The Life Cycle of a Star.

The Life Cycle of a Star. From a Dust Cloud to a Black Hole, Here’s What You Need to Know about a Star’s Life Cycle. ChaptersWhat is a Star?How is a Star Formed?What are the Stages in the Life of a Star?What Happens Next? It Depends on the Size of the Star.Look up into the sky. That thing shining there â€" or ‘burning ferociously’ as might be a better description â€" has been doing its thing for four and a half billion years. Throughout the course of human history, it’s been there the whole time, burning away and giving life to our planet.4.6 billion years. That’s a number that we can hardly even imagine. Yet, scientists reckon that our Sun is about halfway through its lifecycle.Give it another four billion years and it will, like all stars do, ultimately ‘die’ â€" changing from a main sequence star into a supernova or into what is known as a planetary nebula. We ain’t gonna be alive to see any of this happen â€" and, by we, we mean planet earth.Here, we’re going to be talking about the life of a star. We’re going to be talking about the forces that bring it together and that forc e it ultimately apart. And we’re going to be talking about the things that help it keep its shape and size for the unimaginably long time that occurs in between.We’ll be talking about those things with which you’ve probably heard in passing: the red giant, neutron stars, black holes â€" and white dwarfs and supernovas.All of these things make up phases in the life of a star. But let’s take a closer look.In the meantime, find out everything about astronomy, and the differences between astronomy, astrophysics, and cosmology. The sun is the centre of our solar system - not the Earth PetarPhysics Teacher 5.00 (11) £40/h1st lesson free!Discover all our tutors MyriamPhysics Teacher 5.00 (13) £20/h1st lesson free!Discover all our tutors Dr parikhPhysics Teacher 5.00 (8) £40/h1st lesson free!Discover all our tutors RubenPhysics Teacher 5.00 (1) £15/h1st lesson free!Discover all our tutors FrankPhysics Teacher 5.00 (8) £90/h1st lesson free!Discover all our tutors JidePhysics Teacher 4.80 (5) £30/h1st lesson free!Discover all our tutors DorothyPhysics Teacher 5.00 (5) £40/h1st lesson free!Discover all our tutors RahulPhysics Teacher £20/h1st lesson free!Discover all our tutorsWhat is a Star?Now we’re all quite familiar with the Sun â€" from a distance at least. And, unless you live in the biggest of cities and have never looked up, it is pretty likely that you will have seen some stars in your life too.But do you know what a star actually is? Apart from the fact that it ‘twinkles’ and is in ‘the sky’.A star is a massive object in space, held together by gravitational forces, that is distinguished from a planet by its luminosity â€" or the fact that it produces light.That’s the short answer. Now to the long answer.A star is a ball of plasma and gas that radiates energy in the form of heat and light. This radiation is due to the thermonuclear fusion of hydrogen into helium that occurs at its core.All this wouldn’t happen if stars weren’t so big. But, under the force of gravity, and under all sorts of other molecular forces, atoms are smashed together, and new elements are formed. All of this releases energy. This process, which we call nuclear fusion, is, incidentally, something we hope to replicate on Earth â€" because of the sheer amount of energy that it produces.But no, stars aren’t really ‘burning’ or ‘on fire’ or any of these words we use to describe them. Rather, the processes that are happening mean that the sun is much hotter and much more en ergetic than any fire we’ve ever seen.Learn more about our solar system!How is a Star Formed?But why does all this happen? One of the most amazing things about our universe is that there is anything at all in it. As the philosopher Gottfried Wilhelm Leibniz once asked, how come there is something rather than nothing? This question is a little relevant when it comes to thinking that stars produce the very conditions that support life.Nebulae.Imagine an empty, desperately cold space filled with dust and gases that are the debris of old planets and stars. Star formation begins when, in this intense cold, all of this interstellar dust and gas slowly starts to clump together. Gases reach higher densities in the cold, whilst the atoms bind together.This is the first step in the life cycle of stars: the planetary nebulae, these molecular clouds that drift across the universe.As soon as higher densities are reached, the gravitational forces get stronger, meaning that all of the gases and particles in the nebula slowly start coming together. These great big molecular clouds then start collapsing and, as they start moving in on each other, the heat increases.With all this stuff clumping together, the core becomes what will later be the star â€" or often even two or three stars known as star clusters. Meanwhile, different parts of the cloud might become planets or might just stay as dust â€" as in our solar system.(All of this, by the way, takes about ten million years. As a comparison, humans have been about for only two hundred thousand years.)Learn about some of the major astronomical discoveries. Nebulae are some of the most beautiful things in the universe.What are the Stages in the Life of a Star?So far we have seen how stars are created â€" from the big messy clouds of dust and gas in the universe. But what these nebulae create are hardly even stars just yet. Rather, they are protostars, which are the very beginning of the star life cycle.Protostars.After the initial phase as a nebula, the start of stellar evolution is in the protostar. This is when the star is essentially still growing â€" when it is still gathering dust and material from the cloud that formed it.The protostar begins with only one percent of the mass of its future self. But, with all of the mass that is ‘infalling’ due to the core’s gravity, it builds up relatively quickly.Only when thermonuclear fusion begins at the core does the star stop being a protostar and becomes instead a main sequence star. At this point the star’s mass is stable â€" as it produces a ‘stellar wind’ that prevents the i nfall of further mass.Brown Dwarfs.If protostars don’t become big enough, however â€" and by that we mean about eight percent of the size of the sun â€" they never really become stars at all. Instead, they become brown dwarfs, sort of failed stars in which thermonuclear fusion does not take place.Find and take classes with a maths physics tutor here.What Happens Next? It Depends on the Size of the Star.After the protostar stage, the nature and processes of a star’s life cycle depends on the particular star’s mass. And so we’ll split this here into two separate streams.There are those stars that have roughly the mass of the Sun â€" the sun being fairly ‘normal’-sized as far as stars go. Then there are those that are much bigger. The bigger stars are, the quicker they burn. So, whilst Sun-sized stars remain as main sequence stars for about ten billion years, a massive star would live less long.Even so, about ninety percent of a star’s life is as a main sequence star â€" in which it will casually fuse hydrogen into helium. When the hydrogen in its core runs out, the core will begin to collapse and will get much hotter.As the core increases in heat, it pushes the rest of the star outwards, meaning that the outer edges cool.Stars the Size of the Sun â€" Roughly.The most commonly sized stars are stars the size of the Sun. After about ten billion years, once they have run out of hydrogen, they slowly become white dwarfs.White Dwarfs.White dwarfs are cool little things that have perplexed scientists despite their commonness. Imagine the mass of the sun all in the space of the earth and you’ve got yourself a white dwarf. And bizarrely, they are denser the smaller they are â€" meaning the bigger stars would form the smallest white dwarfs.They are hugely dense things that keep themselves from collapsing further due to the activity of electrons. However, with no way of producing energy, there is nothing really that keeps them together. So, gradually coolin g, they just tend to fade away.Massive Stars.Massive stars have a different end in store for them.If a star is about eight times bigger than the sun, you can expect it to end in a massive explosion known as a supernova.Remember that the bigger the star the quicker they burn through hydrogen. And when they have run out of hydrogen, they produce iron as the result of a long series of chemical reactions. When that happens, the core collapses in a matter of seconds from five thousand miles across to just twelve.Temperatures reach a hundred billion degrees and the supernova becomes brighter than a whole galaxy. This is the first picture ever taken of a black hole.What is a Black Hole?Particularly dense stars produce one of the most fascinating phenomena in the universe when they die. These are black holes.Rather than exploding outwards, these stars implode, collapsing into themselves to form an object so dense that nothing â€" not even light â€" can escape it.These things pull everything around them into themselves â€" whilst emitting huge amounts of radiation.Is there life in the universe?

Brush Up Your Math Skills with Online Math Tutoring

Brush Up Your Math Skills with Online Math Tutoring 0SHARESShare Any new grade baffles student mind and needs some strong support in that stage to get used to concepts and do better in performances. Math online tutoring is an essential server for this purpose to do Math without anxiety and trouble. 4Th Grade Math demands the basic strength of Math skills of previous year classes to prep for the present grade. Hence , 4th  grade Math tutoring online is the right support for the student to   brush up his previous year skills and get ready for the proper applications of them in the present year studies Same way, 5Th Grade Math expects a culmination of all the Math skills learnt so far for doing Whole numbers, Fraction, Measurements and Decimals. Here too, 5th  grade Math tutoring online comes handy to help the students in recollecting their past Math  skills and come out successful in coping with the new environment in Math studies Math tutoring online is required for the following reasons Accumulate the past skills with a new touch of energy Replenish the skills where you lack Get more accuracy in calculation Strengthen the knowledge of concepts with repetitive practice Gain a positive approach towards Math doing Tutor Pace.com provides opportunities for the students for acquiring all these skills with the help of its capable tutors. [starbox id=admin]

Mary Cassatt

Mary Cassatt Mary Cassatt: Give me Pastels; Give me Gouache! ChaptersYoung Mary: Getting a Feel for ArtThe Life of an Artist in ParisOn the Quest for AcceptanceMary Cassatt, ImpressionistMary to the EndWhen one thinks of great art, French painting and anything from the Italian Renaissance period comes readily to mind.If one is inclined toward van Gogh â€" impressionist painting as well as expressionism, works by the likes of Paul Klee or Edvard Munch easily spring forth.Does one dare throw The Boating Party in the ‘great art’ mix? What about Mother and Child or In The Box?These and other great artworks by the American painter in Paris are often overlooked in favour of other, more renown tableaux. Mostly, the name Cassatt draws... at best puzzled stares, at worse, blank looks.The sad part is that that is the treatment Marie Cassatt suffered for just about her entire artistic career.Today, we put an end to that by understanding who that American woman with the French-sounding name was. Why was she so set on painting? Why couldn’t she see h erself marrying, settling down and raising children, as was expected of her?We look at the tender scenes she painted that belied her struggle for acceptance as an artist of merit. We uncover just how much she was willing to put on the line in the name of art and, in the end, how she was undone by a certain magnificent art installation.Superprof now presents Mary Cassatt. FernandoPainting Teacher 5.00 (8) £30/h1st lesson free!Discover all our tutors EmmaPainting Teacher 5.00 (5) £75/h1st lesson free!Discover all our tutors DavidPainting Teacher 4.75 (4) £40/h1st lesson free!Discover all our tutors NatashaPainting Teacher 5.00 (7) £37/h1st lesson free!Discover all our tutors StefaniaPainting Teacher 5.00 (4) £40/h1st lesson free!Discover all our tutors SakinaPainting Teacher 5.00 (1) £12/h1st lesson free!Discover all our tutors LorrenPainting Teacher 5.00 (2) £18/h1st lesson free!Discover all our tutors Off2themoonholliePainting Teacher £8/h1st lesson free!Discover all our tutorsYoung Mary: Getting a Feel for Art Mary often asked her neighbours in France to pose for her By Mary Cassatt via WikipediaMary was the second-born in a large, upper-class family living on the outskirts of Pittsburgh. When she entered the world on May 1844, her father was a successful stockbroker and land speculator.The family had a rather elevated pedigree; Dad was a descendant of the original Dutch settlers in Manhattan. Mum too came from a privileged background; she was well-read, with a taste for refinement.It was her mother who introduced her to art.She believed that proper education involved exposure to the wider world. Thus, she took her brood to the capitals of Europe: Berlin, Paris, London and Rome. Little Mary drank it all in, even learning to speak German and French.She also had her first art lessons â€" drawing and music, while in Europe.Before Mary was of school age, the family moved east, ultimately settling in Philadelphia, where she attended school.Upon completion of her compulsory education, Mary and h er father endured a lot of friction.Mary wanted to study art and her father could see no point in it. Her mother may have interceded on her behalf; at age 15, she was enroled in the Pennsylvania Academy of the Fine Arts.Her mother and certainly her father must have had reservations about their teen-aged daughter being exposed to feminist ideas and other vagaries that may corrupt her; they especially feared predatory males.Their fears were for nought. Although the student body was overwhelmingly male, those students nevertheless believed that women and men could create art equally well and should enjoy the same freedom to do so.Being exposed to such liberalism at such a tender age shaped Mary’s worldview. For her entire life, she and all of her classmates remained advocates of equal rights.While the temperament suited Mary well, she grew ever more frustrated with two aspects of her art studies:1. About 20% of the student body was female but, unlike her, they were only learning how to paint as a social skill; they did not have any intention of pursuing a career in art.2. Because she was most serious about her art, she was subjected to patronising attitudes from both her classmates and instructors.That last was most injurious and, added to her frustration with the slow pace of the instruction and gender-based restrictions such as being forbidden to work with male models, Mary soon found she had no choice but to set off on her own.By contrast, Artemisia Gentileschi had no such credibility problems!The Life of an Artist in ParisAs there would be no certificate or degree for all of the academic art she created, Mary again tussled with her family, finally wrangling permission to establish herself in Europe, to learn how to paint from the masters.So, she set off, at the age of 22, with a rotation of chaperones keeping her company.However progressive Mary thought the Paris art scene would be, she must have been sorely disappointed to learn that females were barred fr om the School of Fine Arts. Or maybe she was angry!Undeterred, she petitioned for private lessons with the art teachers. Finally, Jean-Leon Jerome, a painter and sculptor in his own right, agreed to take her on.He might have been surprised at her dedication to learning; not only did she attend every session without fail but, when not in the studio, she was haunting the Louvre, copying the original paintings hanging there.She may have been dismayed at the number of female artists in that art museum, all busily sketching, as she was. She soon discovered that it was quite common for a female artist to sell their sketches to tourists because of the many roadblocks to their being an artist.One such block for  women artists in Paris must have been that they were denied entry to the cafés where all of the great (male) painters tarried. Nevertheless, Marie met and built ties with some of the greatest names in western art: Edouard Manet, Auguste Renoir and, most importantly, Edgar Degas. Wi th him, she enjoyed a lifelong artistic partnership.Do you know who Frida Kahlo’s mentor was?And then, war broke out. Her early impressionist tableaux showed social scenes such as theater and carnivals Source: WikipediaAlso learn about Georgia O’Keeffe, another great American painter... In her later years as an artist, Mary depicted mainly maternal scenes Source: Mary Cassatt.org via WikipediaMary to the EndMary had decided, early in her career as an artist, that she would never marry. Not only was marriage incompatible with the life of an artist but she felt she simply did not have the temperament for it.Indeed, Mary could be ill-tempered, self-centred and narcissistic. Perhaps those traits are a job requirement for artists.Still, she painted scenes of motherhood with such tenderness and, one might think, so much longing that some say she enjoyed motherhood through her depictions.Still, if she missed being a mother or even being married, she gave no outward signs of it; she continued to travel and paint well into old age.At 74, on a journey through Egypt and, upon seeing the ancient works on display in their museum of art, she became so overwhelmed by their beauty and magnitude that her creativity was stifled, at least for a time.Finally, at 80 years old, she gave up painting â€" but only because she could no longer see and it pained her greatly to hold a brush. She lived another 12 years, tortured by the fact that she could not see her beautiful world, much less paint it.Mary Cassatt is but one of many famous female painters who gave up everything for art, making our world more beautiful in the process.

Breeze Through College Math via Online College Math Tutors

Breeze Through College Math via Online College Math Tutors Math is regarded as a tough spot in advanced course of learning for the complexity the subject provides. One has to choose Algebra, Calculus or Geometry- which are all branches of Math involving lots of mental exercise, practice and hard work. Gathering subject skills in all these courses is not an easy thing. It also needs fundamental Math skills to ace College Math. Reasons for college Math being troublesome Students involve themselves in many social interactions and most of them earn and learn that they find no adequate time to concentrate on their Math courses at college level and thus are unable to meet the expectations of their professors and teachers while submitting their homework tasks, projects, assignments and various other educational enterprises. The subject is tough and it needs much concentration as well. Math homework help from renowned tutoring sites like Tutor Pace can be the right solution for such students. Many of the students lack in the fundamental skills in topics as in the case of Algebra or Calculus and thus find the classes extremely out of their reach and alien to their knowledge. At college level, they cannot expect their teachers to spoon feed them at every instance and thus homework assignments pose threats that are overwhelming and unmanageable. The solution in the hands of virtual college Math tutors Seeking remedies for learning issues in college Math at the earliest is good and beneficial for students and they can get this support from the hands of virtual Math tutors who prove their best in times of necessity. The virtual tutors with their expertise are able to thwart any kind of challenge in college Math homework and they also brush up students’ ground knowledge in topics for their betterment in learning advanced topics in subjects like Algebra. Algebra homework help from e-tutoring sites can bring forth suitable solutions for solving Algebra equations and formulas through personalized methods and one on one attention from the tutors. Why Tutor Pace is your right destination for college Math? Subject experts with easy to understand methods and solutions 24/7 services without geographic and  time boundaries Excellent customer care with instant response Screened and certified tutors with experience to handle any tough aspect of college Math Custom made strategies allowing one to learn at his pace Worksheets and practice sheets for strengthening fundamental Math skills Cloud based tech support for uninterrupted communication Affordable prices Conclusion So, it is in your hands to go in for the right destination for your college Math. Seek remedies on time for excelling in your subject through online college Math tutors. Math waits for no time as Math learning is a continuous process. It is good to start seeking tutoring help once you feel that you miss something in your college Math.

10 major struggles when learning the Dutch language

10 major struggles when learning the Dutch language Difficulties when learning the Dutch language ChaptersThe ten main struggles when learning the Dutch languageIn general, the Dutch language is viewed as one that is exceptionally difficult to learn. Here, however, we attempt to make it as simple as possible for you. Fortunately, the alphabet is similar to the English one. That’s a good start. The Dutch language has also borrowed many words from various languages including Italian, French, English and German. Therefore, you might recognize a few words here and there. So, if you speak any of the other languages mentioned, that can be an advantage for you.Nonetheless, Dutch is still a difficult language. Especially the pronunciation and the grammar can be pure horror. Just like with any other language there are certain common mistakes that are made. In this blog, we will discuss ten of the most frustrating issues that can occur when you are trying to learn Dutch. We hope you can put the information provided to good use. Good luck with practising and we are sure that you will mana ge in the end!Some beautiful Dutch flowers. JessicaDutch Teacher 5.00 (5) £21/h1st lesson free!Discover all our tutors NaomiDutch Teacher £17/h1st lesson free!Discover all our tutors EliseDutch Teacher £45/h1st lesson free!Discover all our tutors MichelleDutch Teacher £20/h1st lesson free!Discover all our tutors CéliaDutch Teacher £15/h1st lesson free!Discover all our tutors VanshikaDutch Teacher £25/h1st lesson free!Discover all our tutors MichelleDutch Teacher £20/h1st lesson free!Discover all our tutors LizaDutch Teacher £15/h1st lesson free!Discover all our tutors

Learn English Vocabulary connected to Losing a Job - Real Life English

Learn English Vocabulary connected to Losing a Job - Real Life English Here you will learn useful English vocabulary connected to losing a job. What other words can you use in English to say she has just lost her job or he lost his job last week? Learn English Vocabulary connected to Losing a Job SACKEDThe most common way to describe leaving your job. It usually suggests it was not your decision.He was late too many times for work so his boss  gave him the sack  (or he got the sack).FIREDMore formal way to describe losing your jobI got fired from work two days ago.RESIGNEDUsed when you are in control of the decision to leave.I had had enough of his orders I decided to try something different so resigned  yesterday.TO BE PUSHED OUT/PUSHED ASIDE  When you are no longer needed or you are over looked for promotion.He was unsettled when the company was taken over. The new bosses were different from the previous owners. They really wanted him to leave. He got the feeling that he was being pushed out. He was no longer needed and was pushed aside when it came to promotions and his assistant was promoted above him.SURPLUS TO REQUIREMENTSThis refers to someone who is no longer seen as useful or important to a business.He was called in to his bosses office. They said that due to the lac k of new business they needed to reorganise my unit and unfortunately I  was now surplus to requirements and would be made redundant by the following month.A BIT LONG IN THE TOOTH FOR THAT WORK  When someone has been working for many years in the one place and is almost part of the furniture. The bosses want to make way for younger more energetic staff.They asked me if I  thought i was a little long in the tooth for this business, after all I  had been there for over 20 years. Other Words for Losing a Job - Infographic Enjoyed this infographic? Here's what you can do next: REDUNDANTCompanies reorganise all the time and in bad times or due now to improved efficiencies many people lose their jobs in this way. They are  made redundant. Their job no longer exists.DOWN SIZEDOne of these wonderful euphemisms (ways to say something bad in a softer way to ease the pain!!). The company had to reduce its workforce due to the economic situation.They down sized  and over 100 people lost their jobs.LET GOA more casual or informal way to refer to a job loss.Poor Michael lost his job last week that’s the second time this year he has been “let go”.JUMPED BEFORE HE WAS PUSHEDWhen someone decides it is best to leave a company on his own terms before they ask him to leave.He had been unsettled for a few months. He had the impression that there were going to be changes in his are. He handed in his resignation and thought it was best to jump before he was pushed out.LEFT OF HIS OWN RECORD  Sometimes when people leave companies the market place is never sure whether t he decided to leave or were they asked to leave. When someone decides themselves that it is time to move on,  then  we can say he/she left of their own accord. It was their decision.David had had enough.Late nights early mornings and no appreciation for his efforts. He handed in his notice and left of his own accord three weeks later.TOOK UP ANOTHER POSITIONPeople leave companies every day and for many different reasons. Without giving any great detail such announcements often simply state thatMichael has decided to leave the company to take up another position. We wish him the best of luck in his future career.PUT OUT TO PASTURETo be forced to retire. It’s what they do to old horses and donkeys. When a horse can no longer run it is put in a filed  (pasture)  where it can spend its remaining life eating grass.Mark was good at his job.Everybody acknowledged that. However, if they did not make room for the younger guys they would leave. “They dont need me any more,  I  am being pu t out to pasture” he told his wife.There are many others. To be re-engineered! Be wary when your boss tells you he is bringing in some consultants to look at some re-engineering  (changes) to the business.

Our first verified tutor in Manchester!

Our first verified tutor in Manchester! We've had an amazing 2015 here at Tutorfair thanks to our fantastic community of tutors and students!  We asked a few of our favourites to share their adventures with us.  Here,  Frances tells us about becoming Manchester's first verified tutor! Frances G Having finished the Teach First Leadership Development program this summer and starting work for Parent Hub, I wanted to keep involved with the  Teach First  mission that no child's academic success should be limited by their socio-economic background.  Tutorfair provide an excellent opportunity  to work with young people in my local area and also contribute improving educational disadvantage across the UK! A few weeks  ago I attended a TutorFair Fast Track Event in London â€" I saw it was a great way to get more clients and I really wanted to improve my TF profile. I was the first tutor from Manchester to attend, and I’m really glad I did. It really made me feel part of the Tutor Fair community, and it was great to meet the staff in person â€" Annabel and Sam were so lovely and gave me lots of top tips to improve my profile even further. I hadn’t worked with any clients before the Fast Track event, but in the last two weeks I’ve booked two new clients and I’m in the process of booking lessons with 3 more! Working as a tutor is brilliant, and I love how easy it is to use the Tutor Fair website â€" from booking a lesson to getting paid, I feel like the hard work is all done for me! I’m looking forward to taking on more clients in the New Year, and I’ve had really positive feedback for the lesson I’ve already taught. I can definitely recommend attending the Fast Track event, (even making the video wasn’t as cringey as I thought it would be!) and tutoring with Tutor Fair. Want to  book Frances  for  Science  revision?  Or perhaps you want to  come along to one of Fast Track events? There's something for everyone at Tutorfair!

Tips from an Anaheim Math Tutor Should You Cross Multiply

Tips from an Anaheim Math Tutor Should You Cross Multiply Anaheim Math Tutor Tips: Should You Cross Multiply? Anaheim Math Tutor Tips: Should You Cross Multiply? Cross multiply is an often-used term in mathematics that often sticks in peoples memories long after they graduate high school and stop taking math classes. However, most of those people dont really know what it is and why it works. So many students are just taught to cross multiply when they see an equation with fractions. Unlike many other memorized phrases in math, cross multiplication is actually a useful tool that often makes problems with fractions a little bit easier sign up for your private Anaheim math tutor. But just like anything else you memorize in math, you should understand what youre doing rather than just memorizing a seemingly arbitrary procedure. Cross multiplication is just the undoing of division that is being done. If you have two fractions, that is the same as dividing by something on each side. And, just like we can subtract to undo addition, we can multiply to undo the division that is making those pesky fractions. Here are three examples that we will solve both with and without cross multiplying. You can see that there are always other ways to solve these problems, as well as why cross multiplying works and is a little bit simpler. As a general rule of thumb, I do not teach cross multiplication without making sure my students know how to solve the problems without it first. That way I help promote a stronger understanding of how to solve simple algebraic equations that will hopefully stick with them and translate to other math skills rather than leaving them thinking cross multiply is how you have to solve fractions. Here is the first example: In this example, we have two fractions and need to solve for x. To help find x, we can notice that the fraction on the right can be reduced: Now we have successfully eliminated the fraction on the right. To solve for x, we now only have to divide both sides by 2. There are a few ways to think about doing this, including using your calculator or converting the left fraction to a decimal. Ill not that two-fifths divided by two leaves us with one fifth: We get a solution of one fifth (1/5) which is equal to 0.2. No cross multiplying was needed, only our standard multiplication and division. Lets do the same with cross multiplication: Here, we didnt think about reducing the initial fraction. We simply cross multiplied: taking the diagonal terms in our fractions and multiplying them together (one set in green and the other in purple). This makes the cross of our cross multiplication. After we do this, we get a fairly trivial equation of 8 = 40x to solved. We divide both sides by 40 and get the same answer we did above. Example 2: Now the x is on the bottom of the fraction. To solve this, we could multiply both sides by x to undo the 7 being divided by an x on the left. However, another way would be to make the right side of the equation look like the left. We notice that the left side has a 7 on top, but the right side has a 14. We can change the 14 into a 7 by halving the top and bottom of the right: Let’s use cross multiplication to solve the same problem: No extra thinking here or even trying to get the x out of the bottom of the fraction. We simply follow the cross multiplying procedure and multiply the green together and the purples to get our final, simple equation that yields the same 1.5 answer. Example 3: Our last example gets a little more complex with terms in parentheses. Cross multiplying here would probably be the best choice, but its never the only choice. The fraction on the left is a term being divided by 6. The fraction on the right is a term being divided by 9. We can distribute the division (just like you would distribute multiplication into parentheses) to get an equation we can solve: We could have kept our work as fractions instead of decimals, but this is likely how it would look if you were using a calculator to help you solve the problem, and decimals are equally as valid as long as we dont round them. Lets see how it looks using cross multiplication: Here we get the same answer as above by following the exact same cross multiplication procedure. Notice that we still had to distribute when we multiplied the terms in parentheses. Cross multiplying can be a very helpful tool when solving equations with fractions. However, it should not be thought of as the only way to do so, and it should not be the only way students are taught. Thinking this way will continue to lead to many students being confused about fractions and unsure of how to work with them. Cross multiply doesnt have to go away, but the blind memorization of it should. Michael C. is currently a private math, science, and standardized test tutor with TutorNerds in Irvine and Anaheim. All blog entries, with the exception of guest bloggers, are written by Tutor Nerds. Are you an education professional? If so, email us at pr@tutornerds.com for guest blogging and collaborations. We want to make this the best free education resource in SoCal, so feel free to suggest what you would like to see us write.