Mixture & Alligation · Topic 1 of 6

⚖️ The Alligation Rule

3 exam question types, fully solved

Alligation is just a fast way to do a weighted average backwards. When two things at different prices are blended into one mixture, the cross rule hands you the mixing ratio in seconds — no equations, no and .

The alligation cross

Write the cheaper price on the left, the dearer price on the right, and the mean (mixture) price in the middle. Then cross-subtract:

Each ingredient's share is the distance from the other price to the mean — so the ratio comes out as the inverse of the distances. The ingredient that sits nearer the mean is needed in the larger amount.

• Mean from a ratio: mix in ratio ⇒ mean .

• Same idea, any quantity: ages, salaries and marks alligate exactly like prices.

• Midpoint shortcut: if the mean is exactly halfway between the two prices, the ratio is always .

⚖️Alligation LabCross rule & replacement

Cheaper (c)

Dearer (d)

Mean (m)

Try a classic

cheaper : dearer=6 : 4=3 : 2

Mix 3 parts cheaper with 2 parts dearer

Distance intuition

m sits CLOSER to the cheaper (gap 4 vs 6) → the mix is mostly CHEAPER. Each ingredient's share is the distance to the OTHER value — the inverse-distance rule.

Mix ₹20 and ₹30 rice to a mean of ₹26: distances are and , so cheaper : dearer .

Q1In what ratio must rice at ₹50/kg and ₹70/kg be mixed to get a mixture worth ₹65/kg?

Q2Two sugars at ₹30/kg and ₹45/kg are mixed in the ratio 2:3. The average cost per kg is:

Q38 kg of ₹15/kg tea is mixed with 4 kg of another tea to make a mixture worth ₹20/kg. The dearer tea costs:

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Real exam questions — The Alligation Rule

3 question types · 9 solved examples from real SSC papers

One tool runs this whole chapter — the alligation cross. Put cheaper left, dearer right, the mean in the middle, and read the ratio off the cross-distances.

How to solve this type

This is the heart of the whole chapter, the alligation cross. Put the cheaper price on the left, the dearer price on the right, and the mean (mixture) price in the middle. The rule is:

\dfrac{\text{Cheaper quantity}}{\text{Dearer quantity}}=\dfrac{\text{Dearer}-\text{Mean}}{\text{Mean}-\text{Cheaper}}

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In words: each ingredient's share is the distance from the OTHER ingredient to the mean. The ingredient nearer the mean is needed in the larger amount, so the ratio comes out as the cross-distances — and it is always the inverse of the distances.

In what ratio must two types of rice worth ₹60 per kg and ₹80 per kg be mixed so that the average cost of the mixture is ₹75 per kg?

A1 : 3B1 : 2C2 : 3D3 : 2

A merchant mixes two types of tea priced at ₹300 per kg and ₹240 per kg to sell the mixture at ₹270 per kg without profit or loss. In what ratio are they mixed?

A1 : 2B1 : 1C2 : 1D3 : 2

In what ratio must a grocer mix two varieties of pulses costing ₹15 and ₹20 per kg so as to get a mixture worth ₹18 per kg?

A3 : 2B3 : 4C2 : 3D2 : 5