Last reviewed 22 Sept 2026 · 6 min read
The rule of alligation
Alligation answers one question: in what ratio must two things be mixed so that the mixture has a required mean value? The "value" may be a price per kg, a strength (percent of alcohol or milk), a speed, or a marks average — anything that averages by weight.
For a cheaper ingredient of value , a dearer one of value and a mean value (with ):
The ratio is the opposite difference: the quantity of the cheaper is proportional to how far the dearer is above the mean.
The alligation cross
| Cheaper | Dearer | |
|---|---|---|
| Mean | ||
Write the two values on top, the mean below, and cross-subtract; the bottom row is the ratio cheaper : dearer.
Mixing two mixtures
When two mixtures already contain the same two ingredients in known ratios, work with the fraction of one ingredient in each and then alligate. For milk : water of in the first and in the second, the milk fractions are and , and the required mean fraction sits between them.
Replacement — drawing off and topping up
A vessel holds units of pure liquid. If units are drawn off and replaced by water, and this is done times:
and the liquid : water ratio at the end is .
If the vessel starts as a mixture, apply the same factor to the quantity of each ingredient — replacement removes both in their current proportion.
Changing a ratio by adding one ingredient
Adding water does not change the quantity of milk. So fix the milk quantity, write the new total the required ratio demands, and the difference is the water to add. The same reasoning covers adding pure milk, or removing part of the mixture and topping up.
Worked examples
Tea costing ₹ 180 per kg is mixed with tea costing ₹ 280 per kg so that the mixture is worth ₹ 220 per kg. Find the ratio.
Solution. 3 : 2 (cheaper : dearer).
In what ratio must a 20% alcohol solution be mixed with a 50% solution to get 60 litres of a 30% solution?
Solution. Ratio . Of 60 litres: 40 litres of the 20% solution and 20 litres of the 50%.
Vessel A has milk and water in the ratio 5 : 2, vessel B in the ratio 3 : 4. Equal quantities are taken from each and mixed. Find the milk : water ratio.
Solution. Milk fractions are and ; equal quantities give milk . Ratio 4 : 3.
In what ratio must the two vessels of Example 3 be mixed to get milk : water ?
Solution. Milk fractions and , required mean . , so A : B 1 : 3.
A 60-litre mixture has milk and water in the ratio 2 : 1. How much water must be added to make it 1 : 1?
Solution. Milk L, water L. For 1 : 1 the water must also be 40 L, so add 20 litres.
A vessel holds 40 litres of milk. 8 litres are drawn off and replaced by water; this is done twice. How much milk is left?
Solution. 25.6 litres.
From 16 litres of milk, 4 litres are drawn off and replaced by water three times over. Find the milk left and the final ratio.
Solution. 6.75 litres of milk, so water L and the ratio is 27 : 37.
Rice at ₹ 42 per kg and ₹ 54 per kg are mixed in the ratio 3 : 1. Find the price of the mixture.
Solution. ₹ 45 per kg.
A shopkeeper mixes rice at ₹ 42 with rice at ₹ 54 in the ratio 3 : 1 and sells the mixture at ₹ 54 per kg. Find the profit percent.
Solution. The mixture costs ₹ 45 per kg (Example 8), so the profit is 20%.
A 45-litre mixture has milk and water in the ratio 7 : 2. Nine litres are removed and replaced by water. Find the new ratio.
Solution. Removing 9 of 45 litres leaves of each: milk L, water L, plus 9 L of water → 17 L. Ratio .
Three varieties costing ₹ 60, ₹ 75 and ₹ 100 per kg are to be mixed so that the mixture costs ₹ 80 per kg, using the first two in equal quantity.
Solution. Treat ₹ 60 and ₹ 75 mixed equally as one ingredient at ₹ 67.50. Then . So the varieties are in the ratio 4 : 4 : 5.
Frequently tested points
- Alligation ratio is the opposite difference: cheaper : dearer .
- The mean must lie strictly between the two values, or no mixture is possible.
- Alligation works for any quantity that averages by weight — price, strength, speed, marks.
- Two mixtures combine through the fraction of one ingredient, not through their ratios directly.
- Repeated replacement: liquid left .
- Replacement removes ingredients in their current proportion; adding water leaves the milk untouched.
- To change a ratio by adding one ingredient, hold the other ingredient's quantity fixed.
- Writing the alligation ratio the right way round but assigning it to the wrong ingredient.
- Using when the vessel is topped up with the same liquid rather than water.
- Mixing two mixtures by averaging their ratios (5:2 and 3:4 do not give 8:6).
- Forgetting that after removing part of a mixture, both ingredients have shrunk in the same proportion.
- Alligation gives the mixing ratio from the two values and the required mean, as opposite differences.
- The cross is a layout, not a separate method, and it extends to three ingredients by grouping two of them first.
- Two mixtures are combined through the fraction of one ingredient in each.
- Drawing off and topping up times leaves of the original liquid.
- Adding one ingredient changes the total but not the other ingredient, which is the quickest way to set up ratio-change problems.