“How much of each?” problems scare students because the setup is invisible. Here it is not: understand → set up the two equations → solve → verify against both. Choose percent or price, enter your numbers, and watch the setup happen.
Mix two solutions of different strengths to hit an exact target strength.
In short
A mixture problem asks how much of two solutions or ingredients combine to hit an exact target percent or price. The setup is always two equations: total amounts, and total pure substance or cost. VerifySolve builds both in stages, solves the system, and verifies the blend actually produces the target amount and concentration you asked for.
Amounts add. Values add. That is genuinely all of it — every mixture problem ever set is these two statements with story clothes on.
Amount equation
Value equation (percent)
Value equation (price)
The mixture must land between
Notice the verification checks the amount total and the value total separately. Both must pass — that is what makes a mixture answer certain.
How much 10% and 50% solution make 30 units of 30% solution?
15 of the 10% solution and 15 of the 50% solution
Verified both ways: 15 + 15 = 30 ✓ and 1.5 + 7.5 = 9 units of pure substance ✓. Equal split — because 30% sits exactly halfway.
Blend nuts at $2/lb and $6/lb into 40 lb at $4.50/lb
15 lb of the $2 nuts and 25 lb of the $6 nuts
Verified: 15 + 25 = 40 lb ✓ and $30 + $150 = $180 ✓. More of the pricey kind — the target sits above the midpoint.
Make 20 litres of 40% acid from 20% and 70% stock
12 litres of the 20% stock and 8 litres of the 70% stock
Verified: 12 + 8 = 20 ✓ and 2.4 + 5.6 = 8 litres of pure acid ✓.
“10% and 50% average to 30%” only works when the amounts are equal. With unequal amounts the stronger ingredient pulls harder. The value equation is the weighted average — set it up and let it decide the split.
Writing 10x + 50y on one side and 0.30·30 on the other mixes percents-as-numbers with percents-as-decimals. Pick one form (the solver uses decimals) and use it in every term.
If a problem asks you to blend 10% and 50% into 60%, the correct response is “impossible” — not a negative amount of one ingredient. Negative answers in mixture problems are the algebra telling you the story cannot happen.
With two equations. The amount equation: the parts add to the total (x + y = T). The value equation: each part contributes its strength or price times its amount, and those contributions add to the target amount (p₁x + p₂y = pT·T). Solve the pair — that is the whole method, and it is exactly what the staged solver shows.
Mixing can only land between what you started with — you cannot blend 10% and 50% solutions into 60%. If the target sits outside the range, no mixture exists, and the solver says the story is impossible instead of returning negative amounts.
Both equations are re-checked with the answer: the amounts must sum to the total AND the weighted value must hit the target exactly. One equation passing is not enough — a wrong split can satisfy either one alone, never both.
Only the meaning of the numbers. Percent mixtures track pure substance (acid, alcohol, antifreeze); price mixtures track money per pound (nuts, coffee, candy). The equations are identical — which is why one solver handles both.
Not yet — this solver covers the classic two-ingredient case. Three-ingredient and evaporation/dilution variants (where one “ingredient” is pure water or pure substance — actually a special case of two) are on the roadmap; pure water is 0% and pure acid is 100% if you want to try those in the current solver.
Yes — free, no sign-up, no app, full staged steps and the two-equation verification included.
The other classic staged family: d = rt stories, verified against the story.
The staged method explained, plus every family we cover.
Mixtures are systems in disguise — practise the raw skill here.
Already have your equations? Solve them directly with plug-back proof.
A note on honest use: VerifySolve is built for learning. Use the steps to understand the method, then close the tab and try one on your own — that is where the grade actually comes from. If your teacher or school has a policy on calculators, follow it.