# Introduction

• Mixture, Alligation and Alloy questions routinely appear in Aptitude exams for Government jobs, Bank PO (IBPS) andĀ· MBA entrance exams (CMAT, CAT).
• The concept is very easy, You can master this concept, after barely 2 hours practice, unless you try to complicate it by yourself by mugging up the formulas from R.S.Agarwalās (Most bogus) book on Quantitative aptitude.

## A typical problem runs like this

• there is a cheap liquid (water), and there is an expensive liquid (Milk, Wine)
• sometimes, instead of liquid, solids are given: rice / wheat of different variety, gold, silver, zinc and iron alloys etc.
• both are given in different quantities, and mixed together. You are asked to calculate the concentration of final mixture or its selling price.
• Or you can be asked to find the amount of water or milk to be added in given mixture to bring the concentration Ā·to 50% etc.
• The same concept can be extended for finding average speed, average height, interest rates etc.
• Mathematically speaking, this is a problem of āweighted averageā

# Case: Easy Wine and Water mixture Problem

Ā·

• you have a big bottle of Blue water 10 lit.
• and youāve a glass of Pure Red wine 2 lit. (assuming that a glass can be that big !)
• You mix them both, and you get a purple colour solution. (10+2=12 lit)
• (Mixture) what is the concentration of wine in the final mixture?

## Step 1: Arrange them in a straight line

Ā·

• First arrange these three bottles in a straight line, in ascending order of their price / quality / concentration.
• Water is cheapest, and wine is costliest. The Mixture is going to be not as cheap as water and not as costly as wine, so we put it in between these two bottle.

(alternative logic:), since weāve to find the concentration of wine, : Water bottle has 0% wine in it, so in terms of concentration it has 0% wine. So it goes in the left, wine is pure, so it has 100% wine, and the mixture will have wine between the range of 0 to 100%.

## Step 2: Write down the Quality /concentration /price

• On the head (top) of each item, write down its concentration, price, speed (Whatever is given in the sum).
• Since we are asked about wineās concentration in mixture, it looks like this
• Blue Water =0% wine
• Purple Mixture= m% wine (weāve to find out)
• Red glass= 100% wine

## Step 4: Divide and Rule!

Now our table looks like this

You donāt need to mug up any formulas for ācheaper liquid and expensive liquidā. Just remember this āvisual-moveā

100 minus m = 10 litā¦..(i)
Same way

(M minus 0) =2 litā¦..(ii)
We cannot do (0 minus M) because M is concentration of wine in the final mixture. It is bigger than 0% and smaller than 100%.

Matter is over. Sum is solved. Game Finished.
Just divide equation (i) from equation (ii) or you can do reverse divide (ii) from (i), youāll get the value of ām%ā in either case. See this image for actual calculation

Final answer= the concentration of Wine in given mixture is 16.66%
Which also means, concentration of Water in this given mixture
=100 ā 16.6% ;because conc. Of water + conc. Of wine =100%
=83.4% water.
Ofcourse real exam questions wonāt be this easy so lets take a few cases.

# Case: Average price and profit after mixing two solutions

16 lit. of Soda is mixed with 5 lit. of Wine. Price of this Soda is Rs.12 / lit and price of wine Rs.33/lit. What is the average price of this mixture.
And if the bartender wishes to make 25% profit on his investment, at what price should he sell this mixture?

Our method remains the same. Arrange them in ascending order, put values on the top and bottom of each item. Itāll look like this

Now do the visual move. And you get two equations
33 minus m = 16
M minus 12 = 5 ; very important. Donot make mistake. M is bigger than 12.
Divide them
(33-m)/(m-12)=16/5

• Manually solving this equation
• (33-m)x5=16(m-12)
• Now you can manually solve this equation to get the value of m, but as You can understand, this can be time consuming method to solve equation because weāve to multiply 33 with 5 and 16 with 12 and then do addition, subtraction ā might make mistake in calculation.
• So better apply the

## Sidenotes

• Answer for average must be between the two extremes: 12 and 33.
• So if you get the answer outside this [12-33] range, know that youāve made mistake somewhere in calculation.
• Whenever possible, do this componendo method to solve the sum quickly without making mistakes in lengthy multiplications. At time youāll have to use āDividendoā principle i.e. same but instead of adding (+) bottom to top, you subtract (-) bottom from top.

## Back to the question:

• Price of Final mixture is 17 Rs. Per litre. But this is the ācost-priceā for the bartender. HE wants to make 25% profit on this.
• What is 25% of 17? = 17 x (25/100) = Rs.4.25
• So his Selling price = Cost price + profit of 25% =17 + 4.25 = 21.25

## Quicker method for profit calculation

• 25% =25/100 = 1/4
• You add this one fourth part to the total one part of given cost price.
• So 1 plus Ā¼ =5/4 parts.
• Multiply (5/4) with 17 and you get 21.25 = our selling price.
• Final Answer : Bartender should sell this mixture at 21.25 rupees per liter, if he wants to make 25% profit.
• Same way, if he had asked to find selling price for 50% profit, multiply 17 with 3/2. (because 50%=1/2)

# Case: Add water to decrease the concentration (dilution)

A 75 liter mixture of wine and water contains 80% wine. How much water should be added to decrease the concentration of wine to 75%?

• We are already given a mixture and weāve to add water and create a new diluted mixture.
• Assume that purple bottle contains this 75 liter mixture of 80% wine.
• Weāll add āvā liters of pure water into this purple bottle to dilute it and get the middle mixture of 75% concentration.
• The process is same, arrange them in ascending order, and then add values at top and bottom.

Now apply the visual move:
80-75=V lit. āeq.(1)
75-0=75 lit āeq.(2)
Very easy, divide eq. 1 with 2 and you get V=5 liters directly.

# Case: Average profit or interest Rate

The owner of Gada Electronics, Jethalal sold total 108 mobiles of two companies last month. Samsung at 36% profit and Nokia at 9% profit. If he made total 17% profit on total sales of these mobile phones, how many Samsung phones did he sell?

It is same wine and water problem, but instead weāve phones.
108 phones = total volume of final mixture containing wine + water.
Our procedure remains the same, first arrange them in a straight line, in ascending order of their profit (Value or whatever).
Assume that weāve āVā number of Samsung mobiles
Since total mobiles =108 = Samsung + Nokia,
hence Nokia mobiles = 108-V

Now do the visual method:
36-17=108-V
17-9=V

Divide them and apply componendo principle of ratio

Final Answer= Jethalal soldĀ  32 samsung mobiles.
Which also means he sold 108 minus 32 =76 Nokia phones.

# Case: average weight of group

400 people live in Gokuldham society. Average weight of men is 80kg and women is 60 kg. If the average weight of all people combined is 65 kg, how many women live in this society ?

First arrange them in a straight line, in ascending order of their average weight.

• Assume that Number of women = V. and since total residents are 400, men are 400-V.
• The the simple Visual step and division of two equations
• (80-65)/(65-60)=v/(400-v)
• Solve it and you get