gamesblog

Mixtures

Dilute an acid, blend two prices, draw off milk and refill with water — find what is conserved, and the mixture works itself out.

  • Quantitative reasoning
  • Untimed
  • Endless levels

How it plays

Each question mixes things: water into a solution, two strengths or two prices together, salt water boiled down, a cask drawn off and topped up several times. Find the litres, the strength or the ratio.

New kinds unlock as you climb — alligation, repeated replacement, three batches at once, pure stuff added — and the quantities grow without end. Answers are built to come out whole or to one decimal place.

The wrong options are the real slips: the plain average where a weighted one is needed, the alligation ratio upside down, the milk taken off as if the same amount left every round. The working names what stays fixed.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This makes mixture and concentration problems of these kinds quicker and surer. It claims nothing wider.

Learn

How to get better at mixtures: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Ask what does not change. Adding water leaves the acid alone; boiling leaves the salt alone; adding pure acid leaves the water alone. Find that amount first, in litres.
  2. Work out how much of it there is now: the percentage times the volume. 20% of 10 L is 2 L.
  3. Then ask what volume that same amount makes at the new strength, and compare with what you have. The difference is the water to add, the water to boil off, or the stuff to add.
  4. For two prices or two strengths mixed together, skip the algebra and use alligation, below. Check any answer by putting it back: the new mixture must have the strength the question asked for.

Techniques and tricks

  • Hold the unchanged part fixed

    Whatever is not added or removed is the same before and after. Write it as litres, then divide by the new percentage to get the new total volume.

    e.g. 10 L at 20% acid, diluted to 10%: 2 L of acid stays; 2 L is 10% of 20 L; so add 10 L of water

  • Boiling is dilution backwards

    Boiling off water keeps the salt. Find the salt, find the volume it makes at the higher strength, and the rest boiled away.

    e.g. 20 L of 5% salt water to 20%: 1 L of salt; 1 L is 20% of 5 L; so 15 L boils off

  • Adding the pure stuff: hold the water

    When pure acid is added, the water is what stays fixed. Work with the water’s percentage: 100 minus the strength.

    e.g. 30 L at 20% acid to 50%: 24 L of water; 24 L is 50% of 48 L; so add 18 L of acid

  • Alligation: cross the distances

    To mix a cheap and a dear thing to a mean, the ratio cheap : dear is (dear − mean) : (mean − cheap). Each one’s share is the other one’s distance from the mean.

    e.g. ₹40 and ₹70 rice to ₹50: (70 − 50) : (50 − 40) = 20 : 10 = 2 : 1

  • Weighted mean: weight by the litres

    To mix two or three strengths, multiply each strength by its litres, add, and divide by the total litres. The result lies nearer the strength there is more of.

    e.g. 2 L at 10% with 3 L at 35%: (20 + 105) ÷ 5 = 25%

  • Draw off and refill: multiply by the fraction kept

    Each round keeps the same fraction of whatever milk is in the cask, because later rounds draw off some water too. After n rounds the milk is the start times (kept fraction)ⁿ.

    e.g. 25 L of milk, 5 L drawn off and refilled twice: 25 × (4/5)² = 16 L

As the levels climb

  • The kinds unlock in this order: adding water and finding the new strength; then mixing two prices (alligation) and two strengths; then drawing off and refilling and boiling some off; then three strengths at once and adding the pure stuff. Four in ten questions come from the newest kind.
  • The quantities grow as you climb — more litres, more acid, finer ratios — but every answer is built to come out whole or to one decimal place. If yours has a long tail of decimals, you have slipped somewhere.
  • Draw off and refill goes from two rounds to three and then four as the size grows. Do not subtract each round’s draw; raise the kept fraction to the number of rounds.
  • Each step of size adds 8% to the time, up to twelve steps. Typing the answer gives a quarter more again.

Traps to avoid

  • The plain average where a weighted one is needed: 2 L at 10% with 3 L at 35% is 25%, not 22.5%. Unequal amounts pull the mixture towards the bigger one.
  • The alligation ratio upside down: giving 1 : 2 where 2 : 1 is right. The cheap one’s share is the dear one’s distance from the mean — cross over.
  • The same amount of milk taken off every round: 25 − 5 − 5 = 15 L. After the first refill the cask is part water, so the second draw takes less milk. It is 25 × (4/5)² = 16 L.
  • The final volume given as the water to add: 20 L is the new total, not the water. Subtract what you started with.
  • Holding the wrong part fixed when adding pure acid: the acid changes, so it cannot be the anchor. The water is the part that stays the same.

How to practise

  • Use “Kinds of question” to drill “Mix two prices” and “Two strengths together” as a pair. They are one rule — alligation — seen from both ends.
  • Switch “Answer by” to “Typing the answer”. Litres and percentages count within 0.05, and alligation ratios are typed in lowest terms, cheap : dear.
  • Raise “Options shown” towards 8 to put the plain average, the upside-down ratio and the subtract-each-round answer on screen together.
  • Read the first line of the working after a miss. It names what was conserved, which is the step most misses skip.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More quantitative reasoning

All games →