Mixtures & Alligations - Aptitude Test Tricks, Formulas & Concepts

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Every aptitude exam that you take will contain at least one question from mixtures and alligation. CAT, XAT, MAT, GMAT, Bank PO tests, campus placement tests do give a lot of focus on mixtures and alligations. Questions from this section are nothing but word problems on ratio and proportion or percentages or weighted averages. In this article we will have a look at top aptitude questions on Mixtures and Alligations, basic concepts, important formulas and how to solve Mixtures and Alligations problems easily and quickly.

There are different ways you can approach questions from mixtures and alligations : use direct formulas as suggested below or do calculation by converting the data into percentage form. Take a look at questions commonly asked from this section, you will see that the data easily fits into some pattern under percentages or ratios. Like any other quantitative aptitude section, the mantra to solve questions under this section is to solve as much questions as possible. 

Most students try to ignore questions from mixtures and alligations, simply because they find that there is too much content in the question or it will take time to solve it. Don't Worry, Here we will focus on the formulas that can be used to solve questions from mixtures and alligations easily. We will also tell you how we derived those formulas, for those who cannot remember the formulas. These derivations will tell you how to approach quantitative aptitude questions on mixtures and alligations.

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 eg: 1 liter of a mixture contains 250ml water and 750 ml milk. That means, ¼ of mixture is water and ¾ 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.

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What are some frequently asked questions on mixtures and alligations?

1. You will be given quantity of ingredients, and their price. You will be asked to find mean price of resultant mixture.
2. You will be given a desired quantity/price of mixture and also the price of ingredients. You have to find out in what ratio/percentage of ingredients should be mixed to get the desired quantity.
3. Find the resultant quantity when you concentrate/dilute the mixture.
4. Find the ratio in which 2 quantities should be mixed so that the resultant mixture can be sold mixture can be sold to gain x% profit.

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Core Concepts

  • 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 = qc,
Cost price of cheaper ingredient = pc,
Quantity of dearer or costlier ingredient = qd,
Cost price of costlier or dearer ingredient = pd.
Consider, mean price of mixture as pm and quantity of mixture as qm.
We know, qm = qc + qd
Then we get,
    (qc * pc + qd * pd) = qm * pm = (qc + qd) * pm
qc ( pm – pc) = qd (pd – pc)
qc / qd =  (pd – pc) / ( pm – pc)

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 p1 : q1, p2 : q2, ……, pn : qn 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 = p1 * x / (p1 + q1)
Quantity of P in vessel 2 = p2 * x / (p2 + q2)
Quantity of P in vessel n = pn * x / (pn + qn)... and so on
Similarly,
Quantity of Q in vessel 1 = q1 * x / (p1 + q1)
Quantity of Q in vessel 2 = q2 * x / (p2 + q2)
Quantity of Q in vessel n = qn * x / (pn + qn)... 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 x1, x2, …, xn are filled with the mixture of P and Q in the ratio p1 : q1, p2 : q2, ……, pn : qn and content of all these vessels are mixed in one large vessel, then
Quantity of P in vessel 1 = p1 * x1/(p1 + q1)
Quantity of P in vessel 2 = p2 * x2/(p2 + q2)
Quantity of P in vessel n = pn * xn/(pn + qn)... and so on
Similarly,
Quantity of Q in vessel 1 = q1 * x1/(p1 + q1)
Quantity of Q in vessel 2 = q2 * x2/(p2 + q2)
Quantity of Q in vessel n = qn * xn/(p+ qn)
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,

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