From the 20% solution: 0.20 × 150 = 30 mL. From the 30% solution: 0.30 × 350 = 105 mL.
30 + 105 = 135 mL.
135 ÷ 500 = 0.27. Answer: the mixture is 27% lidocaine.
27% lies between 20% and 30%, and since 350 mL of the 500 mL is the stronger solution (70%), we expect 20 + 0.70 × 10 = 27 ✓.
Let x = mL of the 20% solution = mL of the 30% solution. Then the 40% solution is 600 − 2x mL.
0.325 × 600 = 195 mL.
0.20x + 0.30x + 0.40(600 − 2x) = 195
0.50x + 240 − 0.80x = 195, so 240 − 0.30x = 195, giving x = 150. The 40% solution is 600 − 300 = 300 mL. Answer: 150 mL of 20%, 150 mL of 30%, 300 mL of 40%.
Volumes: 150 + 150 + 300 = 600 ✓. Pure lidocaine: 30 + 45 + 120 = 195, and 195 ÷ 600 = 32.5% ✓. Without the "equal volumes" condition there would be infinitely many solutions, so that extra condition is what makes the answer unique.
Let y = mL of 30% solution added. The final volume is 60 + y mL. This time the total is not given, so it must be written in terms of y.
Pure lidocaine in must equal pure lidocaine out: 0.20(60) + 0.30y = 0.28(60 + y)
12 + 0.30y = 16.8 + 0.28y
Collect the y terms on the left and the constants on the right: 0.02y = 4.8, so y = 240. The final volume is 60 + 240 = 300 mL. Answer: add 240 mL of 30% solution, for a total of 300 mL.
Pure lidocaine: 0.20(60) + 0.30(240) = 12 + 72 = 84 mL, and 84 ÷ 300 = 28% ✓. Alligation agrees: the 20% : 30% ratio must be 2 : 8 = 1 : 4, and 60 : 240 = 1 : 4 ✓. The ratio from the original problem is a fixed property of 20%, 30%, and 28%, so the batch size can change without altering it.
Frequently Asked Questions
x be the volume of one solution, so the other is (total − x). Then write a balance equation for the pure ingredient: (concentration 1 × x) + (concentration 2 × (total − x)) = (target concentration × total). In this example, mixing 20% and 30% solutions to get 500 mL of 28% gives 0.20x + 0.30(500 − x) = 0.28(500), which simplifies to 150 − 0.10x = 140, so x = 100 mL of the 20% solution and 400 mL of the 30% solution.2026-10-29