Mixtures and Alligation
Alligation is a see-saw: the mix price sits between the cheap and dear prices, and the quantities are in the inverse of the distances.
Why this topic matters
Mix tea at Rs 200 a kg with tea at Rs 300 a kg and sell the mix at Rs 240. How much of each did you use? Alligation answers that with one cross.
Mixture questions (milk and water, two kinds of rice, alloys, even a class's average) all reduce to the alligation cross or the removal formula.
The alligation cross
Write the cheaper and dearer prices at the top, the mixture's price in the middle. Cross-subtract: cheaper : dearer = (dearer - mean) : (mean - cheaper).
It works for any weighted average: prices, percentages of milk, speeds over equal times, marks of two classes.
- Cross: (300 - 240) : (240 - 200) = 60 : 40.Distance of the mean from the OTHER price.
- 3 : 2.
Removal and replacement
A vessel has × litres of milk. y litres are taken out and replaced with water, n times. Milk left = x (1 - y/x)n.
Each time, the same fraction of whatever milk is left goes out.
- Each time 4/40 = 1/10 of the liquid goes, so 9/10 of the milk stays.
- After 3 times: 40 × (9/10)3 = 40 × 0.729.
- = 29.16 litres.
Mixing two mixtures
Add up each ingredient separately. Mixture 1 has milk : water = 3 : 1 and mixture 2 has 5 : 3; take equal amounts, say 8 litres each: milk 6 + 5 = 11, water 2 + 3 = 5. New ratio 11 : 5.
- Take 8 litres of each (LCM of 4 and 8).Pick a size that splits both ratios evenly.
- Milk: 6 + 5 = 11. Water: 2 + 3 = 5.
- 11 : 5.
Formula summary
Shortcuts and tricks
Cross (see-saw) diagram
Use it when: Two prices, one mean.
- Write cheap and dear on top, mean in the middle.
- Cross-subtract to get the ratio.
- (75 - 65):(65 - 60) = 10:5.
Before you move on, you should be able to...
- use the cross for any two-ingredient average
- apply x(1 - y/x)n for repeated removal
- combine two mixtures ingredient by ingredient
Practice questions (10)
Try each one before opening the solution. Or practise them one by one so your score is saved.
easy A milkman adds water and sells at cost price, gaining 25%. Water:milk?
- 25 : 100.
Answer: 1:4
easy How much water must be added to 60 L of 80% milk solution so that milk becomes 60%?
- Milk 48 = 60% of 80 L.
Answer: 20 L
medium A 40 L can of milk: 4 L removed and replaced with water, three times. Milk left (L)?
- 40 × (0.9)3.
Answer: 29.16
medium 60 L mixture of milk:water = 2:1. Water to add to make it 1:2?
- Milk 40 fixed. Need water 80. Add 60.
Answer: 60 L
hard Alloys gold:copper 7:2 and 7:11 are mixed in equal amounts. Gold:copper in the new alloy?
- Gold fractions 14/18 and 7/18 → average 21/36.
- Copper 15/36.
Answer: 7:5
hard Teas at Rs 126 and 135 per kg are mixed with a third variety in 1:1:2. The mix is worth Rs 153/kg. Price of the third?
- 126 + 135 + 2x = 612 → x = 175.5.
Answer: Rs 175.5
hard 729 L of milk and water is 7:2. How much water must be added to make it 7:3?
- Milk 567, water 162.
- For 7:3 water must be 243.
Answer: 81 L
hard From a can of pure milk, 10% is removed and replaced with water, four times. % of milk left?
- 0.94 = 0.6561.
Answer: 65.61%
hard Vessel A is milk:water 5:3, vessel B is 1:3. In what ratio must they be mixed to get 1:1?
- Milk fractions 5/8 and 1/4, target 1/2.
- (1/2 - 1/4) : (5/8 - 1/2) = 1/4 : 1/8.
Answer: 2:1
hard Sugar at Rs 20/kg and Rs 30/kg is mixed and sold at Rs 27.60, making 15% profit. Ratio?
- Cost of mix = 27.60/1.15 = 24.
- (30 - 24) : (24 - 20) = 6 : 4.
Answer: 3:2
Class plan (2 hours, for trainers)
A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.
| 10 min | Warm-up. |
| 30 min | Alligation cross: prices, percentages, averages. |
| 25 min | Removal and replacement. |
| 20 min | Mixing mixtures. |
| 30 min | Practice set. |
| 5 min | Recap. |