Mixture & Alligation

1Rule of alligation

A mixture problem has two ingredients at different values — say tea at ₹60/kg and tea at ₹90/kg — and a target mean for the blend (₹70/kg). You need the quantity ratio that hits that mean. The mean is a weighted average of the two values; alligation is the bookkeeping device that turns the three numbers into that ratio without rearranging the weighted-mean equation.

Name the three values in one unit: cheaper c (the lower value), dearer d (the higher), and mean m (the blend you want). Then \dfrac{Q_{\text{cheap}}}{Q_{\text{dear}}} = \dfrac{d-m}{m-c}. Each ingredient's share equals how far the other value sits from the mean — so d-m goes with the cheaper quantity and m-c with the dearer. Never take d-c in one step, and never swap the labels.

Figure. The mean sits on the segment joining the two prices. The longer arm (20) belongs to the cheaper tea and the shorter arm (10) to the dearer, so the mix is 2 : 1 — each quantity matches the distance on the opposite side of the mean.

How it works

1. Picture the mixtureTwo ingredient values and one target mean — find the quantity ratio that produces that blend. That is the problem class alligation solves.
2. Name cheaper, dearer and meanWrite c (lower value), d (higher) and m (target blend) in the same unit — price per kg, percent purity, or whatever the question uses.
3. Take the two distancesd-m goes with the cheaper quantity; m-c goes with the dearer. That opposite-distance swap is the whole rule.
4. Reduce the ratioCancel a common factor and state cheaper : dearer. Check by recomputing the weighted mean.

2Mean lies between the components

The mixture mean is a weighted average of the two component values, so m lies strictly between c and d. A claimed mean outside that open interval cannot be produced by any positive mix of those two ingredients — the arithmetic has already failed before any ratio is written.

Equal quantities put m at the midpoint; more of the cheaper pulls m toward c; more of the dearer pulls it toward d. Hitting c or d would require using only one ingredient, which is not a mixture.

Figure. The shaded stretch between ₹60 and ₹90 is every mean a positive mix of the two teas can produce — ₹70 and ₹75 both sit inside it. The claimed ₹100 sits outside the interval, so no ratio of the two ingredients reaches it; the check fails before any alligation cross is drawn. The endpoints themselves are excluded too: hitting 60 or 90 exactly means using only one tea, which is not a mixture.

How it works

1. Order the three numbersConfirm c < m < d (or d < m < c if the labels were swapped). If not, stop.
2. Only then apply alligationDistances d-m and m-c are both positive precisely when the mean sits between the components.

3Same unit of value

Alligation compares values of the same kind per the same unit of quantity. Price per kg with price per kg is fine; percent purity with percent purity is fine; mixing ₹/kg with a percentage without converting is not a mean at all.

Convert every ingredient to one 'value per unit' — ₹/litre, % acid, g/L — before writing c, d and m. The ratio that comes out is then a ratio of those same units of quantity.

No diagram — the rule is about units, not positions: convert every ingredient to one value per unit before writing c, d and m. The rule-of-alligation number line already owns the geometry; the worked ledger here carries the purity reading.

How it works

1. Pick the value unitWhatever the question measures — price, purity, concentration — write every ingredient in that unit.
2. Apply alligation in that unitThe formula does not know whether the numbers are rupees or percents; it only needs them comparable.

4Repeated replacement

A vessel of volume V starts full of pure liquid. Each time, x units are drawn out and replaced with water. After one replacement the pure liquid left is V\left(1-\dfrac{x}{V}\right); after n identical replacements it is V\left(1-\dfrac{x}{V}\right)^n.

The fraction remaining multiplies by the same factor every time — do not subtract x from the pure amount n times, because each draw removes a mixture, not pure liquid after the first step.

Figure. Each replacement multiplies the pure milk by 4/5: 40 → 32 → 25.6. The bars are to scale on milk volume; the water that fills the vessel back to 40 L is not drawn.

How it works

1. Write the retention factor1-x/V is the fraction of the current contents that stays after one draw-and-replace.
2. Raise to the number of replacementsPure left = V(1-x/V)^n. The same factor applies whether the liquid is milk, acid or spirit.
3. Read what was askedPure amount, or concentration as a fraction of original — both sit in that one expression.

5Three or more ingredients

Alligation's cross reads two components at a time. For three or more values, either apply alligation pairwise — fix a mean, mix two, then mix that result with the third — or write the weighted-average equation m = (c_1 Q_1 + c_2 Q_2 + c_3 Q_3)/(Q_1+Q_2+Q_3) and solve for the unknown quantity.

Pairwise alligation needs an intermediate mean you are free to choose; the equation needs no such choice and is usually shorter when one quantity is missing and the mean is given.

Figure. Contribution bars 4\times 10, 6\times 20 and 6\times 40 total 400 over mass 16, so the mean is 25. With three or more ingredients, write the weighted-average equation m=(c_1Q_1+c_2Q_2+c_3Q_3)/(Q_1+Q_2+Q_3) instead of pairwise alligation crosses.

How it works

1. Prefer the weighted-average equationWhen the mean and all but one quantity are known, one equation in one unknown finishes it.
2. Use pairwise alligation when ratios are wantedMix two ingredients to an intermediate mean, then alligate that mean against the third.

6Alligation beyond price

The same distances-across-the-mean rule applies wherever a quantity is a weighted average of two pools: price per kg, percent purity, alcohol strength, even average speed when two parts of a trip are mixed by time. The labels on c, d and m change; the ratio formula does not.

What must stay consistent is the unit. Once every ingredient is expressed as the same kind of value per the same unit of quantity, the problem is ordinary alligation.

Figure. Price per kg, percent purity, alcohol strength and even average speed mixed by time all use the same distances-across-the-mean rule. Only the unit names on c, d and m change.