Mixture and Alligation (RRB)

Alligation is a fast technique to find the ratio in which two ingredients of different prices/concentrations must be mixed to obtain a given target.

Alligation rule:

   cheaper           dearer
   price (c)         price (d)
            mean (m)
   (d − m)           (m − c)

The mixing ratio (cheaper : dearer) = (d − m) : (m − c).

Example: Mix rice at ₹20/kg and ₹30/kg to get a mixture costing ₹24/kg.
Ratio = (30 − 24) : (24 − 20) = 6 : 4 = 3:2 (3 parts cheaper, 2 parts dearer).

Applications:

1. Mixing two liquids of different concentrations to get a target concentration.
2. Adding water (concentration = 0%) to dilute a solution.
3. Income/expense averaging problems.
4. Speed problems with two phases of motion.

Replacement problem:
A container has V litres of liquid X. You remove r litres and replace with water. Repeat n times.
Final amount of X = V × (1 − r/V)ⁿ.

Example: 80 L of milk, 8 L is replaced with water 3 times. Final milk = 80 × (72/80)³ = 80 × 729/1000 = 58.32 L.

Mixing two milk solutions:
Two cans contain milk-water in ratios 5:3 and 7:5. They are mixed in ratio 2:3. Find milk-water ratio in the final.

Milk fraction in 1st = 5/8; in 2nd = 7/12. Mean milk fraction = (2×5/8 + 3×7/12)/5 = (10/8 + 21/12)/5 = (30/24 + 42/24)/5 = 72/(24×5) = 72/120 = 3/5. So milk:water = 3:2.

Example 1 — Concentration:
In what ratio must a 20% sugar solution be mixed with a 35% sugar solution to get a 25% solution?
Method: Alligation with concentrations as percentages.
Ratio = (35 − 25) : (25 − 20) = 10 : 5 = 2 : 1 (more of the weaker).

Example 2 — Adding water (dilution):
A 60-L solution is 40% milk. How much water to add to reduce concentration to 30%?
Method: The milk quantity is fixed: 60 × 40% = 24 L milk. After dilution, 24 L = 30% of new volume ⟹ new volume = 80 L. Water to add = 20 L.

Alligation approach: mix 60 L of "40%" with x L of "0%" to get 30%. Ratio = (40−30):(30−0) = 10:30 = 1:3. So 60 : x = 1 : 3 ⟹ x = 180… No wait — alligation gives the ratio of quantities mixed. 60 : water = 30 : (40−30) = 30:10 = 3:1 ⟹ water = 60/3 = 20 L. (Always double-check direction — bigger gap from target attracts more of the other part.)

Example 3 — Three-stage replacement:
A 64-L pure-milk tank, 8 L removed and replaced with water 3 times. Concentration of milk after?
Method: milk = 64 × (1 − 8/64)³ = 64 × (7/8)³ = 64 × 343/512 = 42.875 L ≈ 67% concentration.

Example 4 — Two-grade pricing:
A trader has tea worth ₹18/kg and ₹20/kg. He mixes in 3:5 ratio. What is the cost of the mixture per kg?
Method: Cost = (3×18 + 5×20)/8 = (54 + 100)/8 = 154/8 = ₹19.25/kg.

Trap to watch: in alligation, the outputs (cheaper:dearer ratio) tell you the quantities mixed — they do NOT tell you which is heavier in the mixture; that's revealed by which side of mean each price sits.