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How did Jared Morgenstern get recruited to Facebook?
In late 2003, I started a social networking company with Eddie Lim and Eoin Matthews called Metails.com, short for "Details about Me." If you were looking a girl's profile, it was her Shetails; if you were looking at a guy's profile, his Hetails. The concept was to enable individuals to express the products and brands they loved - their style - as part of their identity. By doing so, our goal was to facilitate product and brand discovery free from any advertising bias. Metails enabled you to import your purchases from dozens of e-commerce partners. These products became part of your profile and when your friends purchased an item, our site got a commission and we redistributed the commission between the buyer and the recommender. The concept was to enable buying based on word of mouth, cut out the middle man advertisers, and redistribute value to those involved. In addition to products, your profile included many of the aspects of the then popular Friendster: photos, blogs, and comments. We launched on January 15th, 2004. Facebook launched less than three weeks later, as TheFacebook.com, on February 4th, 2004. I studied it hard. At first only current students could sign up, but there was a sign-up form for alumni to get notified as soon as the site was opened up. I filled that out and waited. Several days or weeks later, I was notified that I could sign up. I was user number 703. I was impressed with the speed and the simplicity of the site, including the 3 bullet points that described its purpose on the main login page. Several friends encouraged me to try to talk with Mark Zuckerberg about possibly working together. I did not do this immediately. I waited four weeks to see how it would grow. I discovered that we were using the same hosting provider and that [sitename]/stats yielded traffic charts for both of our sites. Facebook's dense college network-based growth (at that time, Harvard only) was surpassing our product interest graph growth rapidly.Then, two things caused me to signNow out. The first was that I realized they were going to expand to more schools. The second was that I read Mark's comments defending the site's privacy model and controls on a yahoo Harvard alumni group. It was clear to me that he was very sophisticated in his thinking. A friend of mine, I believe it was Tali Rapaport, introduced us via email and Mark and I planned to meet at Finagle Bagel in Harvard Square. Mark was 10 minutes late to the meeting, and I recall being frustrated. I'm not sure what I was expecting, but within 5 minutes of our interaction, I switched immediately to selling our vision and team members and a way we could work together. This was not what I had planned to do, in fact, I thought quite the opposite - that Mark would try to pitch me on why Facebook was the next best thing. But it was clear from the start that Mark was different. He wasn't reacting to normal social cues - jokes, pauses - he was very concentrated on evaluating the content of my communication. And he was very excited that our monetization approach - friend recommendations - was built into the fabric of the site, a stark contrast to the banner ad companies that had been pitching him.After that meeting, I focused more than 50% of my effort creating screenshots to fill a proposal for how Metails could power word of mouth recommendations and advertising for local businesses, national products and global brands, including how we'd walk from door to door to all the merchants in Harvard Square to get started. The email dialog around the proposals gave me an opportunity to understand a little more about how Mark thought about the world and the same of me for him. Previously we had seen each other in the computer science labs. The proposal was never accepted, but Mark and I became email pen pals and bounced ideas off of one another about once every quarter. He was rapidly opening up school after school my team was acquired by Buy.com, one of our partner merchants. We both had moved from our college towns to the west coast.After a year at Buy.com, it was clear to me that the innovation in the social space was occurring elsewhere. I had learned a lot about myself - both my limitations and my strengths - transitioning Metails into Yub.com (http://www.yub.com/people/?userID=1 - it remains largely the same today as when I left). At each step of the way I was observing what we were doing and how Facebook was approaching a similar problem. It was clear I wanted to start something new, work for Google, or work for Facebook, and Facebook was at the top of my list.I signNowed back out to Mark directly in November of 2005:Mark said, you should come here, and here's our recruiter. My interviews included Dustin Moskovitz, Aaron Sittig, and Soleio. Dustin and I talked about mobile, his desk side by side with Zuck's in the corner of 156. Aaron wore a very intimidating asymmetric sweater and was very hard to read. Soleio had standing desk and made me draw wireframes on photocopied outlines of the main components of the site. I received 2 homework assignments that night, and I stayed up late - powering through the start of a cold - to make sure they had it by 2:00am (assignment 1: http://www.facebook.com/photo.ph... and assignment 2: http://www.facebook.com/photo.ph...).I started on February 13th, 2006 with Eddie Lim, Jon Warman, and James Wang, about two years after Facebook launched. Since then, my focus has never strayed too far from the original conversation I had with Mark at Finagle Bagel in Harvard Square - aligning company monetization with the interests of our users.
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Why doesn't Microsoft understand tablets?
Look: the devices known as "Tablet PC" and the "iPad" are fundamentally different classes of product, and they need to be analyzed differently. I would even say that Microsoft is king of a device that happens to have a niche audience.The device that Apple has made is a "consumer product" that is pretty much only capable (at this time) of "consuming". The typical use cases for it are sitting around reading your e-mail, browsing the web, watching videos, and giving presentations.Despite it being theoretically possible, you don't sit down with an iPad and attempt to actually use it to writ...
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What are some of the best Coursera courses? Why?
I wanted to learn machine learning. So I spent a week scanning the internet for the best learning resources.Coursera kept popping up. I had never seen it before. But it was familiar. A girl I served at the Apple Store once was telling me how she was watching videos on her phone whilst on the train. She showed me the app. It was blue.'What for?' I asked.'No reason, I just think it's always good to be learning something.'You know how those moments someone says one thing. One thing and it sticks with you forever.She left the store.When I found Coursera I remembered her. Her words stuck.I had some free time since I quit Apple and the startup we were working on failed.I collected a few courses together and created my own AI Masters Degree.[1] Whilst building the website for our startup, all I'd heard about was machine learning and deep learning. Disruption here, innovation there. All thanks to computers learning things on their own. On their own? How? I had to know.A) Andrew Ng's Machine Learning Course[2]This came first. It was tough. Really tough. I had to Google the answers a few times. Otherwise, I would've burned out.I remember hardly anything from it. None of the code because it was in MatLab and Octave. I wasn't a fan of the MatLab interface. Plus, I was learning Python on the side.But what I did take away was Andrew's passion for teaching. His energy for machine learning. His enthusiasm for artificial intelligence. It put a smile on my face and wore off on me.Sometimes a teacher sparking an interest in a student is more important than the knowledge itself.If you want to get into machine learning, this course teaches some of the fundamentals. I wouldn't worry too much if you can't pass the coding assignments, I'd watch the lectures, get an intuition and use it to fuel your next learning step. A course with Python.B) Applied Data Science with Python by the University of Michigan[3]When I started as a machine learning engineer, my skills for exploring a dataset were subpar. I'd spent too much time building deep learning networks for prepared datasets rather than getting into the trenches and exploring data from scratch.This course helped me learn the fundamental steps of exploratory data analysis (EDA).If you want to get into data science and machine learning, you'll probably find yourself spending a lot of time using the Python library Pandas doing EDA. And this is exactly what part 1 of the course is about.After I finished, I used the skills to enter my first Kaggle competition.[4]Out of all the courses I've done, Applied Data Science with Python has been the most aligned with what I do day-to-day as a machine learning engineer.In hindsight, this course should've been one of the first I took when starting to learn machine learning/data science.C) deeplearning.ai with Andrew Ng[5]Check out how stoked I was for this one.[6]Anything Andrew Ng releases, I engulf. My learning style is very aligned with his teaching style.I learn best with a combination of text and visuals. Andrew does both of these in his lectures. Often drawing examples over the printed lecture notes.This course taught me the fundamentals of deep learning as well as how to apply them. If the Applied Data Science with Python is the Everest Base Camp of data science, deeplearning.ai is the peak.It gives an overview and working examples of the concepts involved in computer vision (teaching computers to see), natural language processing (teaching computers to understand language) and spoken language detection (teaching computers to hear).In between each of these topics, Andrew shares his wealth of experience and dives deep into some of the math and statistics behind deep learning.D) Learning How to Learn by Barbara Oakley[7]This course should be compulsory for all students. I've left it until last but if I was to start again, I'd do this first. It's worth doing yearly to update yourself.Why?Because learning how to learn is the ultimate skill.If you know the techniques and best practices for learning itself, you can apply them to anything else.Some of my favourites (and ones I use daily) were:Pomodoro timers to avoid procrastination — set a timer for 25-minutes and do nothing except the single task in front of you until the timer is up.Focused and diffused thinking — focused thinking happens when you’re focusing on a single task, diffused thinking happens when you aren't doing anything in particular. The best learning happens at the crossover at these two. It's why it's valuable to have breaks and periods of time where you do nothing at all. It's why some of your greatest ideas come to you in the shower.No need for genius envy — everyone has to start a learning journey somewhere. And the thing is, learning never stops, it's always day one. Charles Darwin was a college dropout who later went on to discover the theory of evolution.Knowing these things in high school probably would've helped my grades a little. And maybe prevented me from failing my first two years at university.I loved this course so much I made a video about it summarising the rest of my favourite takeaways.[8]The rest of the courses here have been biased by my interest in machine learning and data science.But I think you'll gain something from the Learning How to Learn course no matter what you're into.E) Bioinformatics by the University of California San Diego[9]I haven't done this one yet but I'm going to start it tomorrow.Why?Because nothing fascinates me more than the crossover of health and technology. That's bioinformatics in a nutshell.It's amazing to have such resources available online.I can be sitting in my room in Brisbane, Australia and learning exactly what I what I want to be learning. Something originally taught over 5000-miles away.And if you want to learn something new, you can. All you have to do is sign up and then keep showing up.You don't need a reason either.'No reason, I just think it's always good to be learning something.'Keep learning.Footnotes[1] My Self-Created Artificial Intelligence Masters Degree[2] Machine Learning | Coursera[3] Applied Data Science with Python | Coursera[4] Code with me (live): How to make your first Kaggle submission from scratch![5] Home - deeplearning.ai[6] Graduating from the deeplearning.ai Coursera Specialization | Learning Intelligence 22[7] Learning How to Learn: Powerful mental tools to help you master tough subjects | Coursera[8] The Ultimate Skill: Learning How to Learn (course review)[9] Bioinformatics | Coursera
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What is an intuitive explanation of Bayes' Rule?
Your roommate, who's a bit of a slacker, is trying to convince you that money can't buy happiness, citing a Harvard study showing that only 10% of happy people are rich.After giving it some thought, it occurs to you that this statistic isn't very compelling. What you really want to know is what percent of rich people are happy. This would give a better idea of whether becoming rich might make you happy.Bayes' Theorem tells you how to calculate this other, reversed statistic using two additional pieces of information:The percent of people overall who are happyThe percent of people overall who are richThe key idea of Bayes' theorem is reversing the statistic using the overall rates. It says that the fraction of rich people who are happy is the fraction of happy people who are rich, times the overall fraction who are happy, divided by the overall fraction who are rich.So if40% of people are happy; and5% of people are richAnd if the Harvard study is correct, then the fraction of rich people who are happy is:[math] 10\% \times \frac{40\%}{5\%} = 80\%[/math]So a pretty strong majority of rich people are happy.It's not hard to see why this arithmetic works out if we just plug in some specific numbers. Let's say the population of the whole world is 1000, just to keep it easy. Then Fact 1 tells us there are 400 happy people, and the Harvard study tells us that 40 of these people are rich. So there are 40 people who are both rich and happy. According to Fact 2, there are 50 rich people altogether, so the fraction of them who are happy is 40/50, or 80%.
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What is the most efficient way of studying to score a 700 on the GMAT?
I took the GMAT on the 12th of May, 2016 and ended up getting a score of 760. Here is a detailed debrief which may help you prepare for the GMAT. Feel free to get in touch with me in case you need help with your GMAT prep.Practice Materials Used:Verbal:1. Sentence Correction: Manhattan,Aristotle Prep2. Critical Reasoning:Manhattan, Powerscore CR Bible (few chapters)3. Reading Comprehension: GIN’sRC notesQuant:1. Manhattan BooksOthers:1. Kaplan 8002. Official Guide 133. Official Guide 164. Official Guide forVerbal 16Forums:1. GMATClub2. Manhattan3. BeattheGMATApps:1. Veritas Question Bank2. Manhattan AppFor SC, the Manhattan Book is without doubt the best there is. I used a couple of days right at the start of my prep to go through all the rules. I also used Aristotle Prep to supplement my understanding of the rules. According to me, this worked better than simply re-reading the Manhattan Book, as a different approach with varied examples helped drill the rules in my head better. I also completed all questions from OG13 and OG Verbal 16 Guide, and the tougher questions from OG16.For CR, I again referred to the corresponding Manhattan Book. I found it useful in helping me break down the argument. I also used the Powerscore CR bible which is undoubtedly a great book. However, I wanted to practice more questions instead of reading theory again and hence I left the book midway. I practiced mainly from OG 13, OG 16 and OG 16 Verbal Guide.For Reading Comprehension,I found this incredible document called ‘Gin’s RC Notes’ on GMATClub. It certainly helped me in comprehending the passages better. Practice materials were the same as the ones used for SC and CR.As regards Quant, I didn’t spend a lot of time reading theory. I simply went through all the Quant Manhattan Guides in 2-3 days.I also used the Kaplan 800 book to revise all topics. The book is quite good for revision and for practice.I also maintained a notebook to record my answers and to write down key points. Sample shown below:Mock ScoresAs can be seen from the mock scores, I was pretty ok at quant. However, I realized that my verbal performance was way below par and hence decided to almost entirely focus on verbal from that point on.My performance in the last mock was quite good and that score convinced to register for the GMAT. I checked the test center calendar and scheduled my GMAT on 12th May,2016.Manhattan: In my opinion, the quant section in these mocks is way tougher than the one in the actual exam. As can be seen from the pic, my performance in quant in the first mock was a fiasco. However, once I started getting used to the difficulty, I got better at it. The verbal section is a pretty good representative of the section in the actual exam.Veritas: I found the quant section in the Veritas mocks to be ridiculously simple. The verbal questions were fairly difficult and are a bit different as compared to Manhattan. I would suggest taking a few mocks of Veritas in addition to Manhattan just to get a different perspective of verbal questions.GMAT Prep: The best indicator of your current level as can be seen from the fact that my final score was a neat average of my 2 GMAT prep mock scores.Test Day Experience:My test slot was at 9 am. To avoid the deadly Mumbai traffic, I decided to head out early and signNowed the test center at 8 am itself. Since, the test center was closed, I had to wait till 8:30 before I was allowed inside. The formalities were completed quickly post this and the friendly people at the center allotted a computer to me.AWA: I breezed through the instructions and started with AWA. The argument was simple and it didn’t take me much time to find out the flaws in the argument. I took around 5 minutes initially to frame the structure and took nearly 15 minutes to type in the entire thing. I spent a few more minutes proof-reading the entire essay before submitting it.IR: I had hardly prepared for the IR section. All that I practiced were the questions from the practice tests. On the whole, the questions in this section were not too difficult.However, I got bogged down on one particular set with a lot of data and hence couldn’t answer the last question in this section.Quant: The first 3-4 questions were extremely simple. However, post this I started getting a few tricky ones. Most of the mistakes I made in my mocks were in DS and so I forced myself to take a little more time to check for all conditions in DS questions.Few of the questions around Q.20 ate up a lot of time and so I had to rush a bit at the end. The questions from Q.30 onward were not that difficult because of which I thought I had screwed up the Quant section. Finished the section with around a minute to spare.Verbal: Similar to quant,the first few questions were straight forward and I could spare a little more time double checking the answer. By the time I was done with Q.10, I had spent only 15 minutes and so I was bang on time to finish the section before time.Unfortunately, I got distracted by another candidate who seemingly had started off with test at that point in his time and who seemed to be venting his anger on the keyboard. The noise from his typing completely threw me off-guard and as I had not taken earplugs, I simply tried to concentrate harder. The next 10 odd questions took me a lot more time. At the end, I had 10 minutes remaining with around 6 questions to go. These questions were moderately difficult and I could finish the test with time to spare.Score: I was worried about my quant score but I genuinely thought I had done well in verbal. However, when I saw the score (760 with 50 Q and 44V) I was a bit disappointed. I had expected a 780 score especially considering my performance in the last mock. I was however relieved to know that 760 is still a 99th percentile score.Strategy Advice - My 2 cents:Problem Solving: Surely the easier section in Quant. It is pretty easy to get almost all questions right in PS with the right preparation and strategy. Also, in PS you have the added advantage of either eliminating options or choosing appropriate numbers. A good way of getting better is to try solving every question (wherever possible) using options in addition to the normal way of solving. The more numbers of ways you have to solve a question, the quicker you can kill it during the exam.Data Sufficiency: The major headache for me in Quant was in DS and it is probably the key to getting a 50/51 in Quant. In the initial few mocks, I kept missing simple traps (e.g. The number need not be an integer, for comparison of numbers include fractions as well, etc.). As I solved more questions, I got better at it. With practice this section can be easily cracked. I would suggest checking Bunuel’s problems at GMATClub which are segregated neatly topic-wise.Sentence Correction: This is the major bugbear for candidates. At the same time, this is the section in which the most improvement can be made. I went from around 10 wrong in SC in my first mock to all correct in my last 2 mocks. The easiest way to improve in this section is to analyze the errors you are committing and try and eliminate those. My strategy was to search for the question on Manhattan Forums and check if there was an explanation by Ron. I usually found his explanations to be the best. The second best resource was GMATClub. I would try and check if there was an explanation by an expert in that thread. Only if I understood the logic would I move on to the next incorrect question.Critical Reasoning: The Manhattan book was useful in delineating the different types of questions asked. The thing with Critical Reasoning is that the more questions you solve, the better you get. I concentrated on solving questions from all possible sources. I exhausted all questions from the OG13 as well as from OG16 Verbal. I also solved almost all questions from the Manhattan App (which is free for a limited time period) and from the Veritas Question Bank.Reading Comprehension: In the first few mocks, my abysmal performance in SC hid the fact that I was making errors in RC as well. However,in the later ones, I realized I was making at least 1 error per RC. Wanting to improve on this, I searched for a few RC strategies online and found the brilliant Gin’s RC Tips on GMATClub. The point which helped me the most was the part about agreeing or disagreeing vehemently with the author. This helped me grasp the RC better and solving the questions post that was an easy task. I would recommend everyone to check the document out.Parting thoughts:1. Don’t overdo the prep: Since I had less time to take the GMAT, I went a bit overboard with my prep initially. I solved the entire CR of OG in a day and had CR phobia the next few days. So my suggestion is to take it easy and to stop studying if your mind is exhausted.2. Get accustomed to the exam: Try and mimic the actual GMAT as much as you can. Solve the AWA and IR section as well wherever possible. Take the obligatory breaks if you want and eat the same stuff you would eat during the final exam.3. Keep a pair of earplugs with you as standby: The good folks at the Mumbai center provided me with one. However, since I had never taken an exam with earplugs, I didn’t find it necessary. However, the inside of the test center is pretty quiet. So any noise made by any candidate is amplified many fold and could distract you from the exam. Having the earplugs would be handy in such a case.4. Enjoy the experience: Enjoy the prep as much as you can. If you enjoy what you are doing, things get a bit easier. If you are someone who usually doesn’t perform well under pressure, then try and diminish the significance of the exam as much as possible. For example, forcing yourself to think that the GMAT is simply one component of the selection process will probably reduce the significance of it in your mind. This may help you be more relaxed during the test.5. Be patient if you feel your prep isn’t going anywhere: I’ll end with this point about patience. During my prep, in the initial few mocks, I was stuck in the 700-730 level. No matter how many questions I practiced, I kept getting questions wrong. However, once I kept at it, the mocks suddenly started showing results. As can be seen from the mock scores image, post Manhattan mock3, the scores kept increasing each time. So my advice is to keep at it even if you feel you are not improving a lot.The same message from Rahul Dravid - “My wife and I have built a new home with a lovely garden which houses lovely bamboo trees. I got reading on the Chinese bamboo and learned that the tree takes 5 years, 3 months to grow to its whole height of 80 feet. Yet, for the first 5 years, you only see a tiny green shoot, but in the next 90 days, it grows into a full-fledged tree. But in those first 60 months, it is growing its strong network of roots underground,to support the tree. In an era of instant gratification, we settle for shorter trees, but remember patience has its reward. These are your years of growing that strong network of roots but be sure when you finally achieve your success, people will call it “overnight success”. If only they knew of the Chinese bamboo!”All the best!P.S: I also teach aspirants for the GMAT. If you are interested, then do mail me at sriramk1986[at] gmail.com
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What are the new fads of technology?
What is Green Building?Green building (also known as green construction or sustainable building) refers to both a structure and the application of processes that areenvironmentally responsibleandresource-efficientthroughout a building's life-cycle. It uses less energy, water and other natural resources creates less waste & Green House Gases and is healthy for people during living or working inside as compared to a standard Building. Our traditional architecture is much greener.IntroductionWe all know the effects of global warming on earth. The dying of rivers, climate change, rising sea level etc all are threats to not only humans but also many other species living on earth. The cause of global warming is largely contributed by human activities. Building industry is also causing a large amount of pollution. Several resources are depleted for the construction of buildings. Throughout the life of a building the waste generated by it and its demolition waste pollutes the environment.Characteristics of Green BuildingBuilding construction and its upkeep for livable conditions requires huge energy in lighting, air-conditioning, operation of appliances etc. Green Building i.e. energy efficient building is the one which can reduce energy consumption by at least 40% as compared to conventional building. The cost of constructing energy efficient building is estimated to be 15 – 20% higher as compared to conventional building without energy efficiency. However, this is more than compensated over the period of time i.e during life cycle cost and operation & living. Using green building materials and products, promotes conservation of non renewable resources internationally. In addition, integrating green building materials into building projects can help reduce the environmental impacts associated with the extraction, transport, processing, fabrication, installation, reuse, recycling, and disposal of these building industry source materials.Elements of Green Building·Materials for Green Building-Materials for a green building are obtained from natural, renewable sources that have been managed and harvested in a sustainable way; or they are obtained locally to reduce the embedded energy costs of transportation; or salvaged from reclaimed materials at nearby sites. Materials are assessed using green specifications that look at their Life Cycle Analysis (LCA) in terms of their embodied energy, durability, recycled content, waste minimization, and their ability to be reused or recycled.·Energy systems in Green Building-Green buildings incorporate energy-efficient lighting, low energy appliances, and renewable energy technologies such as wind turbines and solar panels. Passive solar design, higher levels of insulation and natural daylight design reduce the electricity cost.·Water Management in Green Building– There are various systems to minimize the water usage and to recycle waste water.Installing greywater and rainwater catchment systems that recycle water for irrigation or toilet flushing; water-efficient appliances, such as low flow showerheads, self-closing or spray taps; low-flush toilets, or waterless composting toilets can help achieving the goal.·Health Components of Green Building– Installing non-toxic materials, improving the indoor air quality and using emission-free appliances could improve the health condition of the occupants.Advantages of Green Building MaterialsGreen building materials offer some or all of the following benefits to the building owner and building occupants:Reduced maintenance/ replacement costs over the life of the buildingEnergy conservationImproved occupant health and productivityLife cycle cost savingsLower costs associated with changing space configurations.Greater design flexibilityRegulations and operationsGreen building rating systems such as BREEAM (United Kingdom), LEED (United States and Canada), DGNB (Germany), CASBEE (Japan), and VERDEGBCe (Spain), GRIHA (India) help consumers determine a structure’s level of environmental performance. They award credits for optional building features that support green design in categories such as location and maintenance of building site, conservation of water, energy, and building materials, and occupant comfort and health.ApplicationsTaipei101, the tallest and largest green building ofLEEDPlatinum certification in the world since 2011.Hanging gardens of One Central Park, SydneyConclusionsGreen Buildings can effectively reduce the environmental impact by Building Constructions. The energy saved can be ploughed back for further development which creates a large employment opportunity. The technologies and the materials used for development should complement the use of local and waste resources.If you want to know more about interesting engineering topics you can visit my blog: Civil Blog
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Why don't physicists write human-readable formulas, like developers do? For example, energy = mass * speed_of_light^2 instead of
Here is an example from everyday life:Problem:My family consists of cats, each with four legs and eighteen nails, and humans, each with two legs and twenty nails. In total, my family contains one hundred fifty-two nails and twenty-four legs. How many of each are there?Solution 1:[Hint: I couldn’t even write Solution 1 until I did Solution 2, so don’t be disheartened if you get lost in Solution 1; just skip to Solution 2 when that happens.]Suppose we consider nine times as many legs. That would imply that thirty-six times the number of cats added to eighteen times the number of humans adds up to two hundred sixteen legs.Now suppose we consider two times as many nails. That would imply that thirty-six times the number of cats added to forty times the number of humans adds up to three hundred four nails.Now let us consider the difference in implications of those previous two paragraphs. Two paragraphs back we considered thirty-six cats, and one paragraph back we considered thirty-six cats, so the difference in the number of cats in those two paragraphs is zero. For the number of humans, we have twenty-two more humans in the second paragraph than in the first. The difference that makes in the total number of items is eighty-eight. That tells us that twenty-two times the number of humans is eighty-eight. (I can’t quite understand the reasoning that leads to that previous sentence; I am only translating into human language the equivalent statement in Solution 2, which I can understand.)If we consider the difference between the conclusion of the previous paragraph and the original statement about legs, it is a statement about four times the number of cats plus zero times the number of humans, and the difference in the number of items is sixteen.If we divide the numbers in the conclusion of two paragraphs back by two, we find that the number of humans is four.If we divide the numbers in the conclusion of two paragraphs back by four, we find that the number of cats is four.Solution 2:Define the following symbols:C = Number of cats in my familyH = Number of humans in my familyL = Number of legs in my familyN = Number of nails in my familyThe statement of the problem isL = 4 C + 2 H = 24N = 18 C + 20 H = 152What are C and H?The statement of the problem forms the augmented matrix[math]\left[\begin{array}{cc|c} 4,&2&24\\ 18,&20&152\\ \end{array} \right ][/math].Multiply the first row by 9 and the second row by 2:[math]\left[\begin{array}{cc|c} 36,&18&216\\ 36,&40&304\\ \end{array} \right ][/math].Subtract the first row from the second:[math]\left[\begin{array}{cc|c} 36,&18&216\\ 0,&22&88\\ \end{array} \right ][/math].Restore the original first row and divide the second row by 11:[math]\left[\begin{array}{cc|c} 4,&2&24\\ 0,&2&8\\ \end{array} \right ][/math].Subtract the second row from the first row:[math]\left[\begin{array}{cc|c} 4,&0&16\\ 0,&2&8\\ \end{array} \right ][/math].Divide the first row by 4 and the second row by 2:[math]\left[\begin{array}{cc|c} 1,&0&4\\ 0,&1&4\\ \end{array} \right ][/math].The left-hand side is an identity matrix. Therefore,C=4,H=4.Which do you find easier to understand?
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