Mixtures and Alligation Questions for SSC CGL

Core Concept: What is Alligation?

Alligation is a shortcut method to find the average (mean) price, concentration, or ratio when two or more quantities are mixed. Instead of using weighted average formula every time, alligation gives you a visual box method that's faster. The key insight: if you mix a cheaper item with a costlier item, the average price will always lie between them. Alligation helps you find that exact average and the ratio in which they were mixed.

• Alligation is NOT a new formula—it's a visual shortcut for weighted average
• Works for price, concentration, ratio, speed, or any measurable quantity
• Two types: Alligation Medial (find average) and Alligation Alternate (find ratio)
• SSC CGL mostly tests Alligation Alternate (reverse problem: given average, find ratio)
• Always draw the box diagram—it prevents careless errors
• The method works only when mixing exactly 2 quantities (or treating 3+ as pairs)

Alligation Medial: Finding the Average

This is the straightforward direction. You have two quantities with known prices/concentrations, and you want the average. Example: 10 kg of rice at Rs 50/kg mixed with 15 kg at Rs 60/kg—what's the average price? Use weighted average. Alligation Medial is rarely tested directly in SSC CGL because it's just basic weighted average, but understanding it helps you reverse-engineer Alligation Alternate problems.

• Formula: Average = (Q1 × P1 + Q2 × P2) / (Q1 + Q2)
• Q = quantity, P = price/concentration
• Useful as a check when you solve Alligation Alternate
• Less common in SSC CGL exams—mostly appears as a verification step

Alligation Alternate: Finding the Ratio (The Real SSC CGL Pattern)

This is what SSC CGL tests 90% of the time. You're given the average price and the two original prices, and you must find the ratio in which they were mixed. The box method is your weapon. Draw a box with the mean (average) in the centre. Write the two prices at the top corners. Subtract diagonally: (Mean - Lower Price) and (Higher Price - Mean). These differences are your ratio. The beauty: it's always inverse—the quantity of cheaper item is proportional to (Mean - Cheaper), and vice versa.

• Box method: Centre = Mean, Corners = Two prices
• Subtract diagonally: differences become the ratio
• Ratio is INVERSE: cheaper item's ratio = (Mean - Cheaper Price), costlier item's ratio = (Costlier Price - Mean)
• Always simplify the ratio to lowest terms
• Works for any measurable quantity: price, concentration, speed, purity
• SSC CGL loves this because it's fast and tests logical thinking

Three-Way Mixtures and Replacement Problems

Occasionally SSC CGL throws a 3-way mixture or a replacement problem. For 3-way: treat it as two separate 2-way problems, or use the weighted average formula directly. For replacement: a quantity is removed and replaced with another—this creates a series. If you remove and replace x units from a total of n units, after k operations, the remaining fraction of original = (1 - x/n)^k. These are less frequent but high-value if you know them.

• Three-way mixtures: use weighted average or break into pairs
• Replacement formula: Remaining = Original × (1 - Removed/Total)^(Number of operations)
• Replacement is common in concentration/purity problems
• Always check if the problem is asking for amount removed, amount remaining, or final concentration
• Draw a timeline for replacement problems—prevents confusion

Concentration and Purity Problems

Alligation works identically for concentration (% of pure substance) and purity (% of metal in alloy). A 50% alcohol solution mixed with 80% alcohol solution to get 60% solution—same box method applies. The 'price' is replaced by 'concentration' or 'purity %'. This variant appears 1-2 times per SSC CGL paper. The logic is identical; only the units change.

• Concentration problems use the same alligation box method
• Purity of alloys (gold, silver) follows the same pattern
• Always express concentration as a percentage or decimal before applying the method
• Common trap: forgetting to convert % to decimal or vice versa
• Reverse problem: given final concentration and two original concentrations, find ratio

• Forgetting the ratio is INVERSE: students often write (Mean - Cheaper) : (Costlier - Mean) instead of reversing it. Remember: the cheaper item's quantity is proportional to the gap between mean and cheaper price.
• Not simplifying the ratio: SSC CGL expects simplified form. If you get 10:15, reduce to 2:3. Unsimplified answers are marked wrong.
• Confusing Alligation Medial with Alternate: Medial is forward (find average), Alternate is reverse (find ratio). 90% of SSC CGL tests Alternate.
• Careless arithmetic in the box: students subtract in the wrong direction or forget to subtract diagonally. Always draw the box and label clearly: Mean in centre, prices at corners.
• Misidentifying which price is higher: if you swap P1 and P2, your ratio flips. Always check: is the mean between the two prices? If not, re-read the problem.
• Ignoring units: if the problem mixes kg and litres, convert first. Alligation works on quantities, so units must match.

• BOX METHOD = ALLIGATION ALTERNATE. Draw a box, put mean in centre, prices at corners, subtract diagonally, ratio is the differences. Inverse relationship: cheaper item gets the larger difference.
• MEAN LIES BETWEEN: The average price always lies between the two original prices. If it doesn't, you've misread the problem.
• DIAGONAL SUBTRACTION: Top-left minus centre, centre minus bottom-left (or vice versa). The differences are your ratio. Think of it as 'distance from mean'.
• REPLACEMENT POWER RULE: (1 - x/n)^k. Each operation multiplies by the same factor. Useful for milk-water, concentration, purity problems over multiple steps.

• SSC CGL Quant papers (Tier 1) typically have 1-2 Alligation Alternate problems in the 25-question set. They appear in the medium difficulty range (Q 12-18 approx). Expect 45-60 seconds per question if you know the method.
• Recent papers (2022-2024) show a shift toward 3-way or replacement variants. If you see 'milk and water' or 'alloy purity', immediately think Alligation.
• Tier 2 (Descriptive) occasionally asks for detailed working. Always show the box diagram—it demonstrates clarity and earns partial credit if arithmetic is slightly off.
• Common SSC CGL twist: 'A mixture of two types of oil costs Rs X. Oil A costs Rs Y, Oil B costs Rs Z. If 10 kg of A is used, how much of B?' Recognize this as Alligation Alternate, find the ratio, then scale to the given quantity.
• Time-saving tip: Once you draw the box and find the ratio, don't recalculate. If the ratio is 3:2 and you need 10 kg total, immediately: 3/(3+2) × 10 = 6 kg of first, 4 kg of second. No need to re-solve.