Mixtures & Alligations • Topic 2 of 4

Alligation Rule

Alligation is the weighted average run in reverse: given the mean and the two strengths, it hands you the RATIO of quantities directly. The rule is q₁ : q₂ = (c₂ − M) : (M − c₁). Picture the cross: put the cheaper value c₁ at the top-left, the dearer c₂ at the top-right, the mean M in the middle, and subtract diagonally — each quantity is proportional to the DISTANCE of the OTHER ingredient from the mean. So the quantity of the cheaper item is proportional to (dearer − mean), and the dearer to (mean − cheaper). This is the single most time-saving trick in CAT Arithmetic: average-price, average-speed, average-age, even SI vs CI blending all yield to the same cross. A crucial caveat — alligation gives a RATIO, not an absolute amount, so you still need one more datum (a total, a difference, or one quantity) to find actual values.

✅ Solved examples

1. In what ratio must rice at ₹30/kg be mixed with rice at ₹45/kg to get a mixture worth ₹36/kg?
Ratio = (45 − 36) : (36 − 30) = 9 : 6 = 3 : 2 (cheaper : dearer).
2. How many litres of water (₹0) must be added to 40 L of milk worth ₹20/L so the mixture sells at ₹16/L?
Water : milk = (20 − 16) : (16 − 0) = 4 : 16 = 1 : 4. Milk is 40 L (4 parts) ⇒ water = 1 part = 10 L.
3. A 90% pure and a 60% pure gold are mixed to give 80% purity. Find the ratio of the two.
Ratio (60-part : 90-part) = (90 − 80) : (80 − 60) = 10 : 20 = 1 : 2.
4. A grocer mixes two rice types costing ₹50 and ₹66 per kg and sells at ₹66/kg (no loss) after a 10% profit. In what ratio were they mixed?
Selling at ₹66 with 10% profit ⇒ cost of mixture = 66/1.1 = ₹60/kg. Ratio = (66 − 60) : (60 − 50) = 6 : 10 = 3 : 5.

✏️ Practice — try these, take hints as needed

1. Mix tea at ₹60 and ₹100 per kg to get ₹75/kg. Find the ratio.
Alligation cross around 75.
(100 − 75) : (75 − 60).
25 : 15.
5 : 3
2. How much water must be mixed with 60 L of milk (₹15/L) to bring the rate to ₹12/L?
Water value = ₹0.
water : milk = (15−12) : (12−0).
3 : 12 = 1 : 4; milk = 60 L.
15 L
3. Two solutions of 15% and 40% salt give a 30% solution. Ratio of the two?
(40 − 30) : (30 − 15).
10 : 15.
Simplify.
2 : 3
4. Coffee at ₹120 and ₹180 per kg is blended; the blend costs ₹150/kg. Ratio?
Mean 150 is exactly midway.
(180−150) : (150−120).
30 : 30.
1 : 1
5. A merchant mixes ₹70 and ₹100 per kg dal and sells at ₹99 with 10% profit. Mixing ratio?
Cost of mixture = 99/1.1.
Mean = ₹90.
(100−90) : (90−70).
1 : 2

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Mixing Two Substances Concentration & Dilution →