Master the W-Method, Cross-Method, and Successive Replacement Formula to solve complex mixture problems in seconds ā 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 ā¹40/kg is mixed with Rice at ā¹60/kg. The mixture costs ā¹55/kg. Find the mixing ratio.
Solution:
ā¢ Cheaper (A) = ā¹40, Dearer (B) = ā¹60, Mean (M) = ā¹55
ā¢ Difference on Left = (60 - 55) = 5
ā¢ Difference on Right = (55 - 40) = 15
ā¢ Ratio = 5 : 15 = 1 : 3
ā
Answer: Mix 1 part of ā¹40 rice with 3 parts of ā¹60 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:
ā¢ Initial milk concentration = 7/10 = 70%
ā¢ Final milk concentration = 3/10 = 30%
ā¢ Water concentration = 0%
ā¢ Using Alligation: Ratio of Original : Water = 30 : 40 = 3:4
ā¢ 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:
ā¢ Initial = 80 L, Removed = 8 L each time, n = 3
ā¢ Remaining = 80 Ć [1 - 8/80]Ā³
ā¢ Remaining = 80 Ć [1 - 0.1]Ā³
ā¢ Remaining = 80 Ć (0.9)Ā³
ā¢ Remaining = 80 Ć 0.729 = 58.32 litres
Memorize these for faster calculation:
ā¢ (0.9)Ā² = 0.81
ā¢ (0.9)Ā³ = 0.729
ā¢ (0.9)ā“ = 0.6561
Similarly: (0.8)Ā² = 0.64, (0.8)Ā³ = 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 ā¹50/kg (Cost) is mixed with Tea at ā¹80/kg (Cost). Mixture sold at ā¹72/kg with 20% profit. Find ratio.
Solution:
ā¢ SP = ā¹72, Profit = 20% ā CP = 72 Ć· 1.2 = ā¹60
ā¢ Now apply alligation: A=50, B=80, M=60
ā¢ 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 ā¹/kg and B is in ā¹/100g, convert first! Many students lose marks here.
Alligation becomes instinct after 50 problems. Spend 2 days practicing ā you'll never struggle again!
Ratio = (B-M) : (M-A)
R = I Ć [1 - r/T]āæ
M = (AĆa + BĆb)/(a+b)
(0.9)Ā³ = 0.729
Ab aapko Mixture & Alligation ki poori samajh aa gayi! Time to test your skills with real exam-level questions!
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