In this guide
Mixture questions come in two kinds. Some ask about ratios inside one mixture: how much water to add to change milk : water from 3 : 1 to 1 : 1. Others ask how to combine two things at different values to get a target value in between. The second kind is where alligation comes in, and it is one of the most useful shortcuts in the whole Numerical Ability section.
Alligation is not only about rice and milk. It works for any two groups with an average: prices, concentrations, marks, salaries, speeds over equal times, even a sum lent at two interest rates. This guide explains the rule, why it works, and the question types built on it.
The alligation rule
Mix a cheaper item (value c) with a dearer one (value d) to get a mean value m, where c < m < d. Then
quantity of cheaper : quantity of dearer = (d − m) : (m − c)
Set it out as a cross, and it takes seconds:
| Cheaper (c) | Mean (m) | Dearer (d) |
|---|---|---|
| ₹30 | ₹36 | ₹40 |
| Takes d − m = 4 | Takes m − c = 6 |
Ratio 4 : 6 = 2 : 3.
Why it works: each kilo of the cheaper item sits m − c below the mean, and each kilo of the dearer item sits d − m above it. For the mixture to average exactly m, the total shortfall must equal the total excess. If you take x kg cheap and y kg dear, x(m − c) = y(d − m), so x : y = (d − m) : (m − c). It is a balance, like a see-saw: the item further from the mean needs a smaller quantity.
Where alligation applies
| Mixing | c and d are | m is |
|---|---|---|
| Two grades of rice or tea | Price per kg | Price of the mixture |
| Two acid solutions | Concentration % | Target concentration |
| Two mixtures of milk and water | Fraction of milk in each | Fraction of milk wanted |
| Two groups of people | Group averages | Overall average |
| A sum at two interest rates | The two rates | Average rate earned |
A useful special case: pure water is a "solution" with 0% milk (or acid), and pure milk is 100%.
Ratios inside one mixture
When you add only water to milk and water, the quantity of milk does not change. Fix the unchanging part first, then work out what the other part must become. This beats setting up an equation in almost every question.
Repeated replacement
A container holds V litres of pure liquid. Each time, x litres are taken out and replaced with water. After n such operations,
pure liquid left = V × (1 − x/V)ⁿ
Why: each operation removes a fraction x/V of whatever is in the container, including the pure liquid. So the pure liquid is multiplied by (1 − x/V) each time. It is the same idea as compound interest with a fall instead of a rise.
Worked examples
Example 1 (two prices). Rice at ₹30/kg is mixed with rice at ₹40/kg so that the mixture is worth ₹36/kg. In what ratio are they mixed, and how much of the dearer rice goes with 20 kg of the cheaper?
- Ratio = (40 − 36) : (36 − 30) = 4 : 6 = 2 : 3.
- 20 kg of cheap rice needs 20 × 3/2 = 30 kg of the dearer rice.
- Check: (20 × 30 + 30 × 40) ÷ 50 = 1,800 ÷ 50 = ₹36.
Example 2 (adding water). 40 litres of a mixture have milk and water in the ratio 3 : 1. How much water must be added to make the ratio 1 : 1?
- Milk = 30 L and water = 10 L. The milk stays at 30 L.
- For 1 : 1, water must also be 30 L, so add 20 L.
Example 3 (adding milk). 20 litres of a mixture contain 20% water. How much pure milk must be added to bring the water down to 10%?
- Water = 4 L, and it does not change.
- For 4 L to be 10% of the new total, the total must be 40 L. Add 20 L of milk.
Example 4 (replacement). A container holds 40 L of milk. 4 L is taken out and replaced with water. This is done twice more, three times in all. How much milk is left?
- Milk left = 40 × (1 − 4/40)³ = 40 × 0.9³ = 40 × 0.729 = 29.16 L.
- After two operations it would be 40 × 0.81 = 32.4 L.
Example 5 (two mixtures). Vessel A has milk and water in the ratio 3 : 1, and vessel B in the ratio 5 : 3. In what ratio should they be mixed to get milk and water in the ratio 2 : 1?
- Fraction of milk: A = 3/4, B = 5/8, target = 2/3.
- B is "cheaper" (less milk). B : A = (3/4 − 2/3) : (2/3 − 5/8) = 1/12 : 1/24 = 2 : 1.
- So A : B = 1 : 2.
- Check: 1 L of A has 0.75 L milk and 2 L of B has 1.25 L. Total 2 L milk in 3 L, which is 2 : 1.
Example 6 (groups of people). The average salary of 21 workers is ₹8,000. Technicians average ₹12,000 and the rest ₹6,000. How many are technicians?
- Rest : technicians = (12,000 − 8,000) : (8,000 − 6,000) = 4,000 : 2,000 = 2 : 1.
- Technicians = 1/3 of 21 = 7.
Mixing with profit
If a trader sells the mixture at a profit, first strip out the profit to find the mixture's cost price, then use alligation. Sugar at ₹40 and ₹50 a kg, sold at ₹54 for a 20% profit: cost of the mixture = 54 ÷ 1.2 = ₹45, so the ratio is (50 − 45) : (45 − 40) = 1 : 1.
Common mistakes
- Using the selling price as the mean when a profit is involved. Alligation works on cost.
- Forgetting the unchanged part. When water is added, milk is fixed; when milk is added, water is fixed.
- Subtracting replacements as if linear. Removing 4 L three times from 40 L does not remove 12 L of milk; the second and third draws take out some water too.
- Mixing ratios with fractions. Convert 3 : 1 to "3/4 milk" before alligating two mixtures.
Practice set
- Tea at ₹60/kg and ₹80/kg is mixed to get tea worth ₹72/kg. Find the ratio.
- 60 L of a mixture has milk and water in the ratio 2 : 1. How much water must be added to make it 1 : 1?
- Solutions with 10% and 30% acid are mixed to get 25% acid. Find the ratio.
- How many kg of dal at ₹15/kg must be mixed with 30 kg of dal at ₹24/kg to get a mixture worth ₹20/kg?
- A container holds 80 L of milk. 8 L is replaced with water, twice. How much milk is left?
- 30 L of a mixture has alcohol and water in the ratio 7 : 3. How much water must be added to make alcohol 60% of the mixture?
- Two varieties at ₹126/kg and ₹135/kg are mixed with a third in the ratio 1 : 1 : 2. The mixture is worth ₹153/kg. Find the price of the third variety.
- 45 L of a mixture has milk and water in the ratio 4 : 1. How much water must be added to make it 3 : 2?
Answers:
- 2 : 3. (80 − 72) : (72 − 60) = 8 : 12.
- 20 L. Milk 40 L, water 20 L. Water must become 40 L.
- 1 : 3. (30 − 25) : (25 − 10) = 5 : 15.
- 24 kg. Cheaper : dearer = (24 − 20) : (20 − 15) = 4 : 5, and 30 × 4/5 = 24.
- 64.8 L. 80 × 0.9² = 80 × 0.81.
- 5 L. Alcohol = 21 L, fixed. 21 ÷ 0.6 = 35 L total, so add 5 L.
- ₹175.50. (126 + 135 + 2x) ÷ 4 = 153, so 261 + 2x = 612 and x = 175.5.
- 15 L. Milk 36 L, water 9 L. For 3 : 2, water = 36 × 2/3 = 24 L, so add 15 L.
What to do next
- Draw the alligation cross for the next 15 mixture questions you solve, labelling cheaper and dearer each time.
- Learn the replacement formula and test it on one example with a calculator-free check.
- Revise averages and ratio and proportion, then move on to problems on ages.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Indian Air Force website .
Get the next AFCAT guide by email
New guides every week. No spam, unsubscribe any time.