Master the W-Method, Cross-Method, and Successive Replacement Formula to solve complex mixture problems in seconds — trusted by 50,000+ IBPS, SBI, SSC aspirants!
Mixture and Alligation is one of the most scoring topics in Bank PO, SSC CGL, Railway NTPC, and other competitive exams. Once you master this concept, you can solve average, percentage, profit-loss, and ratio problems in just 10-15 seconds!
Mixture problems involve combining two or more ingredients with different properties (like price, concentration, speed, weight) to form a final mixture. Alligation is the shortcut rule to find the exact ratio for mixing without using lengthy algebraic equations.
In IBPS PO 2023, 4-5 questions came directly from Mixture & Alligation. Students who knew the W-Method solved them in under 30 seconds each, saving 5+ minutes for other questions. That's the difference between selection and rejection!
The W-Method (also called Cross-Method or Alligation Rule) is the fastest technique. Here's how to apply it:
Find the Cheaper value (A), Dearer value (B), and Mean/Target value (M). The mean always lies between A and B.
Place A and B at top corners, M in the center. Draw cross lines connecting A-M and B-M.
Find (B - M) and place it under A. Find (M - A) and place it under B. These are your ratio quantities.
Ratio of A : B = (B - M) : (M - A). Simplify if needed!
Question: Rice at ₹40/kg is mixed with Rice at ₹60/kg. The mixture costs ₹55/kg. Find the mixing ratio.
Solution:
• Cheaper (A) = ₹40, Dearer (B) = ₹60, Mean (M) = ₹55
• Difference on Left = (60 - 55) = 5
• Difference on Right = (55 - 40) = 15
• Ratio = 5 : 15 = 1 : 3
👉 Answer: Mix 1 part of ₹40 rice with 3 parts of ₹60 rice!
Question: A container has milk and water in 7:3 ratio. How much mixture should be replaced with water to make the ratio 3:7?
Solution:
• Initial milk concentration = 7/10 = 70%
• Final milk concentration = 3/10 = 30%
• Water concentration = 0%
• Using Alligation: Ratio of Original : Water = 30 : 40 = 3:4
• Replace 4/7 of the mixture!
When a portion of a mixture is repeatedly removed and replaced with another liquid, use the Successive Replacement Formula. This is common in problems involving containers of milk/water.
Question: A container has 80 litres of pure milk. 8 litres is removed and replaced with water. This process is repeated 3 times. How much milk remains?
Solution:
• Initial = 80 L, Removed = 8 L each time, n = 3
• Remaining = 80 × [1 - 8/80]³
• Remaining = 80 × [1 - 0.1]³
• Remaining = 80 × (0.9)³
• Remaining = 80 × 0.729 = 58.32 litres
Memorize these for faster calculation:
• (0.9)² = 0.81
• (0.9)³ = 0.729
• (0.9)⁴ = 0.6561
Similarly: (0.8)² = 0.64, (0.8)³ = 0.512
When applying alligation to cost/profit problems, ensure all values are of the same type. If one value is Selling Price (SP), convert it to Cost Price (CP) first using profit/loss percentage.
Tea at ₹50/kg (Cost) is mixed with Tea at ₹80/kg (Cost). Mixture sold at ₹72/kg with 20% profit. Find ratio.
Solution:
• SP = ₹72, Profit = 20% => CP = 72 / 1.2 = ₹60
• Now apply alligation: A=50, B=80, M=60
• Ratio = (80-60) : (60-50) = 20:10 = 2:1
For 3 or more ingredients, pair values above and below the mean, then combine. Or use weighted average approach.
Alligation also works for average speed problems! If a person travels at speeds A and B for different times, use alligation with time ratios.
The Mean value must always lie between Cheaper and Dearer. If it doesn't, the problem is incorrectly framed or you've identified values wrong. This is your first sanity check!
Even for simple problems, draw the W-diagram. It prevents silly mistakes and takes only 5 seconds. Visual clarity = zero errors!
For faster calculation: 20% = 1/5, 25% = 1/4, 33.33% = 1/3. Working with fractions is faster than decimals in exams.
Ensure all values are in the same units. If A is in ₹/kg and B is in ₹/100g, convert first! Many students lose marks here.
Alligation becomes instinct after 50 problems. Spend 2 days practicing — you'll never struggle again!
Mixture & Alligation isn't just for liquids. You can use it as a master key for Profit & Loss and Simple Interest problems too!
If a shopkeeper sells Part 1 at 10% Profit, Part 2 at 20% Profit, and overall makes 15% Profit, simply put 10 and 20 at the top, 15 in the middle. The resulting ratio (5:5 or 1:1) is the ratio of their Cost Prices (CP), NOT quantities.
Always remember what the base of the ratio represents. In Speed, base is Time. In Profit%, base is Cost Price. If you apply Alligation on Profit%, the final ratio you get is of CP. Memorize the bases!
The real power of Alligation is that it works as a universal weighted average shortcut. Every time you see a "combined average" scenario, Alligation is 3x faster than algebra.
Q: A person covers first half of journey at 40 km/hr and second half at 60 km/hr. Find average speed.
Wrong Method (Trap): (40+60)/2 = 50 km/hr ❌ — This is WRONG because time spent is different!
Alligation Method: Mean = ? Put 40 and 60 at top. Time ratio is inverse of speed ratio = 3:2.
Using weighted avg: Mean = (40×3 + 60×2)/(3+2) = (120+120)/5 = 48 km/hr ✅
Q: Average age of 30 students = 14 years. A teacher aged 44 joins. Find new average.
Alligation Method: Put 14 (student avg) and 44 (teacher) at top. Numbers = 30:1.
New Average = (14×30 + 44×1)/(30+1) = (420+44)/31 = 464/31 = 14.97 ≈ 15 years ✅
A sum is split: part at 6% SI and part at 9% SI. If overall rate is 7%, find how the sum was split. Use Alligation → (9-7):(7-6) = 2:1. So 2/3 of sum at 6% and 1/3 at 9%. This type appears in IBPS PO every year!
This is the most feared type of Mixture question in Bank Prelims. The "khaali karo aur bhar do" (draw out and fill) concept:
Q: A 40-litre vessel contains pure milk. 8 litres is drawn and replaced with water — done 3 times. Find milk left.
Solution:
Milk left = 40 × (1 − 8/40)³ = 40 × (32/40)³ = 40 × (4/5)³
= 40 × 64/125 = 2560/125 = 20.48 litres ✅
💡 Shortcut: (32/40)³ = (0.8)³ = 0.512 → 40 × 0.512 = 20.48 — Done in 10 seconds!
Memorize these values: (0.8)² = 0.64 | (0.8)³ = 0.512 | (0.75)² = 0.5625 | (0.9)² = 0.81 | (0.9)³ = 0.729.
These appear as-is in 80% of successive replacement questions. You'll save 90 seconds per question!
These are based on actual patterns from IBPS PO 2021–2024, SBI Clerk 2023, and SSC CGL 2022:
In what ratio must tea at ₹62/kg be mixed with tea at ₹72/kg to get a blend worth ₹65/kg?
Solution (W-Method): (72−65) : (65−62) = 7 : 3 → Mix in ratio 7:3 ✅
A container has 60L milk. 12L is drawn and replaced by water. Then 12L of this mixture is drawn and replaced by water again. Find concentration of milk.
Solution: 60 × (48/60)² = 60 × (4/5)² = 60 × 16/25 = 38.4 litres ✅
A grocer mixes 25 kg of sugar at ₹20/kg with 35 kg at ₹30/kg. At what price per kg should he sell to gain 25%?
Solution: Total CP = 25×20 + 35×30 = 500 + 1050 = ₹1,550 for 60 kg
CP per kg = 1550/60 = ₹25.83 | SP = 25.83 × 1.25 = ₹32.29/kg ✅
Ratio = (B-M) : (M-A)
R = I × [1 - r/T]ⁿ
M = (A×a + B×b)/(a+b)
(0.9)³ = 0.729
Mastered Mixture & Alligation? Practice full mock tests and sectional tests for your upcoming exams:
Ab aapko Mixture & Alligation ki poori samajh aa gayi! Time to test your skills with real exam-level questions!
Start Practice Now ➔