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Mixture and Alligation

Competitive Mathematics · Arithmetic

6 Part5 worked examples13 practice questions

A tea shop blends leaf at ₹300 a kg with leaf at ₹500 a kg to make a ₹400 blend. The question: in what ratio? Here the answer is obvious — equal. But what if the blend must be ₹420? This is where alligation helps: a small cross-subtraction table that gives the ratio without writing equations. Its magic is that it works on anything “per unit” — price, per cent, speed, average, profit.

The alligation formula ( cheaper, dearer, mean)
1

Mean price and alligation

Cross subtraction

The mean price of a blend and the ratio in which cheap and dear are mixed.

Mean price

Always between the two prices

The cost per unit of the blend — total cost ÷ total quantity.

The mean price of a blend is dearer than the cheap item and cheaper than the dear one. It is the weighted mean from the average chapter: . The more of an item, the closer the mean to its price.

Example 1 · Basic
20 kg of tea at ₹300/kg and 30 kg at ₹400/kg are blended. What is the mean price?
Solution, step by step
  1. 1Total cost ; total quantity .
  2. 2Mean
Answer ₹360
BasicTry it yourself first
Tea of 10 kg at ₹40/kg, 30 kg at ₹60/kg is blended. What is the mean price per kg of the blend?
StandardTry it yourself first
Tea of 24 kg at ₹35/kg, 16 kg at ₹55/kg, 20 kg at ₹60/kg is blended. What is the mean price per kg of the blend?
AdvancedTry it yourself first
Tea of 35 kg at ₹48/kg, 25 kg at ₹72/kg, 40 kg at ₹60/kg is blended. What is the mean price per kg of the blend?
StandardTry it yourself first
20 kg of sugar at ₹24/kg and 30 kg at ₹32/kg are mixed and the blend sold at ₹33/kg. What is the gain or loss per cent?
AdvancedTry it yourself first
40 kg of sugar at ₹36/kg and 60 kg at ₹46/kg are mixed and the blend sold at ₹48/kg. What is the gain or loss per cent?

The alligation method

Cheaper : dearer

Three prices, one ratio — and the quantity when the total is given.

Method: write the cheaper price and the dearer price , with the target mean in the middle. Subtract crosswise: gives the cheaper item’s share and the dearer’s. The ratio always puts more of the item whose price is nearer the mean.

CheckCheck: to make a ₹440 blend from ₹400 and ₹500, 440 is nearer ₹400, so more of the ₹400 item goes in. The ratio shows exactly that.
Example 2 · Basic
In what ratio must rice at ₹45/kg and ₹60/kg be mixed to get ₹48/kg?
Solution, step by step
  1. 1Cheaper’s share ; dearer’s .
  2. 2Ratio
Answer 4 : 1
Example 3 · Standard
Two varieties of rice at ₹36 and ₹52/kg are mixed to make 80 kg worth ₹46/kg. How many kg of each?
Solution, step by step
  1. 1Ratio .
  2. 2Cheaper kg; dearer kg
Answer 30 kg and 50 kg
Example 4 · Advanced
Two varieties at ₹50 and ₹80 per kg are mixed and sold at ₹68.40 per kg, giving a 14% profit. In what ratio were they mixed?
Solution, step by step
  1. 1The blend’s cost price per kg.
  2. 2Cheaper : dearer .
Answer 2 : 1
First get the cost from the selling price, then alligate.
Example 5 · Standard
In what ratio must tea at ₹95/kg be blended with tea at ₹70/kg to get a blend worth ₹80/kg?
Solution, step by step
  1. 1Cheaper : dearer .
  2. 2
Answer 3 : 2
BasicTry it yourself first
In what ratio must rice at ₹30/kg and ₹50/kg be mixed to get a mixture worth ₹42/kg?
BasicTry it yourself first
In what ratio must rice at ₹24/kg and ₹40/kg be mixed to get a mixture worth ₹36/kg?
StandardTry it yourself first
In what ratio must rice at ₹18.5/kg and ₹24.5/kg be mixed to get a mixture worth ₹22/kg?
StandardTry it yourself first
In what ratio must rice at ₹64/kg and ₹96/kg be mixed to get a mixture worth ₹72/kg?
AdvancedTry it yourself first
In what ratio must rice at ₹275/kg and ₹425/kg be mixed to get a mixture worth ₹335/kg?
BasicTry it yourself first
Two varieties of rice at ₹30/kg and ₹50/kg are mixed to make 40 kg worth ₹42/kg. How many kg of the cheaper variety are used?
StandardTry it yourself first
Two varieties of rice at ₹36/kg and ₹52/kg are mixed to make 160 kg worth ₹46/kg. How many kg of the cheaper variety are used?
AdvancedTry it yourself first
Two varieties of rice at ₹120/kg and ₹180/kg are mixed to make 300 kg worth ₹150/kg. How many kg of the cheaper variety are used?
2

Alligation on per cents, profits and concentrations

Anything “per unit” in place of price

Alligation also works when the quantity is not a price but a per cent or a profit.

Concentrations of solutions

A per cent is also a price

The per cents of two solutions and their mixing ratio; adding the pure substance (100%) or water (0%).

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Profit per cent and average speed

Selling parts at different profits

When part of the stock sells at one profit and the rest at another, the overall profit.

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3

Draw-and-refill, ratio changes and vessels

Stories after the mixture is made

Repeated draw-off and refilling, ratio changes on adding water, pouring two vessels together and the dishonest milkman.

Repeated draw-and-refill

The fraction multiplies each time

Draw off litres from litres of pure liquid, refill with water, and repeat times.

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Changing ratios and two vessels

Hold the unchanged item steady

Milk stays fixed when water is added; for two vessels add the real quantities.

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4

Traps and fast tricks

Slips · tricks

Where marks go in mixtures.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Checks and reverse working

Habits that raise your confidence in the answer.

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5

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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6

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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