Mixture and Alligation — SSC CGL Quantitative Aptitude

Alligation is weighted average run backwards: instead of combining two groups to find the mean, you're given the mean and asked for the ratio. The cross rule delivers that ratio in one step. The other half of the chapter is the repeated replacement formula — draw off some liquid, top up, repeat — which is just a geometric decay. Two tools, almost every question.

1. What SSC actually asks

Tier 1: 1–2 Q · Tier 2: 1–2 Q. Types: find the mixing ratio for a target price/concentration, find the mean price of a known blend, milk-and-water problems, repeated draw-and-replace, and disguised alligation (average age, average speed, profit percentages). The numbers are light; recognising which tool applies is the skill.

2. The alligation rule

To mix a cheaper ingredient (cost $c$) and a dearer ingredient (cost $d$) into a mixture of mean value $m$:

$\frac{\text{quantity of cheaper}}{\text{quantity of dearer}}=\frac{d-m}{m-c}$

Picture the cross: each ingredient's share is proportional to its distance from the mean on the far side.

• Rice at ₹30 and ₹40 blended to ₹34: ratio $=(40-34):(34-30)=6:4=3:2$ (more of the cheaper, since 34 is closer to 30).
• Water counts as an ingredient costing ₹0 — the key to "mix water to gain X%" problems.

3. Alligation is a general weighted-average tool

The two "ingredients" can be any averageable quantity — marks, ages, speeds, profit rates:

• Boys average 60 kg, girls 50 kg, class 54 kg → boys : girls $=(54-50):(60-54)=4:6=2:3$.
• A trader gaining 25% on one lot and 15% on another for an overall 18% → lot ratio $=(18-15):(25-18)=3:7$.

Whenever you see "two groups, one combined average, find the ratio", reach for the cross.

4. The replacement formula

Remove $x$ from a vessel of volume $V$ (pure liquid), refill with something else, repeat $n$ times. The pure liquid remaining is:

$\text{Final}=V{\left(\frac{V-x}{V}\right)}^n$

• 40 L pure milk, draw 4 L and refill with water, twice: milk $=40{\left(\frac{36}{40}\right)}^2=40\times 0.81=32.4$ L.
• The fraction ${\left(\frac{V-x}{V}\right)}^n$ is the milk concentration after $n$ rounds — read water off as the complement.

5. Direct milk–water ratios

• A ratio $a:b$ in volume $V$ splits as $\frac{a}{a+b}V$ and $\frac{b}{a+b}V$: milk : water $=3:1$ in 40 L means 30 L and 10 L.
• Adding pure water keeps the milk fixed and changes only the water part — set up the new ratio on the milk that never moved.
• A dishonest seller adding water and selling at cost price gains $\frac{\text{water}}{\text{milk}}$: a 25% "profit" means water : milk $=1:4$.

6. Solved PYQ-style examples

Q1. In what ratio must rice at ₹30/kg be mixed with rice at ₹40/kg to get a blend worth ₹34/kg? Solution. $(40-34):(34-30)=6:4=$ 3 : 2.

Q2. From 40 L of pure milk, 4 L is drawn and replaced with water; this is done twice. Milk remaining? Solution. $40(36/40{)}^2=40\times 0.81=$ 32.4 L.

Q3. In what ratio must water be mixed with milk so that selling the mixture at the cost price of milk yields a 20% gain? Solution. Mixture must cost $\frac{1}{1.2}=\frac{5}{6}$ of the milk price per litre. Water (0) and milk (1) to mean $\frac{5}{6}$: $(1-\frac{5}{6}):(\frac{5}{6}-0)=\frac{1}{6}:\frac{5}{6}=$ 1 : 5.

Q4. The average weight of boys is 60 kg and of girls 50 kg; the class average is 54 kg. The ratio of boys to girls is… Solution. $(54-50):(60-54)=4:6=$ 2 : 3.

Q5. 60 L of a mixture has milk and water in the ratio 2 : 1. How much water must be added to make the ratio 1 : 1? Solution. Milk 40 L, water 20 L. For 1 : 1 the water must equal 40 L, so add $40-20=$ 20 L.

7. Exam protocol

1. "Find the ratio given the mean" → alligation cross; "find the mean given the ratio" → weighted average.
2. Water is an ingredient of cost ₹0 — the hook for profit-by-dilution questions.
3. Repeated draw-and-refill → $V{\left(\frac{V-x}{V}\right)}^n$; the bracket is the surviving concentration.
4. Adding pure water fixes the milk; build the new ratio on the unchanged part.
5. The cheaper ingredient always gets the larger share when the mean sits closer to it — a built-in sanity check.