Alligation or Mixture

Mixing substances of different rates

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Alligation is a rule that enables us to find the ratio in which two or more ingredients at given prices must be mixed to produce a mixture of a desired price. It is the reverse of weighted average and is highly useful in solving mixture problems, especially those involving different concentrations, costs, or qualities being blended together.

Key Formulas

Alligation Rule: (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c)\text{Alligation Rule: (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c)}
Where: c = cost price of cheaper ingredient, d = cost price of dearer ingredient, m = mean price of mixture\text{Where: c = cost price of cheaper ingredient, d = cost price of dearer ingredient, m = mean price of mixture}
Mean Price (m) = (Total cost of mixture) / (Total quantity of mixture)\text{Mean Price (m) = (Total cost of mixture) / (Total quantity of mixture)}
When mixing two varieties: Quantity of A / Quantity of B = (Price of B - Mean Price) / (Mean Price - Price of A)\text{When mixing two varieties: Quantity of A / Quantity of B = (Price of B - Mean Price) / (Mean Price - Price of A)}
Replacement formula: If x litres of pure liquid is replaced with water n times, remaining pure liquid = Initial x (1 - x/Total)^n\text{Replacement formula: If x litres of pure liquid is replaced with water n times, remaining pure liquid = Initial x (1 - x/Total)\textasciicircum{}n}
For dilution problems: Amount of pure substance after n replacements = Original x ((Total - Replaced)/Total)^n\text{For dilution problems: Amount of pure substance after n replacements = Original x ((Total - Replaced)/Total)\textasciicircum{}n}

Key Concepts

Understanding the Alligation Cross

The alligation cross is a visual method where the cheaper price (c) is placed at the bottom left, dearer price (d) at the top left, and mean price (m) in the centre. The differences (d - m) and (m - c) are calculated diagonally to give the mixing ratio. This method is faster than algebraic equations for simple two-ingredient mixtures.

Types of Mixture Problems

1. Mixing two different priced items (grains, liquids) 2. Dilution problems (adding water to milk, acid solutions) 3. Successive replacement (removing and replacing liquid multiple times) 4. Mixing two different concentration solutions (acid-water mixtures) 5. Profit/loss in mixtures when selling at certain rates

The Replacement Formula

When a container has 'T' litres of pure liquid and 'x' litres is taken out and replaced with water, this process repeated 'n' times: Final pure liquid = T x (1 - x/T)^n. This exponential decay formula is crucial for repeated dilution scenarios. For example, if 10 litres is removed from 100 litres and replaced 3 times, pure liquid remaining = 100 x (0.9)^3 = 72.9 litres.

Weighted Average vs Alligation

Weighted average calculates the mean of mixed quantities: Mean = (q p + q p ) / (q + q ). Alligation reverses this - given the mean, it finds the ratio q :q . Alligation is essentially the weighted average formula rearranged to solve for ratios rather than the mean. Use alligation when you know the desired mean and individual prices; use weighted average when you know quantities and want the mean.

Milk and Water Problems

Common pattern: A vessel contains milk and water in ratio a:b. When x litres is removed and replaced with the other liquid, find the new ratio. Method: Calculate amount of milk/water removed, subtract from original, add replacement. For successive operations, apply the replacement formula. Example: In 60 litres with milk:water = 2:1 (40:20), if 10 litres milk added and 10 litres removed, new ratio calculation requires careful step-by-step tracking.

Solved Examples

Problem 1:

In what ratio must rice at Rs. 30 per kg be mixed with rice at Rs. 40 per kg so that the mixture is worth Rs. 35 per kg?

Solution:

Using alligation rule:

Cheaper (c) = Rs. 30, Dearer (d) = Rs. 40, Mean (m) = Rs. 35

Apply formula: (Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c)

= (40 - 35) : (35 - 30)
= 5 : 5
= 1 : 1

Answer: The rice must be mixed in the ratio 1:1.

Problem 2:

A merchant has 100 kg of sugar, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. Find the quantity sold at 18% profit.

Solution:

This is an alligation problem with profit percentages.

Profit on first part = 8%
Profit on second part = 18%
Overall profit = 14%

Using alligation:

(18 - 14) : (14 - 8) = 4 : 6 = 2 : 3

So the ratio of quantities = 2 : 3

Total parts = 2 + 3 = 5

Quantity at 18% profit = (3/5) x 100 kg = 60 kg

Answer: 60 kg was sold at 18% profit.

Problem 3:

A vessel contains 80 litres of milk. 8 litres of milk is taken out and replaced with water. This operation is done three times. Find the amount of milk remaining in the vessel.

Solution:

Using the replacement formula:

Initial milk = 80 litres
Amount replaced each time = 8 litres
Total in vessel = 80 litres
Number of operations (n) = 3

Formula: Final amount = Initial x (1 - x/T)^n

= 80 x (1 - 8/80)^3
= 80 x (1 - 0.1)^3
= 80 x (0.9)^3
= 80 x 0.729
= 58.32 litres

Answer: 58.32 litres of milk remains in the vessel.

Problem 4:

How many kilograms of tea powder costing Rs. 200 per kg must be mixed with 40 kg of tea powder costing Rs. 280 per kg to gain 25% by selling the mixture at Rs. 300 per kg?

Solution:

First, find the cost price of the mixture:

Selling price = Rs. 300 per kg
Profit = 25%

Cost price of mixture = 300 / 1.25 = Rs. 240 per kg

Now use alligation:

Cheaper tea = Rs. 200, Dearer tea = Rs. 280
Mean price = Rs. 240

Ratio = (280 - 240) : (240 - 200)
= 40 : 40
= 1 : 1

Since dearer quantity = 40 kg,
Cheaper quantity = 40 kg (same as ratio 1:1)

Answer: 40 kg of tea powder at Rs. 200 per kg must be mixed.

Tips & Tricks

  • Always identify which value is the 'mean' (target) price and which are the component prices before applying alligation.
  • For dilution problems with repeated removal and replacement, memorise and apply the formula: Final = Initial x (1 - x/T)^n to save time.
  • When dealing with percentages in alligation (like profit%, alcohol%, milk%), treat them as the 'prices' in the formula.
  • For three or more ingredients, solve using simultaneous equations rather than alligation - the cross method only works for two components.
  • In milk-water replacement problems, track only one component (usually milk) and calculate water as the remainder.
  • Check your answer by verifying: (Quantity x Price + Quantity x Price ) / (Quantity + Quantity ) = Mean Price

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