Alligation and Mixture: Speed Tricks for Aptitude Tests

Alligation ratio problems take under 30 seconds with the two-subtraction cross method. The bottleneck for most test-takers is recognising the pattern, not the arithmetic.

This article is the speed-and-tricks companion to the complete alligation guide. The full conceptual treatment, including formula derivation and step-by-step worked examples, is covered in that companion article. This article focuses on what to do in the test room: spot the problem type fast, apply the cross without writing an equation, and avoid the traps that waste three minutes on a one-minute question.

Spot an Alligation Problem in Under 30 Seconds

Three signals appear in every alligation problem. When you see all three, apply the cross immediately.

• Signal 1: Two ingredients with different values. The values can be prices (Rs. 40/kg and Rs. 70/kg), concentrations (30% and 60%), or any other linearly-averaging property (speed, rate of interest, grade percentage).
• Signal 2: A mean or mixture value. Either stated directly (“a mixture worth Rs. 55/kg”) or implied (“sell at Rs. 66 with 10% profit,” meaning you derive the cost mean before applying the cross).
• Signal 3: A ratio or total quantity to find, verify, or split. If you are asked to find “in what ratio” or “how many kg of each,” the cross solves it. If you are given the ratio and asked for the mean, use the weighted average directly instead.

Scan the question for these three signals. If all three are present, do not set up an equation. Go straight to the cross.

The Two-Subtraction Cross: Mental-Math Method

The cross operation has two steps and zero algebra.

Step 1: Arrange the values in a cross pattern mentally (or on scratch paper):

  p1        p2
      mean

Step 2: Take diagonal differences:

• Right diagonal: p2 minus mean. This is the quantity (or ratio share) of the ingredient at p1 (the cheaper or lower-concentration one).
• Left diagonal: mean minus p1. This is the quantity of the ingredient at p2.

That is the complete method. The ratio of cheaper to costlier is (p2 - mean) : (mean - p1).

Mental-math tip for speed: in most test problems the differences are round numbers or simplify to small integers. If your differences come out as 23 and 37, double-check the mean. Most setters use numbers that cancel cleanly to ratios like 1:1, 1:2, 2:3, or 3:4.

Six Speed-Trick Examples

Example 1: Standard price blend

• Problem: Tea at Rs. 40/kg and Rs. 70/kg, blended to a mean of Rs. 50/kg. Find the ratio.
• Cross: (70 - 50) : (50 - 40) = 20 : 10 = 2 : 1
• Verify: (40 × 2 + 70 × 1) / 3 = 150 / 3 = 50 ✓
• Speed note: The ratio 2:1 appears in under 5 seconds. No equation needed.

Example 2: Profit buried in the mean

• Problem: Mix two grades of rice at Rs. 50/kg and Rs. 80/kg, then sell the blend at Rs. 66/kg with a 10% profit. Find the ratio.
• Step 1: Strip profit. Cost price of blend = 66 / 1.1 = Rs. 60/kg
• Step 2: Cross: (80 - 60) : (60 - 50) = 20 : 10 = 2 : 1
• Verify: (50 × 2 + 80 × 1) / 3 = 180 / 3 = 60; 60 × 1.1 = 66 ✓
• Trick: Always convert selling price to cost price BEFORE applying the cross.

Example 3: Concentration blend

• Problem: Acid solution at 20% and at 50%, blended to produce 30% concentration. Find the ratio.
• Cross: (50 - 30) : (30 - 20) = 20 : 10 = 2 : 1
• Verify: (20 × 2 + 50 × 1) / 3 = 90 / 3 = 30% ✓
• Speed note: Percentage concentrations work identically to prices. Same cross, same two subtractions.

Example 4: Water as the zero-value ingredient

• Problem: Pure milk worth Rs. 48/L is mixed with water (Rs. 0) and sold at Rs. 36/L. What fraction is water?
• Cross: p1 = 0, p2 = 48, mean = 36. (48 - 36) : (36 - 0) = 12 : 36 = 1 : 3
• Water fraction: 1 / (1 + 3) = 1/4
• Trick: When one ingredient is water or a pure diluent, set its value to 0. The cross still works.

Example 5: Ratio given, find the mean (reverse)

• Problem: Mix A at Rs. 30/kg and B at Rs. 50/kg in ratio 3:2. What is the mean price?
• Method: Use weighted average, not the cross. Mean = (30 × 3 + 50 × 2) / (3 + 2) = 190 / 5 = Rs. 38/kg
• Trick: The cross finds a ratio from a mean. When the ratio is given and you need the mean, use the weighted average directly. Recognising this switch saves 30 to 45 seconds.

Example 6: Average speed (cross gives time ratio, not distance)

• Problem: A person travels one leg at 30 km/h and another at 60 km/h. The overall average speed is 40 km/h. Find the distance ratio.
• Cross: (60 - 40) : (40 - 30) = 20 : 10 = 2 : 1
• What this means: The time ratio is 2:1 (2 hours at 30 km/h, 1 hour at 60 km/h). Distances are 60 km and 60 km, which are equal.
• Critical trap: The cross gives the TIME ratio when applied to speeds. Do not read it as the distance ratio. Distances are then computed from time × speed.

Time-Traps That Slow Down Test-Takers

Three mistakes account for the majority of wrong answers and lost time on alligation questions in campus tests.

Trap 1: Mean outside the ingredient range. The alligation cross only works when the mean value lies strictly between p1 and p2. If a problem states a mean that is higher than both ingredients or lower than both, the problem is either infeasible or you have read a value incorrectly. Stop, re-read the question, and check your numbers before computing.

Trap 2: Wrong diagonal assignment. The diagonal difference (p2 minus mean) belongs to the LOWER-value ingredient (p1), and (mean minus p1) belongs to the HIGHER-value ingredient (p2). Getting this backwards produces an inverted ratio. The mnemonic: the cross-piece belongs to the ingredient on the opposite side.

Trap 3: Selling price used as mean instead of cost price. Any time a question mentions profit or loss in the same sentence as a selling price, convert to cost price before running the cross. This is the most frequent single source of errors in price-blend questions.

Test-Day Strategy for Mixture Questions

Alligation-pattern questions appear consistently across campus tests:

• TCS NQT Numerical Ability: standard two-ingredient price blend and concentration problems. Difficulty is straightforward; these are high-confidence questions when you apply the cross correctly.
• AMCAT Quantitative: ratio-and-proportion overlap with alligation; expect both the forward (find ratio) and reverse (find mean) variants.
• Mu Sigma MuApt: speed is the primary constraint. The three-signal recognition method pays off here because it lets you categorise and commit to a method in under 10 seconds.
• D.E. Shaw preliminary quant round: harder format. Expect multi-step problems that combine a price-blend with a quantity constraint, or an alligation plus a replacement in sequence.

Attempt Order for Mixed Quant Sections

Take alligation questions in your first pass if the two ingredient values and the mean are all stated explicitly. Skip to your second pass if the mean must be derived (profit/loss strip, or an average-speed problem), since those require one more setup step and are more time-sensitive.

Alligation and time and work problems are the two formula-light clusters in the standard quant syllabus. Both reward pattern recognition over equation setup. Budget 60 to 90 seconds per alligation question in a campus placement aptitude test. More than two minutes on a single question usually signals a misidentified problem type.

The speed-trick in Example 6 (recognising that the cross gives a time ratio, not a distance ratio) is the kind of first-principles check that avoids a wrong answer in under 10 seconds. That same instinct transfers to LLM work: verifying what a model’s output represents before acting on it. At ₹299, TinkerLLM is where that checking habit meets live model calls, not just aptitude shortcuts.