#### MIXTURE & ALLIGATION CONCEPT

**Concept | Practice set 1 | Practice set 2**

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#### Mixture and Alligation

**DETAILS:** Mixture and Alligation concept

- Mixture or alloys contains two or more ingredients of certain quantity mixed together to get a desired quantity. The quantity can be expressed as a ratio or percentage.
*For Ex: 1 liter of a mixture contains 250ml water and 750 ml milk. That means, 1/4 of mixture is water and 3/4 of mixture is milk. In other words, 25% of mixture is water and 75% of mixture is milk.* - Alligation is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of desired price. The cost price of unit quantity of such a mixture is called its Mean Price. Remember the rule that
*cost price of costlier ingredient > cost price of mixture > cost price of cheaper ingredient*.

##### Important formulas and shortcuts for mixtures and alligations

**1) Rule Of Alligation**

Given , Quantity of cheaper ingredient = **q _{c}**,

Cost price of cheaper ingredient =

**p**,

_{c}Quantity of dearer or costlier ingredient =

**q**,

_{d}Cost price of costlier or dearer ingredient =

**p**.

_{d}Consider, mean price of mixture as

**p**and quantity of mixture as

_{m}**q**.

_{m}We know,

**q**

_{m}= q_{c}+ q_{d}Then we get,

(q

_{c}* p

_{c}+ q

_{d}* p

_{d}) = q

_{m}* p

_{m}= (q

_{c}+ q

_{d}) * p

_{m}

**→**q

_{c}( p

_{m}– p

_{c}) = q

_{d}(p

_{d}– p

_{c})

**→**q

_{c}/ q

_{d}= (p

_{d}– p

_{c}) / ( p

_{m}– p

_{c})

**Thus we get the important relation for alligation as** _{}

An easy way to remember this relation,

**2) Quantity of ingredient to be added to increase the content of ingredient in the mixture to y%**

If P liters of a mixture contains x% ingredient in it. Find the quantity of ingredient to be added to increase the content of ingredient in the mixture to y%.

Let the quantity of ingredient to be added = Q liters

Quantity of ingredient in the given mixture = x% of P = x/100 * P

Percentage of ingredient in the final mixture = Quantity of ingredient in final mixture / Total quantity of final mixture.

Quantity of ingredient in final mixture = [x/100 * P] + Q = [ P*x + 100 * Q] / 100

Total quantity of final mixture = P + Q **→** y/100 = [[ P*x + 100 * Q] / 100]/[P + Q] **→ **y[P + Q] = [P*x + 100 * Q]

**The quantity of ingredient to be added**

**3) If n different vessels of equal size are filled with the mixture of P and Q**

If n different vessels of equal size are filled with the mixture of P and Q in the ratio p_{1} : q_{1}, p_{2} : q_{2}, ……, p_{n} : q_{n} and content of all these vessels are mixed in one large vessel, then

Let x liters be the volume of each vessel,

Quantity of P in vessel 1 = p_{1} * x / (p_{1} + q_{1})

Quantity of P in vessel 2 = p_{2} * x / (p_{2} + q_{2})

Quantity of P in vessel n = p_{n} * x / (p_{n} + q_{n})... and so on

Similarly,

Quantity of Q in vessel 1 = q_{1} * x / (p_{1} + q_{1})

Quantity of Q in vessel 2 = q_{2} * x / (p_{2} + q_{2})

Quantity of Q in vessel n = q_{n} * x / (p_{n} + q_{n})... and so on

Therefore, when content of all these vessels are mixed in one large vessel, then

**Quantity of P / Quantity of Q = Sum of quantities of P in different vessels / Sum of quantities of Q in different vessels**

**4) If n different vessels of sizes x1, x2, …, xn are filled with the mixture of P and Q**

If n different vessels of sizes x_{1}, x_{2}, …, x_{n} are filled with the mixture of P and Q in the ratio p_{1} : q_{1}, p_{2} : q_{2}, ……, p_{n} : q_{n} and content of all these vessels are mixed in one large vessel, then

Quantity of P in vessel 1 = p_{1} * x_{1}/(p_{1} + q_{1})

Quantity of P in vessel 2 = p_{2} * x_{2}/(p_{2} + q_{2})

Quantity of P in vessel n = p_{n} * x_{n}/(p_{n} + q_{n})... and so on

Similarly,

Quantity of Q in vessel 1 = q_{1} * x_{1}/(p_{1} + q_{1})

Quantity of Q in vessel 2 = q_{2} * x_{2}/(p_{2} + q_{2})

Quantity of Q in vessel n = q_{n} * x_{n}/(p_{n }+ q_{n})

Therefore, when content of all these vessels are mixed in one large vessel

**Quantity of P / Quantity of Q = Sum of quantities of P in different vessels / Sum of quantities of Q in different vessels**

**5) Quantity of ingredient to be added to change the ratio of ingredients in a mixture**

In a mixture of x liters, the ratio of milk and water is a : b. If the this ratio is to be c : d, then the quantity of water to be further added is:

In original mixture

Quantity of milk = x * a/(a + b) liters

Quantity of water = x * b/(a + b) liters

Let quantity of water to be added further be w litres.

Therefor in new mixture:

Quantity of milk = x * a/(a + b) liters → Equation(1)

Quantity of water = [x * b/(a + b) ] + w liters → Equation (2)

→ c / d = Equation (1) / Equation (2)

**Quantity of water to be added further, **_{}