Online Integer Number Tutors

Online Integer Number Tutors It is the system of numbers; containing negative of natural numbers. The collection of integers is denoted by Z or I. Thus, Z = {, -4, -3, -2, -1, 0, 1, 2, 3, 4, } Zero is neither a negative nor a positive number. It is a neutral number. This system also has infinite numbers towards negative as well as positive sides. Therefore the smallest and the largest integers cannot be obtained.In this system, any number can be obtained by adding 1 to its previous number. Note: - The difference between two consecutive integers is always 1 (one). Example 1: - A man moves 20 m due east and then 16 m due west. Find his position with respect to his starting point. Solution: - If 20 m due east is represented by +20, then -16 represents 16 m due west. On adding + 20 and 16, we get: (+ 20) + (- 16) = + 20 16 = + 4; which is positive Therefore the position of the man with respect to his starting point is 4 m due east. Example 2: - The temperature of a body first rises by 25 degree C and then falls by 32 degree C. Find the final temperature of the body, if its initial temperature is 0 degree C. Solution: - If the rise in temperature is denoted by positive sign, then the fall in temperature will be denoted by a negative sign. Final temperature = Initial temperature + rise in temperature + fall in temperature. = 0 + (+ 25) + (- 32) =-7 degree C.

The Best Educational Apps To Use For Tutoring

The Best Educational Apps To Use For Tutoring What Are The Best Educational Apps For Kids? Chapters1. Numberland2. DragonBox Big Numbers3.Lola's Alphabet Train4.iTooch Elementary School and iTooch Middle School5. World Geography - Quiz Game6.Memrise7.Revision App8.TED“People expect to be bored by eLearningâ€"let’s show them it doesn’t have to be like that!” - Cammy Bean, e-learning instructional designer.The  private tutoring industry quickly realised that new technologies had to be integrated on the normal learning experience of any pupil or student.These days, you can access private lesson or video call classrooms for free on many websites. You can revise all subjects, maths, physics, chemistry, history or geography and even French.It is predicted that 15.5 billion app downloads will occur in the UK in 2018. That is a crazy 236 apps for every single person living in the UK.The app market grew by 6% in 2017 and will keep growing in the years to come.With 96% of the A-level students owning a smartphone, it is no wonder that e-learning has been booming in recent yea rs.We summed up a few apps that you could use to help your students further by providing them with the right tool for an e-education.Children can now use smartphones to play around with their hand drawn art. (Toca Boca)Graduating college is usually the achievement marking the end of studies for most students. The graduation ceremony is particularly important in most countries.8.TEDProbably aimed at an older crowd, the TED app, by the TED Talks organisation is worth mentioning.This library of more than 1,000 videos of some of the most influential speakers in the world, can introduce you to many fascinating subjects.The videos are also proposed in more than 90 subtitle languages and the subjects range from business to technology and education.You can even download them to watch them when you don't have any Internet access, which is ideal if you're taking the tube to school.And best of all, it is free.So is this expanded guide to making lessons fun!

Become an Ace Learner in Chemistry Through Online Chemistry Tutoring

Become an Ace Learner in Chemistry Through Online Chemistry Tutoring 0SHARESShare Studies have revealed that most of the students obtain a below average score in subjects like chemistry or organic chemistry. Thanks to the various advantages of online tutoring services, they have gained immense momentum. Available 24/7, these online tutors provide you with their online tutoring services right to the comfort of your room, thus helping you to get rid of the commuting hassle. Do you often find yourself entrapped in the Chemistry formulas and chemical reactions? Chemistry is one of the most difficult subjects. It requires you to study properties, structure and reactions of organic compounds.  However, with the hiring of an  online Chemistry tutor, you can do away with all of your Chemistry’s confusion and problems. The tutor provides you with the personalized attention and one-on-one tutoring, as per your convenience and requirements. Employed with online tutoring services providers, these certified  online tutors  are extensively experienced and highly qualified. Using their decades of teaching experience, they leave no stone upturned when it comes to a student’s satisfaction and help him/her to become a whiz in the Chemistry. Get proper Chemistry learning from the cozy comfort of your home! Easy to get connected anytime, anywhere, these tutors, having better understood your learning requirements, provide you with the required help. Whether you are looking help for basic concepts or advanced ones, these tutors impart you with the complete Chemistry learning. However, the only thing that you need to do first before hiring the services of  organic Chemistry tutoring, online Chemistry tutoring is to conduct a little RD of online tutors available online by reading reviews of parents, students who have been making use of these services. With the help of these reviews, you can choose the best online tutoring services among the rest. [starbox id=admin]

Discover the History of Yoga

Discover the History of Yoga Yoga: Origins, History and Evolution! ChaptersThe Origins of YogaPast and present: How is Yoga Practiced in the world?Discovering Different Forms of YogaYoga is an age-old practice that has evolved over the centuries.Today, when we talk about yoga, we think of a kind of gentle gymnastics, in which you move your body in a string of easy and more difficult movements.But did you know that this has not always been the case?Going back to the origin of yoga together, we will find that there are several types of yoga, not always practised in the same way...To learn more about yoga, read our article.Sri Krishnamacharia (1888-1989): he founded a yoga school in India;K.V. Desikachar (son of Sri Krishnamacharia): he strove to make the practice known throughout the world;Bellur Krishnamachar Sundararaja Iyengar (1918-2014): founder of Yoga Iyengar;Sri Krishna Pattabhi Jois (1915-2009): teacher of Yoga Ashtanga.Integral Yoga: an example of a modern form of yoga developed by Sri Aurobindo in the early 20th century. Source: Visual hunt In the late 19th century, yoga masters began to travel to the West, sparking interest and new followers.During this period, a large number of federations and associations were created throughout the world.Among these new followers was, of course, Britain. During the mid-twentieth century the art form  became popularised, appealing to the British people due to its emphasis on both physical and spiritual wellbeing.By the end of the twentieth century it had become a mainstream activity and a huge commercialised business.In Britain, the discipline is essentially known in its Hatha Yoga form. But there are many others. Learn about all this philosophy at a yoga class in London or across the country...Discovering Different Forms of YogaLet us turn to the traditional and modern forms of yoga.You will find that there are many types, each with its own objectives, practices and exercises.Hatha YogaIn the West, Hatha Yoga is undoubtedly the most widespread form of this discipline.It consists of linking and maintaining poses, while promoting breathing and concentration. Not to mention relaxation, which is an important part of the exercise.Its goal is to work your flexibility, but also to have better stress management or improve the quality of your sleep.It's a pretty comprehensive variety.Kundalini YogaKundalini Yoga allows you to harmonise your energy centres.To achieve this, it focuses on practising yoga poses accompanied by breathing exercises, meditation and relaxation.The idea is to offer you a better physical and mental balance while strengthening your immune system and letting go of daily stress.Iyengar YogaThe Yoga Iyengar (named after its inventor: BKS Iyengar) was created a short time ago.Its principle is to offer a gentle approach to the exercise by allowing you to help yourself with equipment.Designed primarily for beginners, it also brings many physical and spiritual benefits.Prenatal YogaAs the name suggests, prenatal yoga is for pregnant women who want the r ight aids to deal with their pregnancy and childbirth.Thus, yoga postures are adapted to the changing body. Each yoga session offers different exercises that will help you to:Relax and avoid mood swings;Maintain your flexibility (or gradually improve it);Be in touch with your child and your changing body;Be calm on the delivery day through inspirational/expiratory exercises that will help you on D day;Not to mention avoiding the bodily pains that you experience during this period.Prenatal Yoga is a popular practice for women who want to prepare for childbirth. Source: PixabayBe careful though: this yoga class requires the help of a professional. It is not advisable to practice it at home, if you are not already an expert.Yoga NidraYoga Nidra is a rather unusual practice of yoga since you do not string together movements.In fact, you lie down and you enter a state close to sleep (the same as sophrology).By taking a slow, deep breath, you relax and put aside all negative feelings.If y ou have problems sleeping or if you have trouble managing your emotions, this is the type of Yoga for you!Raja YogaRaja Yoga is also called Ashtanga Yoga. It is based on 8 members who are spiritual precepts to achieve the desired state of unity.Considered the most comprehensive form, it draws from all types of yoga!Vinyasa YogaLet's finish our selection with Vinyasa Yoga, which draws heavily on Astanga Yoga.Yet, while the second consists of always performing the same series of movements, Vinyasa Yoga is much freer.Very energetic, you follow your yogi master who links poses one after the other, in a fluid and rhythmic way.Therefore, you need to be a decent athlete to be able to have a go at it.In any case, it allows you to sculpt your body without building it up (unlike bodybuilding) while eliminating any superfluous fat that has accumulated.Obviously, there are many others that we invite you to discover at your own pace.To start practising today, find associations for yoga near me.W hat do I do if there are no classes for yoga near me? There are also individual teachers or groups you can join.In any case, if you feel overwhelmed by stress and if it affects your body, give yourself a break in life. You will soon see the benefits.

7 Tips for Working as a Freelance Musician

7 Tips for Working as a Freelance Musician Suzy S. Last nights 84th Annual Oscars paid tribute to the best of the best of the past years big screen efforts and, of course, the music involved.   Out of the two music categories, The Artist won for Best Original Score, and Bret McKenzies Man or Muppet won for Best Original Song. So what can the Oscars, more prominently known for recognizing great films, actors and directors, teach the modern day musician?   Lucky for you, as a musician there are several different paths to take.   Whether you see yourself on lead guitar, behind the scenes in the recording studio, as concertmaster in the Philharmonic Orchestra, or penning an Oscar-winning film score, a passion for music can take you to many different careers. For most, freelance work is part of the journey at some point.   Its a great way to establish a network, get your name out there and earn some extra cash.   If youre just getting started, check out these great tips from Ultimate-Guitar.com about surviving as a freelance musician: 1. At least in the beginning, you will need a day job: This is the part that nobody likes, but you will most likely not be able to support yourself by playing music alone. The thing about the music business is that it really is all about who you know. Its such a tight-knit, close community, and a lot of your credibility will come from people you know and recommendations from your peers and other clients. When youre first starting out, you have none of that credit, and may not be called about very many gigs. You may have to solicit yourself to play for things, not the other way around. 2. Make yourself marketable: As odd as it sounds, you are a product that your clients have to buy. Just like an advertising campaign makes a certain product stick in the minds of consumers, you have to work to stick in the minds of your potential clients. And the best way to do that is not through crazy costumes and ridiculous stage antics. Make business cards (you can do this at home with Microsoft Office and other programs) and hand them out any time you have a gig. That way, your name will always be available when people ask for a guitarist. Make sure people know that youre available and willing to play gigs. Be polite, be reliable and work hard. 3. Professionalism, professionalism, professionalism!: This may be the most important aspect that will help your career. Just like with any other job, you will not be hired again if you make the experience an unpleasant one for your employer, nor will you be recommended for any other job. But make it a happy, comfortable experience, and you have a shot at more gigs. Always be on time, always have everything youll need for that day of playing, and always have your parts learned. If for some reason there is a part you cant play at the first rehearsal, make sure that its flawless by the next rehearsal. Take criticism, dont ever lose your temper, and always be the kind of person that people want to work with. 4. Never, ever complain about the check: This isnt the marketplace. You dont get to haggle. What you get paid is what you get paid. If you dont think its fair, then just dont take gigs from that person anymore. Making a scene burns a bridge, and someday, you might need that bridge to get jobs. Of course, if somebody says theyre going to pay you a certain amount, and then the check comes and you get shorted or not paid at all, then you have every right to (politely) discuss the discrepancy just be careful not to lose your temper. 5. Be versatile: You might love to play metal, but if you can play rock, country and jazz, too, youve got a much better shot at getting gigs.   The more things you can do, the more jobs youre going to get. And never turn down a gig just because it isnt your kind of style. When youre making the big bucks, then you can be as picky as you want. 6. Be proactive: If you hear about a gig, pursue it. It is perfectly acceptable to call somebody and say Hey, I heard you might need a musician on such and such a date. Well, my name is such and such, Im very capable, and if you need somebody, heres my number. It cant hurt, as long as youre polite and professional. 7. Dont give up: Even if you cant make a full-time job out of being a musician, continue to take gigs and put yourself out there. You never know which gig could be the break youve been looking for! What have you learned from your own experiences freelancing?   Leave a comment below and share your tips and expertise with the community! Like these posts?   Sign up to receive daily updates right to your inbox!   Click here to subscribe. You might also like: 5 Things to Avoid for Aspiring Musicians How to Break into the Music Biz Without Even Playing a Note More Than Just Music: 7 Secrets of Successful Musicians Photo by Universidad de Navarra.

Entry Requirements for Art School

Entry Requirements for Art School Everything To Know About Getting Into Art Schools ChaptersTypes of Art ProgramsBuilding Your PortfolioWhat You’ll Need for Your ApplicationWhat To Expect From Your Art CoursesWhile it may be tempting to start researching and hunting for the school of art that will be right for you, you should understand the different study programs within art education that will be available to you. CalumDrama School Entrance Teacher 5.00 (15) £50/h1st lesson free!Discover all our tutors ToriSpanish Teacher 5.00 (1) £15/h1st lesson free!Discover all our tutors OliviaSchool support Teacher 5.00 (2) £21/h1st lesson free!Discover all our tutors MarkESOL (English) Teacher 4.76 (17) £20/h1st lesson free!Discover all our tutors YuweiChinese Teacher 4.33 (6) £19/h1st lesson free!Discover all our tutors JenniferMusic reading Teacher 5.00 (1) £30/h1st lesson free!Discover all our tutors LouiseAutoCAD Teacher 5.00 (3) £60/h1st lesson free!Discover all our tutors RickyPercussion Teacher 5.00 (7) £35/h1st lesson free!Discover all our tutors Ni colasGuitar Teacher 5.00 (2) £35/h1st lesson free!Discover all our tutors MyriamOrganic chemistry Teacher 5.00 (13) £20/h1st lesson free!Discover all our tutors JonathanEconomics Teacher 5.00 (9) £40/h1st lesson free!Discover all our tutors Oluwakemi imoleMaths Teacher 5.00 (1) £30/h1st lesson free!Discover all our tutors AlexPhysics Teacher 5.00 (1) £50/h1st lesson free!Discover all our tutors AdamSinging Teacher 5.00 (14) £48/h1st lesson free!Discover all our tutors ValentiniMusic reading Teacher 5.00 (2) £50/h1st lesson free!Discover all our tutors MilenaMaths Teacher 5.00 (5) £25/h1st lesson free!Discover all our tutors RashmiEconomics Teacher 5.00 (1) £35/h1st lesson free!Discover all our tutorsTypes of Art ProgramsArt, in general, is normally broken down into three branches. These include the visual arts, performing arts, and literature. If you’ve ever seen a play or read a comic book, you’ll understand that these three branches have a ton of artistic cro ssover. However, there are some generalizations you can make about all three branches â€" here, we’ll be focusing on the first two.Get ready for an art and design course by preparing earlyFine ArtsYou’ll be hard pressed to find one, all-inclusive definition of fine arts â€" and that is because, as mentioned, fine arts is an umbrella term covering endless amounts of subjects. To give you an idea of how large this breadth is, fine arts can involve anything from lecture-based, art theory courses to classes in modern ceramics.That being said, it can be very rewarding to pursue a bachelor degree or masters degree in fine arts because of this incredible diversity. Here are some of the most common programs and specializations you are likely to find at your future college of art:SculpturePrintmakingArt designDrawing and PaintingStudio artsArchitectureDigital mediaIf you’re interested in studying one or two particular subjects more in-depth â€" you’re in luck. There are many universi ties which specialize in particular aspects of the art world. For example, if you’re interested in fashion, you will be able to find a university with an art center that includes a design program. This is especially helpful for students who’d like to attain scholarships, as having a specific career focus can take your application to the next level.You can take the best art classes near me here.Film SchoolFinding a college of the fine arts that is dedicated solely to film is not only possible, it is one of the more popular destinations of young artists. Whether you’re interested in script writing, want to become a master in technical skills like editing, or would like to make cartoons â€" you’ll find the right master or bachelor in fine arts program in no time.Check out some of the programs you are likely to encounter on your journey to find the right film school:Computer animationFilm and television productionPhotographyGame artArt HistoryNeedless to say, if you’re interes ted in art, you’re also probably interested in the history of your craft. Many students who specialize in the discipline of visual art are often influenced by either contemporary artists or those of the past. On the other hand, students who are curious about the world of museums, art galleries and auctions â€" but don’t necessarily want to create art themselves â€" are often drawn to studying art history too.Luckily, art history courses can be found even without going to an academy of art or an art institute. While there are many different kinds of art history programs you will find at your university, finding one that you like will generally depend on what movement of art interests you the most. If you’re curious about a career in curating or selling art, or interested in preserving priceless works of art, this is definitely a subject you should explore.Building Your PortfolioBuilding a portfolio can be rewarding at best and the most stressful task you’ve ever had to accomp lish at worst. Before you start panicking, it will be extremely useful to go through three simple steps. These will be extremely helpful in getting your application ready, as well as to the interview stage many art colleges require in their admissions stage.What Kind of Artwork Do You Have?The first step on your portfolio-building journey will be to understand the kind of work that you have prepared at the moment you are about to submit your applications. Artists with a lot of experience behind them will normally already have a running portfolio form their art classes or from some pre college courses.Building a portfolio can help you later to find careers in artStudent work, however, is often artwork that is in progress. If there’s anyone who will understand the struggle to finish a work of art, it will be art educators. Because of this, many portfolios choose to include work that is in progress because it shows the direction the candidate wants to take their art and that they’v e already shown the initiative to start it.It might be helpful to as your art teacher, alums, or even your school’s provost for some advice on what kind of pieces are appropriate to submit in a portfolio and the best way to start organizing these pieces.OrganizationThe second step is arguably the most important step and, for the most part, one of the hardest for both high school and undergraduate students alike: organizing your art.Whether your work has already been on exhibit in galleries or you’re the leader of art workshops in your neighbourhood, organization will always be the most difficult part of your craft. Whether your work is studio art or using digital media, here are some general rules you can follow to prepare your art to be added to your portfolio.Being selective: while you may deem every piece that you have executed to be indicative as your progression as an artist, the people who will be reviewing your portfolio are looking for a narrative.Cherry picking your art work can be based on chronological order, on which pieces marked a shift in your values, or on the progression of a specific project (even if it’s not complete).Build a NarrativeThis last step is actually something you should be doing throughout the course of building your portfolio and is what will mark the difference between an average portfolio and a strong one â€" regardless of whether you’re applying to art colleges, to be an artist in residence or are trying to get into an exhibition.Here are some general rules to follow when establishing a narrative.Be unique: while it sounds difficult in theory, in practice it’s much easier than you think. The artwork that inspires and drives you will naturally be different than anybody else’s. Focus on what you like best about your artwork and what you would like to accomplish with it in future works.Motivation: what is the reason you study or create art? While this will be different for many people, try to put a unique spin on your answer.Establishing a storyline: it should be very clear to the people looking through your portfolio why you’ve chosen the pieces within it. While the narrative of your artwork might be very clear to you, you will have to get the main points across to complete strangers â€" so spend some quality introspective time on you, your art, and what it means to the world.Another thing to note is that sometimes portfolio requirements are different for undergraduate degrees or even for international applicants. Think of your portfolio like your CV â€" that is, treat it with the importance of a CV while also making it adaptable to the various different programs and arts schools you will be applying to.Finding an art program of study you like can be easyWhat You’ll Need for Your ApplicationWhile it may seem like applying to art programs and art colleges requires extra steps than applying to other university programs, the process is exactly the same for a prospective student in arts educati on. Applying to a British university requires you to go through the UCAS system, which is the Universities and Colleges Admissions Service. Here’s some information on what this process involves, whether you’re enrolling to a school of art and design or for a program in interior architecture.RegistrationThe registration process is fairly simple but it is super important that all of the details you will give to the UCAS system are accurate.Finding a CourseChoosing a course through UCAS will allow you to find courses that might be of interest to you, but that you perhaps don’t know about yet. While many people search for a university first and then try to find a program that they fit into â€" you can also do this process in reverse.There are many different online portals that allow you to explore different programs for liberal arts, art and design, and interdisciplinary subject. Once you find a program that you think might fit your needs, you can take note of its UCAS code.Findin g a UniversityFind a couple of universities that appear to suit your artistic ability, creativity, and financial concerns â€" a task easier said than done. Research will play an important role in this step, so make sure you give yourself enough time to find the right universities.Fill Out your ApplicationsThe UCAS system allows you to apply either to five different universities in the same program or to five different programs in the same university.TuitionWhether you’ve already found a school through looking at their university art or are still in the messy process of finding the right art schools â€" tuition is always an important factor. While there are several government grants and scholarships for students going to a college of art and design, make sure to check out your university’s website to see if they offer more financial aid.What To Expect From Your Art CoursesOnce you’ve completed your UCAS application, submitted your portfolio, and perhaps interviewed for your art colleges â€" you might now be worrying about what you can expect from your art classes or, going further, your arts degree.Your college of fine arts or design school, whether you’re pursuing a master or bachelor of fine arts, will prepare you for a wide array of careers. Here are some of the jobs you will be able to apply for once you graduate:CuratorGraphic designerOrganizer at a center for the artsTeacher at an art collegeFilm director

The Most Famous Chemists Ten People who Changed Chemistry and the Way we Live

The Most Famous Chemists Ten People who Changed Chemistry and the Way we Live We Owe a lot to These Incredible Chemists ChaptersThe Founders of Modern Chemistry: Famous Chemists from the Eighteenth CenturyPerhaps the most Famous Chemist of all: Dmitri MendeleevRenowned Chemical Scientists: the Nineteenth CenturyTwentieth-Century Chemists You Should KnowOver the last two thousand years, we’ve come a long way in science. We no longer think that everything is made of water or fire, for example, or that the human body is composed of only four things. From this subject â€" one you might sigh about in the classroom, or get frustrated at before exams â€" came some of the most important developments of the human race.The field of chemistry has contributed to this development, and this discipline has given us some of our basic knowledge of the world. All these things we might now take for granted: that the world is made of tiny little particles, that you can burn things like coal to get heat. Not only that, but chemistry has revolutionised the way we live our lives â€" offering us technologies from radiation to solar panels to the dye in jeans. If a lot of people think that chemistry is for the academic, the professor, or the weirdo scientist in his big round glasses and white lab coat, it is important to remember that it is so much more than that. And those weirdo chemists changed your life in more ways than you probably realise.In this article, we’ll look at ten of the most important scientists in the history of chemistry. You can decide just how crucial their discoveries have been.chemistry tutor guide your learning!Alfred NobelYou know this name too. This chemist and businessman, Alfred Nobel (1833-1896), is known in all fields, due to the Nobel Prize â€" the award for excellence in literature, the sciences, and economics.But this man’s biography is not as nice as you might expect. His professional life was in selling weapons â€" explosives, mainly â€" and to chemistry, his contributions were generally in inventing the explosives that he sold.Nobel’s main achievement was the invention of dynamite. Intended as a tool for construction â€" for roads, industry, and canals â€" this chemical discovery was to have a sordid history, as we know.In his later years, as the story goes, he felt a little guilty about producing one of the most destructive chemical substances ever devised. He donated his massive fortune to inaugurating the Nobel Prize in Chemistry, Physics, Medicine, and Peace.Find out what the essential equipment you need in your chemistry set is.Twentieth-Century Chemists You Should KnowIn the twentieth century, the subject of chemistry blossoms even further, branching out into different sub-disciplines and becoming the subject we know today.Linus PaulingLinus Carl Pauling (1901-1994) has often been hailed as one of the most important scientists ever to have lived â€" and if you haven’t heard of him, now is your opportunity.Pauling essentially invented, single-handedly, what we now know as molecular biology â€" the study of things like proteins and acids in the body â€" and quantum chemistry, or the study of the relationships between the smallest things in the atom. And if, at school, you’ve heard of covalent or ionic bonds â€" you have Pauling to thank for understanding them too. Learn more about chemical bonds in our piece on basic chemistry concepts.As a lifelong activist for nuclear disarmament, Pauling was not only a Nobel Laureate in chemistry, but he won the Nobel Peace Prize too. Finally, his research into proteins also inspired the work of Francis Crick and James Watson, who, without Pauling, would never have discovered the structure of DNA.Dorothy HodgkinNow time for a chemist that is often overlooked in the histories of famous scientists: Dorothy Hodgkin (1910-1994). Another Nobel Laureate, Hodgkin is the only British woman to have ever won a Nobel Prize in a science. Hodgkin’s main discoveries were in the structure of molecules and proteins in the body, and her research had a massive impact on biochemistry and medicine. She and her team worked on the structure of penicillin, vitamin B12, and, importantly, insulin. Her career was notable for her advocating for the importance of knowledge of insulin, too â€" for its central role in diabetes.My online chemistry tutor introduced me to her work; she is a truly admirable scientist!Kathleen LonsdaleAnd another twentieth-century chemist that deserves to be more famous than she is: Kathleen Lonsdale (1903-1971). As a scientist, she was one of the first women to be elected as a Fellow of the Royal Society and was awarded numerous titles for her important work in crystallography, or the study of the arrangement of atoms in crystals.Alongside her work on diamonds, and her profoundly influential research into the structure of benzene, she was an advocate for pacifism and an inspiration for women scientists across the world.Still hungry for chemistry? We have an all you need to know about chemistry piece just for you!

GMAT Combinatorics 1.2 Problem Solutions

GMAT Combinatorics 1.2 Problem Solutions GMAT MBA Admissions Blog 1. Suppose you perform the following experiment - You flip a coin 5 times and record the results. How many different outcomes are there? (A result being a 5-character long string of H's and T's, such as HHTHT) Solution: For every flip of the coin, there are two possible outcomes. Because we flip the coin 5 times, according to the fundamental principle of counting the number of possible outcomes is: [latex]2 imes 2 imes 2 imes 2 imes 2 = 32[/latex] Note that this calculation can also be expressed by [latex] 2^5 [/latex]. In fact, if you flip a coin [latex] n [/latex] times, the number of possible outcomes is [latex] 2^n [/latex]. __________ 2.0 You receive a new debit card in the mail and you are asked to create a new PIN (Personal Identification Number) for your debit card. The PIN must be 4 digits long and each digit must be one of the numbers 0 - 9. You can use the same digit more than once. How many possible PIN numbers are there? Solution: Each number in your PIN can be one of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. One digit must be assigned on each of four places. So, by the fundamental counting principle, the number of outcomes is: [latex]10 imes 10 imes 10 imes 10 = 10,000 [/latex] __________ 2.1 Same as 2, but now you can use each digit only once. Solution: Now that we cannot reuse digits, once we assign a digit to the first place, we will only have 9 options for the second place, then 8 options for the 3rd place and so on. As I mentioned before, this is a permutation - the ordering or a subset of elements taken from a larger set. But, as you'll see in the course of these posts on combinatorics, all the technical jargon and formulas you get in other prep material just reduces to a single principle - the fundamental principle of counting. The number of possible PIN numbers is now, [latex] 10 imes 9 imes 8 imes 7 imes [/latex] = 5040 __________ 2.2 Same as 2, but now you can use each digit only once and you cannot begin the PIN with a zero. Solution: This is very similar to the last problem with one exception. We only have nine possibilities for the first digit. So the number of possible PIN numbers is [latex]9 imes 9 imes 8 imes 7 = 4536 [/latex] __________ 3. Sam, Sally, Xanthar, and their parents drive to the park in their Prion (a Prius that has been modified by Xanthar for hydrogen ion technology). If the front of the car has two seats (the drivers seat and the passenger seat), the back seat will hold three people, and the two parents sit in the front and Sam, Sally, and Xanthar sit in the back seat, how many ways can the family be arranged in the car if each seat contains only one passenger. Solution: The total arrangements in the car depends on the arrangements of two groups - the kids and the parents. We will take the product of these two arrangements because they are not mutually exclusive. [latex]Kid Arrangements = 3 imes 2 imes 1 = 6[/latex] [latex]Parent Arrangements = 2 imes 1 = 2[/latex] [latex] Total Arrangements = 6 imes 2 = 12[/latex] __________ 4.0 You roll two die. How many possible outcomes are there? Solution: You can think of the outcome of one roll as an ordered pair or numbers [latex](x, y)[/latex]. For instance, the ways to roll a sum of seven are [latex](1, 6), (2, 5), (3, 4), (4, 3), (5, 2),[/latex] and [latex](6, 1)[/latex]. Obviously, there are six options for the first number in the ordered pair, and six options for the second number in the ordered pair. So, according to the fundamental principle of counting the number of outcomes is: [latex]6 imes 6 = 36[/latex] __________ 4.1 Same as 4.0. How many ways can you get two even numbers? Solution: We can use the same thinking as before. However, this time, because we are only considering even numbers, there are only 3 options for each number. Thus, the number of possibilities is [latex]3 imes 3 = 9[/latex] __________ 4.2 Same as 4.0. How many ways can you get two odd numbers? Solution: This is the exact same problem as the one above. See the solution to 4.1. __________ 4.3 Same as 4.0. How many ways can you get one even and one odd number? Solution: This may seem somewhat complicated as we don't know in what order the even number and odd number appear in our ordered pair. However, if you were paying attention when you read my original combinatorics post, you might be thinking about the 'complement method.' In fact, it's perfect for this problem. On any given roll, one of three things will happen: You'll get two odds You'll get two evens You'll get an even and an odd We already know the number of ways in which the first two outcomes can happen, and we know the total number of outcomes. So, we can use the complement method: Even Odd = Total - Two Odds - Two Evens Even Odd = 36 - 9 - 9 = 18 __________ 4.4 Same as 4.0. Now take the sum of the two upward facing numbers. How many ways can the sum be an even number? Solution: We can only get and even sum in two ways: Even + Even = Even Odd + Odd = Even Using the problems we already solved above, we know that there are nine ways to roll two evens and nine ways to roll two odds. Since these are mutually exclusive events we add to get the total number of outcomes, which is 18. __________ 4.5 Same as 4.0. Now take the product of the two upward facing numbers. How many ways can the product be an even number? Solution: We proceed as above. We get an even product in one of two ways Even X Even = Even Even X Odd = Odd X Even = Even The number of ways the first even can happen is nine and the number of ways the second event can happen is 18. Since they are mutually exclusive, we add, and get 27. 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