Averages and Mixtures Demystified - Road to GMAT Q51

Averages and Mixtures Demystified

Most of us must have at least once in our lives spent some time on the see-saw with our friends. How about using them to solve the problems of Mixtures and Alligation, and Averages. Yes, you heard it right. Math is fun after all. Let us rewind our lives to the point when we used to have fun on the seesaws.

I was quite skinny when I was young (don’t ask me what I look like nowJ). Consequently, I had to use the long handle to balance the “normal” guy on the other side.

Fig

·       

In fig 1,

 a)   A is heavier than B (that’s me).

 b)   L1=L2

Clearly the see-saw is unbalanced.  In order to balance the seesaw. Either B must go further away from the support or fulcrum, which essentially means going out the frame, or A must come closer to fulcrum. See fig 2. 

What is the underlying physics behind this? The equation that governs where one needs to sit in order to balance the seesaw about the support is

(a)  x (L1) = (b)  x  (L2)…………….  Eq (1)

Where

     A, B= weights of A and B respectively

     L1 and L2= Distance of A and B from the fulcrum. 

Now, you can think of the fulcrum or the support as the point where the average weights of A and B is balanced. Let’s call the average weight as ‘M’. If either the length or the weight of our guy A is increased. The equilibrium is disturbed. B must move further from M to maintain M at the same place, or increase his weight in some way to maintain Eq 1.

Now what does this all have to do with averages or mixtures?

Well, we can use Eq 1 and its implications to solve problems of averages and mixtures. Let’s focus on averages first.

    Consider this problem

Q1.) What is the average weight of a fruit bag containing 1 orange of 100gms and 1 apple of 200gms? (Consider the weight of the bag negligible)

It is easy to solve average M= (100+200) /2 = 150gms 

Let us use the graphical implications of the problem. M=150;

And the equation= L1xA=L2xB

                                        Or

                                   50x1=50x2;

What if we didn’t know the value of M? can we still use Eq 1. Well that’s the plan!

                                           (M-150)x1= (200-M)x1

=> 2M=300

=>   M=150

Are we making matters difficult for ourselves? No, we are making it easier. How? Let’s me add complications to the question we just discussed.

Q2.) What is the average weight of a fruit bag containing 10 orange of 100gms and 5 apple of 200gms? (Consider the weight of the bag negligible)

Using Eq 1.  

(Why 200-M and not M-100 you may ask? Well Average can’t be greater than the maximum value of an individual element)

  ð  (M-100)x10= (200-M)x5

  ð  15M=2000

  ð  M=133.33

Now, how do we use this to solve slightly difficult problems? Let’s take a slightly difficult problem.

Q3)  A batsman scores with an average of 49 runs in 99 innings. How many should he score in his 100^th match to take his career average to 50 runs?

To solve this question let us introduce two more variables in place of 100gms and 200gms used in Q2 and Q3. Let’s call them n1 and n2

So

L1=M-n1

L2=n2-M

The eq1 becomes

L1xA=L2xB

=>            (M-n1)x A = (n2-M)xB

And by extension

…………………eq2

……………………………………. using the identity – [a/b]=[c/d] =[(a+c)/(b+d)]

Let us look at the question now armed with eq 2 

So we have by virtue of Eq 2

Solving, you can get n2=149;

Another way to look at this is the extra runs, n2-49, has to be distributed among all the innings (99+1) in such a way that that the average is 50. So he needs 1 more run for each of his 99 innings to compensate for the average. And 50 more in the last inning to maintain the average. Or he needs to score

                              99 +50 =149.

There are many variants of Eq 2 that you may find in literature. However, I like to write them down in a full equation and solve.

Mixtures

How can we use the concept developed for averages in mixtures problems? Well they are not much different to be honest.

Let’s look at a 600 level (according to GMATClub) problem to understand this

Q1) Krisp cereal is 10% sugar by weight. Brano cereal is 2% sugar by weight. If you wanted to make the mixture 4% sugar by weight what should the ratio of Krisp to Brano.

so we have 

Or 6/A=2/B

A/B = 3:1

Note- An interesting thing to note here is that, if A=B, then the average will be 6%. But in the question the average is 4%. 4% is closer to n1 or 2%. So A must be greater than B to balance out. Recall the see-saw, where to counter my heavier friend I needed a longer handle (or Torque for the engineers out there!). Consequently A/B will be greater than 1. If the question asked us to find the ratio if the average % was 8% (closer to n2). Then A/B would have been less than 1. Actually A/B = 1:3. (You can imagine the central point or the strength of the solution as the center or the fulcrum of the see-saw and imagine the equilibrium conditions). This is a very important observation. If a question like this appears on the test. Chances are that you will get half of the ratios less than one and half greater than one. You can eliminate half of the options using this simple analysis. Saves time! J

Let’s take a 700 level question

Q2) A tank holds x gallons of a saltwater solution that is 20% salt by volume. One Fourth of the water is evaporated, leaving all of the salt. When 10 Gallons of water and 20 gallons of salt are added, the resulting mixture is 33 1/3 % salt by volume. What is the value of x?

A. 37.5
B. 75
C. 100
D. 150
E. 175

                                 20% salt implies that = 20 units salt and 80 units water. If ¼ of water evaporates, that is (80/4=) 20 units of water is lost. (Don’t try to evaporate salt please J) You are level with 60 units water. So new strength of the solution after evaporation= 20/ (20 + 60) 0r Y gallons of 1/4 (25%) strength. (It is no longer X gallons. Since part of the water evaporated, right). 

                                 10 gallons of water + 20 gallons of salt = 30 gallons of 2/3 (66.67 %) strength.

So our figure becomes

Or

Y= 120 gallons

So we have 1/4 (25%) strength 120 gallons solution.

Or 30 gallon of salt and 90 gallon water. But 90 gallon is 3/4 of water before evaporation. Or the total water before evaporation

90*4/3=120

So, X= 150 

Note that the question can be solved differently in many ways. One of them is by pure algebra.

x= salt + water

  = 0.2x + 0.8x

When 1/4 of 0.8x evaporates. You are left with 0.6x

Adding 10 gallons of water and 20 gallons of salt. New solution becomes

Solving, X=150

The Idea is to introduce the concept using a fun filled yet powerful analogy, so that the questions can be solved intuitively. Besides, it is a very good way to visualize the problem.

CheersJ

I can be reached on ranjan.rahul885@gmail.com

Conquering GMAT - My story

Part1- A crushing experience-The beginning

                           

Unlike most other tests, GMAT is probably the one test that was not just another obstacle for me but a game changer in the way I took life in general. I know it is a bold statement, but those who have been through this will identify the emotions. I remember the first time I took the test on 12th July 2014.I was so desperate that I couldn't sleep the night before. Trust me, this is the worst thing that can happen to any test taker. I was so tensed that all night that I was visiting and revisiting the strategies!!! (And this is the worst strategy of all) The next day by the time verbal sprang on up on screen my brains had stopped processing data and I was not even in a position to differentiate between Chinese and English on the screen. And voila 650!!- IN MY FACE. For a moment, everything looked unbelievable. I couldn’t imagine that this could happen to me. Dejected, I went to the instructor from Byju’s for consultancy (I was using Byju's tablet for preparation). He was not a consultant but he was willing to listen to my part of the story. He listened to me for 30 mins- my story, where I come from, how I approached the tests. After listening to me with an unperturbed look for the entire duration, he said “You know you are a 740 guy; you are taking the test again”.  He still remembers the expression on my face. Now here was a consultant who didn’t want to offer consultancy to me. He had more faith on me than I had on myself then. For the next 1 hour, he talked about the ways I could improve. He assigned me specific milestones. Not easy, but he earned my respect and trust. I promised I would do whatever it takes, I was willing to walk the talk and make things happen. The best thing he said to me was “Relax, GMAT is the most liberal exam on the planet and you are right up there”. I have no clue what he saw in me. But I wanted to believe in him.

Part 2- Revenge

I must say I gained a lot of heart from what he said. But I didn’t follow him to the letter. (If you are reading this, sir, I must apologize). I wanted to address my issues and I realized that the biggest reasons for the (apparent) failure was I was too nervous. A Great man once told me the secret of taking the test – “take it as if you don’t need it”. Easier said than done! I decided to see what happens, perhaps I took it too seriously!! I didn’t study for the next two months, but I invested on my stamina and anxiety issues. I was convinced that my concepts were fine, all I needed to control my nervousness and I should be fine. Talked myself through it that at the end of the day, it’s just a test and not a matter of life and death. (Best thing I said to myself). When I decided to begin again, I worked on my SC, I had strategies for RC. Interestingly, I found that the Quant, SC and RC were pretty mathematical. Quant came naturally to me (and to almost every other Asian by default) I could eliminate most of the options when I would do it untimed!! But CR was my biggest nemesis. I hated it the most -CR. That was the time I decided to do it the hard way and went through CR bible- the most decorated book on CR for GMAT test takers. But I still had timing issues with CR. So I did something outrageous and began to concentrate more on SC and NOT on CR. This was actually a good strategy. In a verbal test, no matter what any test taking agency would write or say, it consisted of 15-17 questions from SC; and then approximately 14 questions from RC (usually four paragraphs), CR can be around 10-12. So even if you are not so good at CR but just good enough, you can do well. As I was quick with SC and managed my RCs I could give some extra time on CR, I took the test and when 710 sprang up (QA-51, V34), it was moment of triumph not just because anything beyond 700 is cool but also because I felt vindicated! I knew my weakness and I now I had proof that I can overcome it- a personal victory!!

Part 3- Redemption

As fate would have it, I had taken the GMAT on 29th Oct, so most of the deadlines were over and I did not have a lot of time for essays. And I could only apply to ISB (couldn't clear the interview, but that's a topic for another day). So I had time in my hand. The GMAT is a very important part of the application process. I thought -Can I improve? I could hear a loud cry- HELL YEAH!! But I wanted to play it smart. And once again I took a break for a month. Also,I received some brilliant advice from gyan guru Tanisha PatelI, a friend and a wonderful motivator (don't kill me for this ;)). I took the e-GMAT course and although not as great as Manhattan, it was very good on certain aspects. Their SC module is very good, and their CR is good too. They introduced me to the idea pre-thinking. A must if you are looking to score above 700 IMO. But the idea was to prepare according to my strategy and I wanted to test it on Veritas Test- I have reservations about the concepts that Veritas teach. Some of the questions are debatable. Sometimes they can't defend themselves on various forums .Some concepts were good no doubt, others were downright bad! But at least I had a strategy in place!! Also I was very relaxed throughout my preparation- Participated on at least half a dozen football competition, halfway through an international online course, joined clubs, cricket of course, presented an international paper, and a host of other activities. The Goal was always to intersperse my GMAT activities with other stuffs to stay cool. I was very calm before the tests, in fact I remember discussing with a friend that I can’t seem to build any sort of nervousness. (A little bit of nervousness is always good). The day before the test I was sleeping like hell and during the "Awake time"- I must have watched at least 3 movies .To bring things to a perspective.

My last few mock scores were
Veritas- 680,700
Kaplan free test-760
GMAT exam pack-1-710
GMAT exam pack-2-710
GMAT prep 1-720(retake)
GMAT prep 2-760(retake)
Actual Test- 740(OA), V40, Q50(all hail complacency)

Although I am no guru, here is my take on the areas tested on the test
Quant- concentrate on P&C, Statistics, and Probability and sets- The most loved topic for higher level questions. Also, they don’t test obscure theories so I was ok. No real practice-(read complacent)

SC- Followed E-GMAT course, Manhattans and Maghoosh blogs, I was never an avid follower of GMAT club and any such groups because all sources are not great. I would typically abstain from Jamboree, unnamed sources, and used Veritas with caution. I would rather concentrate on GMAT preps and OG questions. The questions matter little. The concepts have greater value. I have seen far too many people concentrate on questions from sources that they have a different concept for a each question. Too many questions guys- Concentrate on all the aspects of the tested question. For GMAT preps you can always copy paste the question on Google and read Manhattan's sols- brilliant! (They hardly give sols to OG questions), Maghoosh has all the sols for OG-please see if you have doubts. 
If you are starting out, do the following. Get used to the terms used. For example, I was stumped when I first heard an explanation that involved substantive clauses!! What the hell were they? But to be honest when I started out I had no idea what clauses were- the most tested base concept. But you don’t have a thousand terms to study. Let me list out the most imp ones:
Clauses- as modifiers, dependent, independent, noun clauses , substantive clauses

Modifiers -ing modifier(with and without a comma), -ed modifier(positioning matters). Which ones modify the entire clause or only the subject, noun phrases (the most confusing initially), noun+ noun modifier (very versatile).

Tenses- not much to say, but sequencing is usually the most tested- Meaning analysis is very important. I have often noticed that grammatically correct answers have laughable meaning.

Noun parallelism - Concrete Nouns, Action Nouns, Gerund and complex noun.

Idioms- Finally! Don't fret too much over it. Not All idioms are tested. The list may not be even more than 50 and that includes idiomatic structures. Don’t try and mug up everything from

MGMAT SC guide-sheer waste of time.

Suggestion- keep a pocket diary, write everything that confuses you in short. Revisit almost every day. This one step can improve your accuracy by 10% at least.

RC- RC IMO is very mathematical. If you can, personalise the argument, it won’t be difficult. When you come across a passage on lysosome activities - Instead of saying- "Ye ki kittan yaar" say, aha!!- Lysosome activities- I have been dying to learn about you. Let's see what you got! This makes it less painful and at the least a little more engaging. Also, every question type must be studied in detail- main theme questions, detail, detail-inference, pure inference, hypothetical, tone and CR based. And of course Reading Principles - that was a game changer from BYJU's. 

CR- Mug up CR bible if you have to-it has some very engaging questions and concepts. Dissect the questions and pre-think!! It will help you think better and it drastically reduces time. Earlier I used to solve CRs in just under 3 min (average, use your statistics to guess the range), pre-thinking helped me to reduce that to just under 2 minutes. (I almost feel that I am anchoring for a stuff advertised on Telebrands, but I can’t emphasize more on that). 
Finally get your strategy right for the tests and a backup plan if things don’t go according to plan.

If any of the things that I have discussed resonates with you, please feel free to discuss, send a message, or mail me. Happy to help when I can.

I can be reached at ranjan.rahul885@gmail.com