Mixture and Alligation Questions for SSC CGL

The Alligation Rule (Cross-Difference Method)

Alligation is a shortcut to find the ratio in which two ingredients must be mixed to get a mixture of a desired mean value. The mean value can be price, concentration, speed, or any rate. You place the cheaper value on the left, the dearer value on the right, and the mean in the middle. Then cross-subtract: (Mean - Cheaper) and (Dearer - Mean). These differences give the ratio of mixing.

• Ratio = (Dearer - Mean) : (Mean - Cheaper)
• Works for price, percentage concentration, profit percentage, average speed, etc.
• Mean value must always lie BETWEEN the two ingredient values, else no valid mixture exists.
• If three or more items are mixed, apply alligation pairwise.
• The result is always a ratio, not an absolute quantity, unless total is given.

Replacement of Mixture (Repeated Dilution)

A classic SSC CGL question type: a container has pure liquid. Some quantity is removed and replaced with another liquid (like water). This is repeated multiple times. After each replacement, the concentration of the original liquid decreases. The formula captures how much pure liquid remains after n such replacements.

• After each replacement, the fraction of original liquid reduces by the factor (1 - removed/total).
• Applies even when the container already has a mixture before replacement.
• Very common with milk-water questions in SSC CGL.
• If question asks for water added, subtract pure liquid remaining from total volume.
• n replacements compound the reduction, so even 2-3 replacements drop concentration sharply.

Mixing Three Ingredients

When three items are mixed, SSC CGL either gives ratios and asks for mean, or gives the mean and asks for a condition. Use pairwise alligation: fix one ingredient as common and apply alligation between (item1, item3) and (item2, item3) separately. Then combine the two pairs to get a three-way ratio.

• Fix the third item as the reference, apply alligation twice.
• Alternatively, set up a weighted average equation if two ratios are given.
• Be careful to identify which ingredient plays the role of cheaper vs. dearer in each pair.
• This type is less frequent than two-ingredient mixing but appears in CGL Tier 1 occasionally.
• Shortcut: if two of three are at same price, treat them as one combined ingredient.

Profit and Partnership Angle of Alligation

SSC CGL sometimes disguises alligation as a profit or partnership problem. For example: a merchant mixes two varieties to earn a certain profit percentage — this is the same as finding the mean cost price. Similarly, questions about average salary, average marks, or average speed of groups follow the same alligation logic. Recognise the structure: two groups, two values, one target mean.

• Alligation works for ANY weighted average problem, not just price or concentration.
• Speed-distance: two trains/buses at different speeds averaging to a combined mean speed — use alligation.
• Marks/salary: two groups with different averages combining to one overall average.
• Always check: is the mean given between the two extremes? If not, the problem is set up incorrectly.
• The ratio found is always (dearer-mean):(mean-cheaper), no exceptions.

• Swapping the ratio: many aspirants write (Mean - Cheaper):(Dearer - Mean) instead of (Dearer - Mean):(Mean - Cheaper). Remember: the BIGGER value gets the SMALLER proportion if mean is closer to it.
• In replacement problems, forgetting to raise the fraction to the power n. They calculate (1 - 8/40) once and multiply by n instead of raising to the nth power.
• Mixing up which quantity corresponds to which ingredient after the cross-difference. Always label Cheaper on the left, Dearer on the right BEFORE subtracting.
• Treating water as having cost Rs 0 but then forgetting to include it in the total volume when computing mean price, leading to a wrong denominator.
• In three-ingredient problems, trying to apply alligation all at once instead of pairwise — leads to wrong ratios.

• The CROSS trick: draw a big X. Put Cheaper top-left, Dearer top-right, Mean in the centre. Subtract along each diagonal: bottom-right = Mean minus Cheaper, bottom-left = Dearer minus Mean. Read bottom-left:bottom-right as Cheaper qty : Dearer qty.
• Mnemonic DM-MC: Dearer Minus Mean gives quantity of Cheaper. Mean Minus Cheaper gives quantity of Dearer. (D-M = C qty, M-C = D qty). Counterintuitive but correct — the farther a value is from the mean, the less of it you need.
• Replacement formula memory hook: TANK formula. Final Pure = Total x (Keep/Total)^n where Keep = Total minus Removed. Like a bank applying compound interest in reverse.
• Alligation is just the seesaw rule: the mean is the fulcrum. The heavier (larger quantity) side must be closer to the fulcrum (mean). So if mean is closer to cheaper, you need MORE of the cheaper.

• SSC CGL Tier 1 typically has 1-2 direct alligation questions (mixing two grades, find ratio) and 1 replacement question. Tier 2 can push this to 3-4 questions including disguised weighted average problems.
• Most questions can be solved in under 60 seconds if you draw the cross/X diagram immediately instead of writing equations. Practice the visual method for speed.
• Watch for the word 'profit' in mixture problems — the merchant may buy at two prices and sell at one. Convert to cost price context, then apply alligation on costs.
• If the answer choices are simple ratios like 2:3 or 3:5, verify by back-calculating the mean — this takes 10 seconds and saves you from silly errors under exam pressure.
• Recent CGL papers (2023-2024) had replacement questions with n=2 or n=3 replacements. Always write out the formula before plugging in numbers to avoid power errors.