[Aptitude] Alligations,Mixtures,Alloys: Water,Milk,Wine mixing: WeightedAverage Made Easy for CSAT,CAT,CMAT,IBPS Aptitude

1. Easy Wine and Water Mixing Problem2. Two Solutions: Selling Price of Wine and Profit3. Dilution by adding Water in the Wine4. Average Profit / Interest Rate: Jethalal’s Mobile shop5. Average weight of Group : Gokuldham society’s case6. Recommended Booklist7. Previous Articles

Introduction

Mixture, Alligation and Alloy questions routinely appear in Aptitude exams forGovernment jobs, Bank PO (IBPS) and· MBA entrance exams (CMAT, CAT).The concept is very easy, You can master this concept, after barely 2 hourspractice, unless you try to complicate it by yourself by mugging up the formulasfrom R.S.Agarwal’s (Most bogus) book on Quantitative aptitude.

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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 tocalculate 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 givenmixture to bring the concentration ·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 “weighted average”Let us start with a very basic (and easy) problem

Case: Easy Wine and Water mixture Problem

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you have a big bottle of Blue water 10 lit.and you’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

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

(alternative logic:), since we’ve to find the concentration of wine, : Water bottle has0% wine in it, so in terms of concentration it has 0% wine. So it goes in the left, wineis pure, so it has 100% wine, and the mixture will have wine between the range of 0 to100%.

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 wine’s concentration in mixture, it looks like thisBlue Water =0% winePurple Mixture= m% wine (we’ve to find out)Red glass= 100% wine

Step 3: Write down the weight / volume.

Step 4: Divide and Rule!

Now our table looks like this

You don’t need to mug up any formulas for ‘cheaper liquid and expensive liquid’. Justremember this ‘visual-move’

100 minus m = 10 lit…..(i)Same way

(M minus 0) =2 lit…..(ii)We cannot do (0 minus M) because M is concentration of wine in the final mixture. Itis 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),you’ll get the value of “m%” 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 – 16.6% ;because conc. Of water + conc. Of wine =100%=83.4% water.Ofcourse real exam questions won’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 / litand 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 whatprice should he sell this mixture?

Our method remains the same. Arrange them in ascending order, put values on the topand bottom of each item. It’ll look like this

Now do the visual move. And you get two equations33 minus m = 16M 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 canunderstand, this can be time consuming method to solve equation because we’veto multiply 33 with 5 and 16 with 12 and then do addition, subtraction – mightmake mistake in calculation.So better apply the

“Componendo” principle of ratio proportion

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 you’ve mademistake somewhere in calculation.Whenever possible, do this componendo method to solve the sum quicklywithout making mistakes in lengthy multiplications. At time you’ll have to use“Dividendo” principle i.e. same but instead of adding (+) bottom to top, yousubtract (-) bottom from top.

Back to the question:

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

Quicker method for profit calculation

25% =25/100 = 1/4You add this one fourth part to the total one part of given cost price.So 1 plus ¼ =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 hewants to make 25% profit.Same way, if he had asked to find selling price for 50% profit, multiply 17 with3/2. (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 watershould be added to decrease the concentration of wine to 75%?

We are already given a mixture and we’ve to add water and create a new dilutedmixture.Assume that purple bottle contains this 75 liter mixture of 80% wine.We’ll add ‘v’ liters of pure water into this purple bottle to dilute it and get themiddle mixture of 75% concentration.The process is same, arrange them in ascending order, and then add values at topand bottom.

Now apply the visual move:80-75=V lit. –eq.(1)75-0=75 lit –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 twocompanies last month. Samsung at 36% profit and Nokia at 9% profit. If hemade total 17% profit on total sales of these mobile phones, how manySamsung phones did he sell?

It is same wine and water problem, but instead we’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 ascendingorder of their profit (Value or whatever).Assume that we’ve “V” number of Samsung mobilesSince total mobiles =108 = Samsung + Nokia,hence Nokia mobiles = 108-V

Now do the visual method:36-17=108-V17-9=V

Divide them and apply componendo principle of ratio

Final Answer= Jethalal sold 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 andwomen 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 are400-V.The the simple Visual step and division of two equations(80-65)/(65-60)=v/(400-v)Solve it and you getNumber of ladies (v)=300.This answer seems plausible too, because the average weight 65 is closer to 60,that means more number of women in the weighting scale, so the balance shifttowards their side. because Men are only 100.

Case : Average speed.

Will be covered under separate article on TSD (Time, Speed, distance)

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URL to article: http://mrunal.org/2012/03/aptitude-alligationsmixturesalloys.html

Posted By Mrunal On 20/03/2012 @ 19:33 In the category Aptitude