Pariksha SaathiMixture and Alligation Questions with Solutions
Alligation is a shortcut for finding the ratio to mix two things of different value so the mixture comes out at a target average — it’s really just the weighted-average formula worked backwards. Here’s the concept, four fully worked examples, and a few to try yourself.
The concept, quickly
- Alligation rule: to mix a cheaper item (price C) and a dearer item (price D) so the mixture’s mean price is M, the required ratio is cheaper : dearer = (D − M) : (M − C).
- This is just the weighted-average equation rearranged — you can always sanity-check an answer by computing the weighted average directly from the ratio you found and confirming it equals the target mean.
- The same rule works for anything with a "concentration" — milk-water mixtures, alloy compositions, even mixing two classes’ average marks — not just prices.
- For "cost price vs selling price" alligation questions (like diluting milk with free water to make a profit), treat water’s cost as ₹0 and work out the mixture’s effective cost price from the profit percentage first, before applying the ratio formula.
Worked examples
1. In what ratio should tea worth ₹60/kg be mixed with tea worth ₹80/kg so that the mixture is worth ₹65/kg?
Solution
- Using the alligation rule: cheaper : dearer = (80 − 65) : (65 − 60) = 15 : 5 = 3 : 1.
- Check: weighted average = (3 × 60 + 1 × 80) / 4 = (180 + 80) / 4 = 260/4 = 65 — matches.
Answer: 3 : 1 (₹60/kg tea : ₹80/kg tea)
2. A trader mixes 20 kg of rice costing ₹40/kg with 30 kg of rice costing ₹60/kg. Find the average price of the mixture.
Solution
- Total cost = (20 × 40) + (30 × 60) = 800 + 1,800 = ₹2,600
- Total quantity = 20 + 30 = 50 kg
- Average price = 2,600 / 50 = ₹52/kg
Answer: ₹52/kg
3. A mixture of 60 litres contains milk and water in the ratio 2 : 1. How much water must be added to make the ratio of milk to water 1 : 2?
Solution
- Milk = (2/3) × 60 = 40 L, Water = (1/3) × 60 = 20 L.
- Adding water doesn’t change the milk, so if x litres of water are added: 40 / (20 + x) = 1 / 2
- 80 = 20 + x → x = 60 litres. Check: 40 : (20 + 60) = 40 : 80 = 1 : 2.
Answer: 60 litres
4. A milkman mixes water (free) with milk and sells the mixture at the same price as pure milk, thereby gaining 20%. Find the ratio of water to milk in the mixture.
Solution
- Let the price of pure milk (and the selling price of the mixture) = ₹6/litre. A 20% gain means the mixture’s own cost price = SP ÷ 1.2 = 6 ÷ 1.2 = ₹5/litre.
- Apply alligation with water (₹0) as the "cheaper" item and milk (₹6) as the "dearer" item, mean = ₹5: water : milk = (6 − 5) : (5 − 0) = 1 : 5.
Answer: 1 : 5 (water : milk)
Try these yourself
Work through these the same way, then check against the answer.
1. In what ratio should rice worth ₹50/kg be mixed with rice worth ₹70/kg so that the mixture is worth ₹55/kg?
Answer: 3 : 1 (₹50/kg rice : ₹70/kg rice)
2. A shopkeeper mixes 15 kg of sugar costing ₹30/kg with 25 kg of sugar costing ₹42/kg. Find the average price of the mixture.
Answer: ₹37.50/kg
3. A mixture of 45 litres contains milk and water in the ratio 4 : 1. How much water must be added to make the ratio 2 : 1?
Answer: 9 litres
Where this comes up
Mixture and Alligation is tested directly in the Quantitative/Numerical Aptitude section of every exam Pariksha Saathi covers — SSC CGL, SSC MTS, SSC CHSL, IBPS PO, IBPS Clerk, SBI PO and SBI Clerk — and builds directly on Ratio and Proportion.
Practice more Mixture and Alligation questions
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