Записывайте свои чувства. Придумайте себе занятие. Живите настоящим. Не забывайте заботиться о себе.

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If you want to keep busy, it may be easier to do so once you get negative feelings out of your head. Get a pen and paper and jot down what you're feeling for a little bit each day. Then, once your feelings are out, you can focus your energy elsewhere.  Make a list of everything he did to disappoint you. For example, write things like, "He never held my hand in public" and "He didn't want to call me his girlfriend." Once you're done, set the list aside and try to focus your energy somewhere else. In the wake of a breakup, you may find yourself constantly thinking about what went wrong and why. It can be helpful to have new goals. This will help you stay busy and avoid thinking about him.  It is important to strike a balance between activity/distraction and rest/reflection. Thinking about it too much can result in depression, but running away from it can keep you from healing. Think about something you always wanted to do. For example, did you always want to take up running? Try developing a workout routine that will teach you to run. People often become obsessive over past lovers. Sometimes, the easiest way to beat one obsession is to replace it with another. Try to stay present instead of lingering over what happened in the past. At any given moment, keep your mind on what you're feeling and experiencing right now.  Try to enjoy the small day to day things. Even something as simple as eating a great lunch should be savored. If you find yourself thinking about your ex, stop and think something like, "That was then. I'm going to focus on what I'm doing and how I'm feeling right now." You can use grounding techniques to get off the hamster wheel of thinking about him and bring you into the present. Look around and name five things you see. Five colors. Five textures. Close your eyes and identify what you hear or smell. Self care is important, especially after someone hurt you. Do not neglect the normal rituals of eating right, exercising, and caring for yourself overall.  Keep up with personal hygiene. Even if...


Записывайте собственные чувства. Улыбайтесь и смейтесь. Попробуйте выплакаться. Постарайтесь посмотреть на сложившуюся ситуацию шире. Отвлеките себя. Начните читать.

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Consider buying a journal that you can designate just for writing down your thoughts and feelings. Often it is helpful to document your feelings of sadness in writing in order to better make sense of them. This will help you get "in tune" with yourself and achieve a deeper level of self-understanding.  Journaling - even for 20 minutes a day - works to clarify and organize your thoughts about your sadness, which may help you to better pinpoint the reasons why you are feeling sad. It also helps you keep track of your behavioral and emotional patterns. Additionally, some research has found that journaling can even help your physical health and strengthen your immune system because it works to reduce stress.  While writing, concentrate on what you are writing, not how you are writing it. In other words, don't focus on your grammar or spelling. An example of a journal entry may look something like this: "I have had a particularly rough day today - my sad thoughts about the divorce will not leave me alone. Sometimes I'm not sure if I should even still feel sad about it because even though it happened a year ago, the marriage had been dead for a long time. I know that. But I do worry that I'm stuck in the past and that my kids are suffering for it. I'm also angry at myself for not being able to let myself let the sadness past go. People get divorced all the time, so why am I having such a hard time? Talking with my sister helped the last time I had a rough day; I think I'll call her. I know tomorrow is another day.” Studies have shown that the very act of smiling can improve your mood and make you feel happier In addition, laughing releases endorphins, those brain chemicals that enhance your mood.  Even though you may not feel like it, even using those small muscles that facilitate laughing and smiling can make you happier. So, try faking it. It may be forced at first, but laughing or smiling could trigger a funny or happy memory and cause a real laugh or smile to occur.  If you want to try smiling or laughing...


Записывайте физические упражнения. Записывайте информацию о пищевой ценности. Следите за своим продвижением к цели. Используйте дневник питания для контроля за расходами.

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If you're keeping your food diary as a way to track calories and get in shape, it makes sense to write down your physical activity, too.   Record the type of activity and time spent doing it.  If you can, also add about how many calories you've burned during that particular exercise. See how your level of exercise affects your hunger level and what you eat.  Make notes whether you've noticed increased hunger overall or if you experience increased hunger immediately after a workout. If you're keeping a food diary to make sure you're getting enough of a certain nutrient, you might want to record the nutrition information for each item. It's easy to find nutrition information for any type of food online and many food journal apps automatically provide this for you.  Examples of nutrients to track include:  Fiber Protein Carbohydrates Iron Vitamin D A food diary can be a motivating tool when you have a goal you want to reach that's related to your diet. Whether you want to lose weight or you're just trying to eat more fruit and vegetables, tracking your progress will inspire you to keep going and show you where there's still room for improvement. Here are a few ways you can track it:  Record your weight. Write it down at the end of each week so you can see how it has fluctuated. Note important milestones. If you successfully eliminated gluten from your diet for a month, note it in your diary. Record how much exercise you can perform. For example, record your progress toward running a 5k. Since you're already writing down everything you eat, why not also write down how much it costs? It's a great way to stay within your food budget for each day, week, and month. You might be surprised when you find out where you tend to spend the most money.  Note how much you spent on each meal. Include meals you make at home as well as the meals you eat out. Look for patterns to determine how much you spend on food each week or month, and find places where you can cut back. It might be helpful to tally up how much you spend on...


Записывайте, на что вы тратите деньги. Создайте бюджет. Решите, какую часть вашего бюджета вы хотите выделить для хобби. Выберите бесплатное или дешевое хобби, если у вас недостаточно средств.

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Take a month to note down where you spend all of your money. You can use a smartphone app to help with this project, or you can rely mostly on your bank account if you don't use much cash. Separate out what you spend into categories. For instance, you can have categories such as "groceries", "gas", "clothing", "eating out", "entertainment", "rent", "bills" and "fees". You could also separate your bills into two categories: ones that are absolutely necessary, such as insurance, and ones that you may want to reduce or remove, such as cable and your phone. On a spreadsheet or app, identify what percentage of your money is taken up by necessary items, such as rent and bills. Also, use your last month's spending to see how much you spend on gas and groceries. Determine how much you have leftover for discretionary spending. If you are starting a new hobby, some money must come from other areas. For instance, you might want to cut down on other entertainment or stop eating out as much. Maybe you could spend less at the grocery store. How much you allocate depends on what hobby you choose, as some are more expensive than others. You have a number of options available to you if you need a cheaper hobby. For instance, you could read or write, take up running, or try gardening or camping.


Запишите ваш чистый получаемый ежемесячный доход. Исключите из вашего чистого ежемесячного дохода часть, направляемую на сбережение. Распишите все ваши ежемесячные расходы по трем отдельным категориям. Вычтем общую сумму расходов из общего дохода за месяц. Если ваш бюджет трещит по швам, пересмотрите  гибкие и дискреционные расходы. Посмотрите, сколько денег сейчас остается от вашего дохода, после покрытия  всех расходов. Пересматривайте бюджетный план в конце каждого доходного периода, для того, чтобы убедиться, что вы находитесь в рамках запланированных цифр.

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Net income is the amount that you actually get to take home, after all the deductions (taxes, health care) have been subtracted. Include other sources of income as well if applicable, such as tips, monthly  bonuses, cost of living increases, dividends, interest income, etc. What do you do if your income is constantly changing? You pretty much have to do things topsy-turvy. Determine all your essential, priority and lifestyle expenses first. Then, with your paycheck, start paying off all your essential needs first, followed by your priority and then lifestyle expenses. The money you have left over can either be set aside for an emergency account or stashed away into savings. Ideally, set up an automatic withdrawal into another account so that you aren't even tempted to touch it. If you never see it, you won't miss it. Squirreling away savings will help you anticipate emergency scenarios and prepare you for retirement.  How much money should you try to save? It mostly depends on your salary, but a good benchmark is 15% to 20%. If you can only afford to save 10% of your annual income, that's fine, provided that you save some of it. Take advantage of matching contributions from your employer, if possible. If your employer matches contributions to your 401(k) — up to a certain percent — take full advantage of their generosity. It's the closest thing you'll get to "free money" in your lifetime. These categories are "fixed," "flexible," and "discretionary."  Fixed expenses remain the same each month, such as a mortgages or rent, a loan payment, insurance or medical premiums, etc. Total up all fixed expenses. Flexible expenses include items that are necessary, but in which you can control the amount of money spent on, such as household and grocery items, clothing, utilities, etc. Total up all flexible expenses. Discretionary expenses are items that are not necessary for survival. These could include expenses for recreation such as movies, travels, and impulse buys. If your expense to income ratio is out of balance...


Запишите ваши баллы. Запишите максимально возможный балл за каждое задание. Сложите числа в каждой колонке. Вычислите средний балл. Полученный средний балл (он будет выражен десятичной дробью) умножьте на 100. Определите итоговый балл.

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Find your scores from each assignment, quiz, homework, etc. throughout the term. Sometimes this will be collected for you online, such as in a system like Blackboard. Sometimes, you'll have to go through your graded work. Write them all in one column on a piece of paper to refer back to. If things like participation or discussion are included as percentages of your final grade, you may have to ask your teacher or professor what your grade on this part is. Refer to the syllabus for the grading system. Teachers employ various systems to determine final grades, but two common systems are points and percentages. Either way, write out the total possible points in the second column next to the score you received.  In a points-based system, there is a maximum number of points you can earn in the class. Each assignment has a certain number of points assigned. For example, there may be 200 total points available in a course, split up into four assignments each worth a maximum of 50 points (4x50=200). In a percentage-based system, each assignment will be worth a certain percentage of your grade. These percentages add up to 100%. For example, you might have four assignments due, each worth 25% of the grade (4x25=100). Note that in these examples, each assignment is worth the same weight overall in your course, even though the numbers are different. Do this whether your assignments are graded on a percentage scale or some other total. Add up all the numbers for the first column and write the total at the bottom. Add up the numbers in the second column and write the total below.  For example, say you have 5 graded course activities total. 2 of these activities were exams worth 20 points each. 2 of these activities were quizzes worth 10 points each. The final activity was an assignment worth 5 points. 20+20+10+10+5= 65. This is total number of points available in the class. Now add up your scores. Let us say you scored an 18/20 on your first exam, a 15/20 on your second exam, a 7/10 on your first quiz, a 9/10 on your...


Запишите ваши положительные качества. Переосмыслите свой негативный жизненный опыт. Примите себя. Смотрите на все с позитивной точки зрения. Напоминайте себе, что вы стараетесь на полную. Цените ваши обстоятельства. Ежедневно записывайте в дневник, в чем именно вы старались по максимуму.

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In order to excel at life, you have to want to excel at life.  You will have to develop discipline, perseverance, resilience, and drive.  The easiest road to developing these qualities is learning to value yourself for your abilities, courage, worth, and valuing your existence  Engage in positive self-talk by writing down all the positive qualities you can think of about yourself.  Add to this list as much as you can. Read this list every morning when you start your day.  You are the agent in creating your life, so if you are going to be mentally and emotionally prepared to strive to excel in your life, you need to nurture your self-confidence to the point that you believe in yourself. Celebrate yourself for even wanting to create an excellent life. You may have received negative messages in the past from your childhood, your social experiences, or the society you live in.  Those messages can be much more ingrained than you think.  Write down every negative thing you have ever been told, or that you think about yourself.  Take some time to look at that list and start reframing negative themes that there may be in your life.  For example, everyone has made mistakes in their life.  Are you still holding on to guilt and shame about these mistakes?  Did someone tell you that you were stupid or worthless while you were growing up?  Do you still somehow hold on to that message and let it hinder you? In order to excel at life, you must begin to shed those negative messages and replacing them with positive ones.  An easy example of this that most people engage in is negative self-talk.  Say you dropped your keys on accident.  What are the first thoughts in your head?  Perhaps you think, “I’m so stupid, I can’t even hold on to keys.”  If you pay attention, you may find you speak very harshly to yourself throughout the day.  In striving to excel, you are the coach, the team, and the star player.  You have to start treating yourself as someone who deserves the excellent life you are building. .  A major factor in...


Запишите возможные варианты. Определите возможные последствия. Взвесьте все плюсы и минусы. Примите решение. Если вы не удовлетворены, то попробуйте другой вариант.

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You now have a good idea of what caused each feeling you have about the person. Now you can determine your choices. Write down every way you might respond in this situation. Even if the choice is not ideal, write it down anyway. This will give you a full picture of your choices. For example, your list for a co-worker or acquaintance may look like this now:  Feeling: Bewildered Situation: I was praised by a friend for a project I completed, but this same person criticized me an hour later. Possible choices: Approach the friend, keep it to myself, talk about it with my parents, start school gossip, tell the situation to my teacher, etc. Next to each choice, write down all the possible consequences or outcomes that you can think of. Your list may look like this:  Choice: Approach friend about the issue  Possible consequence: Friend becomes offended Possible consequence: Friend receives the message well Possible consequence: I feel vulnerable telling how the situation affected me   Choice: Keep it to myself  Possible consequence: The problem will continue Possible consequence: The problem may go away on its own Possible consequence: It will continue to bother me   Choice: Talk to my parents about it  Possible consequence: I will feel better about the situation Possible consequence: Nothing at school changes Evaluate the possible outcomes. Think about your level of comfort with each outcome. Consider how you will feel when you make the choice. Consider how the other person will feel. Based on all the possible outcomes, choose one that you feel most comfortable with. This choice should be the one with the best outcome for yourself and the other person involved. Start with the choice that provides a needed outcome coupled with a consequence you are willing to experience.  In the friend situation, for example, starting school gossip would probably not be ideal. The consequences could be painful or could jeopardize your relationship with other friends. At this moment, maybe you would like to start by keeping it to...


Запишите все ваши цели. Выделите время для пересмотра ваших целей и составления плана. Придерживайтесь составленного вами ежедневного расписания. Составляйте списки задач. Выделите пункты и разделы вашего списка разным цветом.

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The first step in being less impulsive is to determine how you want to prioritize your time.Then, before acting impulsively, check and make sure your action reflects the values you’ve written down for yourself.  This might mean writing a  master task list. It might mean creating a business plan. If you're a student, you might want to create a study plan. The important thing is to record your current goals in order to measure them against your impulses. You might think of this written record as a kind of container for your values, intentions, commitments, best practices for yourself, and tasks that support these goals. You can use a notebook, a spreadsheet, or another method that works for you. The main thing is that the framework is accessible and functional. In order to make the most of your organizational system, you’ll need to commit to reviewing how effective your system is in your life, as well as plan out any needed changes.  You might find you need to sit down once a week to consider your actions of the previous week. Did you follow your listed priorities? Note what went well, what you found challenging, and what you might do better. You might find that you need to do this review more often until you find a system that works for you. Doing a brief daily review is recommended; you might also do a more extensive monthly review. If you have lots of unstructured time in your day, you’re more likely to fill it with impulsive acts. Try writing down a daily schedule for yourself, using 30 minute blocks of time. It’s okay if you spend more than one 30-minute block on a task, but don’t leave any blocks empty and unstructured.  If you don’t know how to plan ahead for certain activities, write in a series of choices for yourself. For instance, if you’re not sure yet whether a friend is coming over or not, write: “Friend or ______.” Your daily schedule should include your social time and free time too. A schedule that doesn’t include any downtime will ultimately fail. If you’re the kind of person who’s prone to...


Запишите все настройки радио и аудио. Заглушите машину и откройте капот. Отключите отрицательную клемму аккумулятора. Подождите 10 секунд, затем повторно подключите клемму. Если плеер по-прежнему не включается, замените предохранитель.

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This method is useful when you can't remove a CD because your CD player will no longer turn on. This method involves disconnecting, then reconnecting the electrical supply to the CD player. For most CD players, this means that any radio presets you've set will be erased and your personal audio settings will be reset to their defaults. If you're an avid listener of music in your car, make sure you record your personal settings so they can easily be restored afterward. When manipulating or altering your car's electrical system, it's important to ensure you're not in danger of electrocution. Turn the car off and remove the keys from the ignition, then open the hood for access to the battery. The negative terminal on a car's battery is colored black, while the positive terminal is colored red. Carefully disconnect the negative terminal. Some terminals may require you to use a small wrench or pliers to loosen a nut before removing the wire connection. After reconnecting the terminal, tun the car on and try to eject the CD as normal. Disconnecting then reconnecting the CD player's power supply can cause the CD player to "reset" to its factory default, which, in some cases, can restore its ejection functionality. Check your owner's manual - often, a car's fuse box will be behind a panel somewhere on the driver's side of the dashboard. Disconnect the battery, remove the fuse box's protective covering, then, consulting your owner's manual, replace any of your CD player's fuses that may have blown.


Запишите все свои симптомы. Сделайте анализ крови на анемию. Пройдите физический осмотр.

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When you're going into a doctor, it's always best to be prepared. Make a physical list of the symptoms you've been experiencing. Try to be as specific as possible. Don't be embarrassed; your doctor has heard it all.  For instance, you might write down, "heavier flow (on heavy days, bleeding through a pad every hour for 3 or 4 hours in a row), more cramping, blood clots the size of quarters, feeling weak and tired, blood flow lasting 12 to 14 days." It can help to count the number of pads or tampons you use while you're bleeding. It's also important to note any major changes in your life, such as big events that have caused you stress and sudden weight gain or loss. Ask around in your family to see if anyone else has similar problems, as menstrual problems can be genetic. If you think you may be suffering from a anemia, ask your doctor about having a blood test. A blood test can determine the level of iron in your blood. If you have low iron, your doctor will likely recommend increased iron in both your diet and the supplements you take. Usually, to help diagnose the problem, your doctor will want to conduct a physical examination, including taking a pap smear. A pap smear is when your doctor takes a small scraping of cells from your cervix to test for any problems.  Your doctor may also take tissue from your uterus to biopsy. You may also need an ultrasound or a hysteroscopy. With a hysteroscopy, a tiny camera is threaded into your uterus through your vagina, allowing the doctor to look for problems.


Запишите все то, что вы можете делать благодаря своему телу. Смотритесь в зеркало с признательностью. Оставляйте себе приятные записки.

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Human bodies are amazing treasures with infinite uses, abilities, and surprises. Your body does so much for you on a daily basis. Take a moment to write a list of all the incredible functions that you gratefully notice about your body. Perhaps you can use your legs to run a 5k, your fingers to deftly strum the guitar, or your strong arms to swing your kids around. Rather than focusing on what you hate about your body, think about all the things you love. When you look at yourself in the mirror, don't search for flaws. Instead, strive to uncover aspects about your body that you appreciate. Maybe you have quirky dimples, a scar that adds character, or striking eyes. Take in the characteristics that make you who you are. After you have gazed at your reflection for a time, give your best smile. Blow yourself a kiss or produce your sexiest wink. Not only will this make you feel happier, but it will also gradually change your relationship with the mirror into a more positive, fulfilling experience. Spread the good vibes. Sometimes, all you need is a gentle reminder that you are beautiful, and that you are enough. Cut out several pieces of paper, a dozen or more. Write down positive statements to tape to your mirror or hide in unusual places so you can reap the benefits of these love notes later. Your personal love notes can say things like:  Hey, there, good-looking! Happiness isn't size-specific. Love the skin you're in. Beyoutiful!


Запишите все числа в ряд. Найдите число, которое чаще всего встречается.

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In this case, you're working with the numbers 2, 4, 5, 5, 4, and 5. It can help to order them from least to greatest here too. Think: "Mode is most." In this example, the number 5 occurs the most, so that is the mode. If there are two numbers that occur the most, then that set is "bimodal," and if more than two numbers occur the most, then the set is "multi-modal."


Запишите все числа, среднее взвешенное которых нужно вычислить. Определите вес (или «ценность») каждого числа. Умножьте каждое число (х) на соответствующий весовой коэффициент (w). Сложите полученные значения, чтобы найти среднее взвешенное.

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You’ll need to start by assembling a list of the numbers for which you’d like to find the weighted average. For example, if you’re trying to find a weighted average for a series of grades in a class, first write down each of the grades. For instance, maybe your total grades are 82 for quizzes, 90 on your exam, and 76 on your term paper. Once you have your numbers, you’ll need to know how much each of them weighs, or is worth, as part of your final average. For instance, in your class, the quizzes might be worth 20% of your total grade, while the exam is 35% and the term paper is 45%. In this case, the weights add up to 1 (or 100%). In order to use these percentages in your calculation, you’ll need to convert them to decimal form. The resulting numbers are called “weighting factors.” Once you have all your numbers, pair up each number (x) with its corresponding weighting factor (w). You’ll be multiplying each set of numbers and weights together, then adding them all up to find the average. For example, if your total quiz score is 82 and quizzes are worth 20% of your grade, multiply 82 x 0.2. In this case, x=82 and w=0.2. The basic formula for a weighted average where the weights add up to 1 is x1(w1) + x2(w2) + x3(w3), and so on, where x is each number in your set and w is the corresponding weighting factor. To find your weighted average, simply multiply each number by its weight factor and then sum the resulting numbers up. For example: The weighted average for your quiz grades, exam, and term paper would be as follows: 82(0.2) + 90(0.35) + 76(0.45) = 16.4 + 31.5 + 34.2 = 82.1. This means you have a grade of 82.1% in the course.


Запишите все, что вы чувствуете к человеку. Вспомните ваше последнее взаимодействие с человеком. Установите, почему вы испытали конкретное чувство.

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Use the Problem Identification, Choices, Consequences (PICC) model to sort through your feelings. The first step is to identify all of the feelings you have about the person. Examples of feelings you might have toward a person might be intrigued, unsure, self-conscious, etc.  Make a list of all the feelings you can identify. Be sure not to categorize these feelings as good or bad, such as in a pros or cons list. Just write them all out. Feelings are not “good or bad”; they all have a purpose. For example, your feelings toward a coworker or acquaintance might include bewildered, respectful, offended or exasperated. Your feelings toward someone close to you, such as your best friend or a family member, might include: love, disappointment, annoyed, comforted, and so on. Identifying feelings can be difficult. It may help to start with a situation you remember and then add the feelings in after. Think of a recent time you spent with this person. Write a list of the feelings you had during this time.  You may find that the feelings you identify are not because of who that person is or their relationship to us, but rather the situation you were in with them, or something specific they said or did. For example, you might go on a first date and have a good first impression. Then your date takes you to a party where you do not know anyone, and you feel uncomfortable or self-conscious. In this example, the situation or unfamiliar environment made you feel uncomfortable, not necessarily the person as a whole. There may be other factors that make you feel a certain way. It might not be entirely this person’s fault. Try to identify the specific source of each emotion.  This is even more specific than identifying the situation. Think back to the moment you felt a certain way. Identify what was said or done immediately prior. For example, if you remember feeling rejected during a date, you might recall that you noticed your date moving away from you during a walk. This may have been the source of your feeling of rejection....


Запишите все, что вы чувствуете. Подумайте, чего вы хотите добиться этим разговором. Подберите подходящее время для разговора. Начинайте разговор, не имея каких-либо ожиданий. Постарайтесь понять точку зрения родителей.

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Trying to explain an issue you're having to your parents can be hard. It can be helpful to write your feelings down ahead of time. This helps you figure out what you want to say, allowing you to have a productive, effective conversation.  To start, just write down what you are feeling. Are you upset over an argument you had with your parents? Do you feel like your parents do not respect or understand you as much as they could? Explain your feelings in detail, and also make note of why you're feeling them.  You should also purge any anger in writing. Going into a conversation mad can be toxic to healthy dialogue. It may be a good idea to write out any anger you're feeling ahead of time rather than expressing it later. Try to find the best way to articulate your feelings. As you write, read over your words. See if there's a way to tweak them to make them more easily understandable. This can help when you sit down and confront your parents. You should consider what your end goal is with this conversation. Do you want your parents to apologize? Do you want them to do something differently in the future? A difficult conversation should have some kind of end goal. Make sure you consider this ahead of time.  To start, you probably want your parents to simply understand where you're coming from. Generational gaps can cause misunderstanding between children and parents. Things change over time, and cultural norms are likely different for you than they were when your parents were your age. It's important to get your parents to understand how you're shaped by the times.  You may be looking for something more specific, however. Maybe you're asking permission to do something, like attend a party. Maybe you want support or guidance in school or your social life. Try to consider what you're asking for, and the best way to make that request. For example, maybe your parents will think you wanting to extend your curfew for prom night is petty. However, maybe you're a senior in high school and this is one of the last nights you...


Запишите выражение. Определите подобные члены (члены с переменной одного порядка, члены с одинаковыми переменными или свободные члены). Приведите подобные члены. Перепишите выражение с учетом приведенных членов. Соблюдайте порядок выполнения операций при приведении подобных членов.

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The simplest algebraic equations, those involving just a few variable terms with whole number coefficients and no fractions, radicals, etc., can often be solved in just a few steps. As with most math problems, the first step to simplifying your equation is to write it out! As an example problem, for the next few steps, let's consider the expression 1 + 2x - 3 + 4x. Next, search your equation for like terms. Remember that like terms have both the same variable(s) and exponent(s). For example, let's identify like terms in our equation 1 + 2x - 3 + 4x. 2x and 4x both have the same variable raised to the same exponent (in this case, the x's aren't raised to any exponent at all). In addition, 1 and -3 are like terms, as neither has any variables. So, in our equation, 2x and 4x and 1 and  -3 are like terms. Now that you've identified like terms, you can combine them to simplify your equation. Add terms together (or subtract in the case  of negative terms) to reduce each set of terms with the same variables and exponents to one singular term. Let's add the like terms in our example.  2x + 4x = 6x  1 + -3 = -2 After combining your like terms, construct an expression from your new, smaller set of terms. You should get a simpler expression that has one term for each different set of variables and exponents in the original expression. This new expression is equal to the first. In our example, our simplified terms are 6x and -2, so our new expression is 6x - 2. This simplified expression is equal to the original (1 + 2x - 3 + 4x), but is shorter and easier to manage. It's also easier to factor, which, as we'll see below, is another important simplifying skill. In extremely simple expressions like the one dealt with in the example problems above, identifying like terms is simple. However, in more complex expressions, like ones that involve terms in parentheses, fractions, and radicals, like terms which can be combined may not be immediately apparent. In these cases, follow the order of operations, performing operations on...


Запишите два возможных выражения. Определите размер конечной матрицы. Найдите значение первого элемента. Продолжите вычислять скалярные произведения, чтобы найти каждый элемент конечной матрицы.

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In "ordinary math" with scalar quantities, multiplication is commutative; 2 x 6 = 6 x 2. This is not true for matrices, so you may need to solve two problems:  [A] * [B]-1 is the solution x for the problem x[B] = [A]. [B]-1 * [A] is the solution x for the problem [B]x = [A]. If this is part of an equation, make sure you are performing the same operation on both sides. If [A] = [C], then [B]-1[A] does not equal [C][B]-1, because the [B]-1 is on the left side of [A] but the right side of [C]. The dimensions of the final matrix are the outer dimensions of the two factors. It has the same number of rows as the first matrix, and the same number of columns as the second matrix.  Returning to our original example, both (13263913){\displaystyle {\begin{pmatrix}13&26\\39&13\end{pmatrix}}} and (313−413−213713){\displaystyle {\begin{pmatrix}{\frac {3}{13}}&{\frac {-4}{13}}\\{\frac {-2}{13}}&{\frac {7}{13}}\end{pmatrix}}} are 2 x 2 matrices so the dimensions of the answer are also 2 x 2. To take a more complicated example, if [A] is a 4 x 3 matrix and [B]-1 is a 3 x 3 matrix, then the matrix [A] * [B]-1 has dimensions 4 x 3. . Refer to the linked article for full instructions, or refresh your memory with this summary:  To find row 1, column 1 of [A][B]-1, find the dot product of [A] row 1 and [B]-1 column 1. That is, for a 2 x 2 matrix, calculate a1,1∗b1,1+a1,2∗b2,1{\displaystyle a_{1,1}*b_{1,1}+a_{1,2}*b_{2,1}}. In our example (13263913)∗(313−413−213713){\displaystyle {\begin{pmatrix}13&26\\39&13\end{pmatrix}}*{\begin{pmatrix}{\frac {3}{13}}&{\frac {-4}{13}}\\{\frac {-2}{13}}&{\frac {7}{13}}\end{pmatrix}}}, row 1 column 1 of our answer is:(13∗313)+(26∗−213){\displaystyle (13*{\frac {3}{13}})+(26*{\frac {-2}{13}})}=3+−4{\displaystyle =3+-4}=−1{\displaystyle =-1} For example, the element at position 2,1 is the dot product of [A] row 2 and [B]-1 column 1. Try to complete the example on your own. You should get the following answers:  (13263913)∗(313−413−213713)=(−1107−5){\displaystyle...


Запишите двоичное число. Начиная слева, удвойте ваш предыдущий итог и добавьте текущую цифру. Удвойте ваш текущий итог и добавьте следующую цифру слева. Повторите предыдущий шаг. Снова повторите предыдущий шаг. Снова повторите предыдущий шаг. Снова повторите ваш предыдущий шаг. Продолжайте удваивать ваш текущий итог и добавлять следующую цифру, пока цифры не закончатся. Запишите ответ вместе с основанием системы счисления (нижним индексом). Используйте данный метод для преобразования из любого основания в десятичное.

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This method does not use powers. As such, it is simpler for converting large numbers in your head because you only need to keep track of a subtotal. The first thing you need to do is to write down the binary number you'll be converting using the doubling method. Let's say the number you're working with is 10110012. Write it down. Since you're working with the binary number 10110012, your first digit all the way on the left is 1. Your previous total is 0 since you haven't started yet. You'll have to double the previous total, 0, and add 1, the current digit. 0 x 2 + 1 = 1, so your new current total is 1. Your current total is now 1 and the new current digit is 0. So, double 1 and add 0. 1 x 2 + 0 = 2. Your new current total is 2. Just keep going. Next, double your current total, and add 1, your next digit. 2 x 2 + 1 = 5. Your current total is now 5. Next, double your current total, 5, and add the next digit, 1. 5 x 2 + 1 = 11. Your new total is 11. Double your current total, 11, and add the next digit, 0. 2 x 11 + 0 = 22. Now, double your current total, 22, and add 0, the next digit. 22 x 2 + 0 = 44. Now, you're down to your last number and are almost done! All you have to do is take your current total, 44, and double it along with adding 1, the last digit. 2 x 44 + 1 = 89. You're all done! You've converted 100110112 to decimal notation to its decimal form, 89. Write your final answer as 8910 to show that you're working with a decimal, which has a base of 10. Doubling is used because the given number is of base 2.  If the given number is of a different base, replace the 2 in the method with the base of the given number.  For example, if the given number is in base 37, you would replace the "x 2" with "x 37".  The final result will always be in decimal (base 10).


Запишите двоичные числа друг под другом так, как вы записываете десятичные числа при их вычитании. Если значность чисел разная, к числу с меньшей значностью слева припишите соответствующее количество 0. В вычитаемом числе поменяйте цифры: К полученному вычитаемому прибавьте 1. Теперь вместо вычитания сложите два двоичных числа. В полученном результате игнорируйте любую цифру, стоящую первой слева (так как вы получили число не той значности). Попробуйте применить этот метод к десятичным числам.

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This method is used by computers to subtract binary numbers, since it uses a more efficient program. For a human used to ordinary decimal subtraction problems, this is probably the more difficult method to use, but could be useful to understand as a programmer. We'll use the example 101 - 11 = ? For example, convert 101-11 to 101-011 so that both have three digits. 101 - 011 = ? Change all the 0s to 1s and all the 1s to 0s in the second term. In our example, the second term becomes: 011 → 100. What we're actually doing is "taking the one's complement," or subtracting each digit in the term from one. The "switching" shortcut works in binary, since the only two possibilities result in switching the term: 1 - 0 = 1 and 1 - 1 = 0. . Once you have the "reversed" term, add one to the result. In our example, we get 100 + 1 = 101. . Use binary adding techniques to add the new term to the original term, instead of subtracting:  101 + 101 = 1010 If this does not make sense to you, review how to add binary numbers. This method should always end up with an answer one digit too long. For example, our example problem involved three-digit numbers (101 + 101), but we ended up with a four digit solution (1010). Simply cross out the first digit, and you'll have the answer to the original subtraction problem:   1010 = 10  Therefore, 101 - 011 = 10  If you don't have an extra digit, you tried to subtract a larger number from a smaller one. See the tips section for how to solve problems like that, and start again. This method is called the "two's complement" method, since the "reverse the digits" steps results in the "one's complement," and then the number 1 is added. If you'd like a more intuitive understanding of why this method works, try it in base ten:  56 - 17 Since we're using base ten, we'll take the "nine's complement" of the second term (17) by subtracting each digit from nine. 99 - 17 = 82. Change this into an addition problem: 56 + 82. If you compare this to the original problem (56 - 17), you can see that we've added...


Запишите двоичные числа друг под другом — меньшее число под большим. Некоторые задачи на вычитание двоичных чисел ничем не отличаются от вычитания десятичных чисел. Рассмотрим более сложную задачу. В первом столбце справа вы получаете разность 0 - 1. Вычтите цифры в правом столбце. Вычтите цифры в оставшихся столбцах. Решите сложную задачу. Проверьте ответ.

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Write the larger number above the smaller number. If the smaller number has fewer digits, line them up on the right, as you would in a decimal (base ten) subtraction problem. Some binary subtraction problems are no different than base ten subtraction. Line up the columns and, starting from the right, find the result for each digit. Here are a few easy examples:  1 - 0 = 1 11 - 10 = 1 1011 - 10 = 1001 You only need to know one special "rule" to complete any binary subtraction problem. This rule tells you how to "borrow" from the digit to the left so you can solve a "0 - 1" column. For the rest of this section, we'll set up a couple example problems and solve them using the borrow method. Here's the first: 110 - 101 = ? Starting from the right column (the ones place), we need to solve the problem "0 - 1." In order to do this, we need to "borrow" from the digit to the left (the twos place). This has two steps to it:  First, cross out the 1 and replace it with a 0, to get this: 1010 - 101 = ? You've subtracted 10 from the first number, so you can add this "borrowed" number to the ones place: 101100 - 101 = ? Now each column can be solved as usual. Here's how to solve the rightmost column (the ones place) in this problem:  101100 - 101 = ? The rightmost column is now: 10 - 1 = 1. If you can't figure out how to reach this answer, here's how to convert the problem back to decimal: 102 = (1 x 2) + (0 x 1) = 210. (The sub numbers indicate which base the number is written in.) 12 = (1x1) = 110. Therefore, in decimal form this problem is 2 - 1 = ?, so the answer is 1. The rest of the problem can now be solved easily. Solve it column by column, moving from right to left: 101100 - 101 = __1 = _01 = 001 = 1. Borrowing comes up a lot in binary multiplication, and sometimes you'll need to borrow multiple times just to solve one column. For example, here's how to solve 11000 -  111. We can't "borrow" from a 0, so we need to keep borrowing from the left until we turn it into something we can borrow from:  10110000 - 111 = ...


Запишите двузначные числа друг под другом. Единицы второго (нижнего) числа умножьте на единицы первого (верхнего) числа. Теперь единицы второго (нижнего) числа умножьте на десятки первого (верхнего) числа. Напишите нуль. Десятки второго (нижнего) числа умножьте на единицы первого (верхнего) числа. Десятки второго (нижнего) числа умножьте на десятки первого (верхнего) числа. Сложите оба результата, чтобы получить окончательный ответ.

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Place 1 of the double-digit numbers on top and the other double-digit number directly below it. While there's no right or wrong way to place the numbers, if you have a double-digit that ends with a zero such as 40, place it on the bottom, and put any number with more (nonzero) digits on top. This will let you skip a calculation. For example, if you're multiplying 22 by 43, you can put either number on the top or bottom. For the moment, don't worry about the bottom tens number. Just use the bottom one's number and multiply it by the one's number directly above it. Write the result directly below the line. So for 22 x 43, you'll multiply 3 by 2 to get 6. Use the same bottom number and multiply it by the number of the top ten. Then write the result under the line so it's directly below the tens space. For example, with 22 x 43, you'll now need to multiply 3 by the other 2 to get 6. The number under your line should be 66. Before you begin multiplying the next part, place a zero under the one's place. This will hold the space so you can begin multiplying the bottom tens number. So if you've got 66 as your first result, put a 0 right under the 6 in the ones place. Now that you've already worked the bottom one's number, multiply the bottom tens number by the top one's number. Write the result next to the zero you wrote. For example, 4 x 2 = 8, so write an 8 next to the 0. Write the result next to the number you just wrote down. For 4 x 2, write another 8 next to the 80 you already have written down. If your numbers don't have any more digits, you're ready to add both rows of your results. The number you get will be the answer. For example, you'll need to add 66 + 880 to get 946.


Запишите делимое и делитель так, как вы записываете обычные числа при их делении в столбик. Разделите первый член делимого на первый член делителя. Умножьте результат из предыдущего шага (х) на делитель. Вычтите результат (из предыдущего шага) из делимого. Повторите предыдущие три шага с промежуточным двучленом (полученным в предыдущем шаге).

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You write out the long division of polynomials the same as you do for dividing numbers. The dividend goes under the long division bar, while the divisor goes to the left. If you’re dividing x2 + 11 x + 10 by x +1, x2 + 11 x + 10 goes under the bar, while x + 1 goes to the left. The result of this division goes on top of the division bar. For our example, dividing x2, the first term of the dividend, by x, the first term of the divisor yields x. You would write an x on the top of the division bar, over the x2. Write the result of the multiplication under the leftmost terms of the dividend. Continuing with our example, multiplying x + 1 by x produces x2 + x. You would write this under the first two terms of the dividend. To do so, first reverse the signs of the product of the multiplication. After subtracting, bring down the remaining terms of the dividend. Reversing the signs of x2 + x gives - x2 - x. Subtracting this from the first two terms of the dividend leaves 10x. After bringing down the remaining term of the dividend, you have 10x + 10 as the interim quotient to continue the division process with. You’ll again divide the first term of the divisor into that of the interim quotient, write that result on top of the division bar after the first term of the quotient, multiply the result by the divisor, and then calculate what to subtract from the interim quotient.  Because x goes into 10x 10 times, you would write “+ 10” after the x in the quotient position on the division bar. Multiplying x +1 by 10 gives 10x + 10. You write this under the interim quotient and reverse the signs for the subtraction, making -10x – 10. When you perform the subtraction, you have a remainder of 0. Thus, dividing x2 + 11 x + 10 by x +1 produces a quotient of x + 10. (You could have gotten the same result by factoring, but this example was chosen to keep the division fairly simple.)


Запишите делимое слева, а делитель справа. Под делителем напишите нуль и поставьте после него запятую, так как результат деления числителя на знаменатель обыкновенной дроби всегда меньше 1. Выполните деление в столбик Запишите ответ:

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Let's say we're working with the problem 3/4. Simply write a "4" outside the division bar with a "3" on the inside. The "4" is your divisor, and the "3" is your dividend. Since you're working with a fraction, you know that your answer will be less than one, so this first step is necessary. After you do this, you'll need to place a decimal after the 3 under the division bracket and write an additional zero after it. Though 3 and "3.0" are the same, adding the extra zero will allow you to divide 4 into the 3.0. . When you do long division, you can pretend that the decimal points don't exist and act like you're just dividing 4 into 30, for now. Here is how you do it:  First, divide "4" into 3.0, which you can think of as 30. The closest you can get is 4 x 7, or 28, which will give you a remainder of 2. So, write a 7 after the "0." above the division bracket and "28" below the "3.0" under the division bracket. Under that, write 2, your remainder, or what you have left when you subtract 28 from 30. Next, add another "0" to the "3.0" so you have "3.00", or what you can think of as "300," under the division bracket. This will allow you to drop down another 0 to the right of the "2" so you can divide "4" into "20." Divide "4" into "20" to get 5. Write the "5" to the right of the "0.7" above the division bracket, so you have "0.75" written there. You'll find that "3" divided by "4" will give you ".75". Write down this answer and you're all done.


Запишите дробь. Используйте один знак корня. Разделите подкоренные выражение. Упростите

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If your expression is not already set up like a fraction, rewrite it this way. This makes it easier to follow all the necessary steps when dividing by a square root. Remember that a fraction bar is also a division bar. For example, if you are calculating 144÷36{\displaystyle {\sqrt {144}}\div {\sqrt {36}}}, rewrite the problem like this: 14436{\displaystyle {\frac {\sqrt {144}}{\sqrt {36}}}}. If your problem has a square root in the numerator and denominator, you can place both radicands under one radical sign. (A radicand is a number under a radical, or square root, sign.) This will simplify the simplifying process. For example, 14436{\displaystyle {\frac {\sqrt {144}}{\sqrt {36}}}} can be rewritten as 14436{\displaystyle {\sqrt {\frac {144}{36}}}}. Divide the numbers as you would any whole number. Make sure to place their quotient under a new radical sign. For example, 14436=4{\displaystyle {\frac {144}{36}}=4}, so 14436=4{\displaystyle {\sqrt {\frac {144}{36}}}={\sqrt {4}}}. , if necessary. If the radicand is a perfect square, or if one of its factors is a perfect square, you need to simplify the expression. A perfect square is the product of a whole number multiplied by itself. For example, 25 is a perfect square, since 5×5=25{\displaystyle 5\times 5=25}. For example, 4 is a perfect square, since 2×2=4{\displaystyle 2\times 2=4}. Thus:4{\displaystyle {\sqrt {4}}}=2×2{\displaystyle ={\sqrt {2\times 2}}}=2{\displaystyle =2}So, 14436=4=2{\displaystyle {\frac {\sqrt {144}}{\sqrt {36}}}={\sqrt {4}}=2}.


Запишите заданное целое число вверху страницы. Найдите два числа (помимо 1), при перемножении которых получается заданное число. Поищите простые множители. Продолжайте раскладывать составные (не простые) числа на множители. Представьте каждый простой множитель в степенной форме. Запишите разложение числа на простые множители.

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You need to leave enough room so that you can set up a factor tree below it. You can use other methods to factor a number. Read Factor a Number for more instructions. For example, if you want to know many divisors, or factors, the number 24 has, write 24{\displaystyle 24} at the top of the page. These are two divisors, or factors, of the number. Draw a split branch coming down from the original number, and write the two factors below it. For example, 12 and 2 are factors of 24, so draw a split branch coming down from 24{\displaystyle 24}, and write the numbers 12{\displaystyle 12} and 2{\displaystyle 2} below it. A prime factor is a number that is only evenly divisible by 1 and itself. For example, 7 is a prime number, because the only numbers that evenly divide into 7 are 1 and 7. Circle any prime factors so that you can keep track of them. For example, 2 is a prime number, so you would circle the 2{\displaystyle 2} on your factor tree. Keep drawing branches down from the non-prime factors until all of your factors are prime. Circle the prime numbers to keep track of them. For example, 12 can be factored into 6{\displaystyle 6} and 2{\displaystyle 2}. Since 2{\displaystyle 2} is a prime number, you would circle it. Next, 6{\displaystyle 6} can be factored into 3{\displaystyle 3} and 2{\displaystyle 2}. Since 3{\displaystyle 3} and 2{\displaystyle 2} are prime numbers, you would circle them. To do this, look for multiples of each prime factor in your factor tree. The number of times the factor appears equals the exponent of the factor in your exponential expression. For example, the prime factor 2{\displaystyle 2} appears three times in your factor tree, so the exponential expression is 23{\displaystyle 2^{3}}. The prime factor 3{\displaystyle 3} appears 1 time in your factor tree, so the exponential expression is 31{\displaystyle 3^{1}}. The original number you are working with is equal to the product of the exponential expressions. For example 24=23×31{\displaystyle 24=2^{3}\times 3^{1}}.


Запишите задачу в формате деления в столбик. Посмотрите на первые две цифры делимого и подумайте, делится ли это число на делитель. Предположите, сколько раз делитель (в нашем случае 15) может уложиться в делимое (в нашем случае 34), и запишите свое предположение. Запишите найденное число. Умножьте предложенное вами число (2) на делитель (в нашем случае 15) и запишите результат под делимым. Вычтите результат умножения из делимого и запишите результат вычитания на следующей (новой) строке. Спустите вниз следующую цифру исходного делимого (так, как вы это делаете при делении в столбик). Чтобы получить следующую цифру частного, повторите шаги, описанные выше (начните с предположения числа). Повторяйте процесс предположение – умножение – вычитание до тех пор, пока не получите в остатке нуль или промежуточный итог меньше делителя. Спустите следующую цифру исходного делимого и повторите описанный процесс. Найдите остаток.

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Write the problem as a long-division problem. Just like a simpler division problem, you can start by looking at the smaller number, and asking "Does it fit into the first digit of the larger number?" Let's say you're solving 3472 ÷ 15. Ask "Does 15 fit into 3?" Since 15 is definitely larger than 3, the answer is "no," and we move on to the next step. Since you can't fit a two-digit number into a one-digit number, we'll look at the first two digits of the dividend instead, just like we would in a regular division problem. If you still have an impossible division problem, you'll need to look at the first three digits instead, but we don't need to in this example: Does 15 fit into 34? Yes, it does, so we can start calculating the answer. (The first number doesn't have to fit perfectly, it just needs to be smaller than the second number.) Find out exactly how many times the first number fits into the other. You might know the answer already, but if you don't, try making a good guess and checking your answer with multiplication. We need to solve 34 ÷ 15, or "how many times does 15 go into 34"?  You're looking for a number you can multiply with 15 to get a number less than 34, but pretty close to it:  Does 1 work? 15 x 1 = 15, which is less than 34, but keep guessing. Does 2 work? 15 x 2 = 30. This is still less than 34, so 2 is a better answer than 1. Does 3 work? 15 x 3 = 45, which is greater than 34. Too high! The answer must be 2. If you set this up like a long division problem, this should feel familiar. Since you were calculating 34 ÷ 15, write the answer, 2, on the answer line above the "4." This is the same as a normal long division problem, except we'll be using a two-digit number. Your answer was 2 and the smaller number in the problem is 15, so we calculate 2 x 15 = 30. Write "30" underneath the "34." The last thing you wrote went underneath the original larger number (or part of it). Treat this as a subtraction problem and write the answer on a new line underneath. Solve 34 - 30 and write the answer...


Запишите задачу на деление. Превратите целое число в десятичную дробь. Переместите десятичную запятую вправо так, чтобы десятичные дроби превратились в целые числа. Перепишите задачу в виде деления в столбик.

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Use pencil in case you want to revise your work.  Example: What is 3 ÷ 1.2? Write a decimal point after the whole number, than write zeroes after the decimal point. Do this until both numbers have the same number of places to the right of the decimal point. This does not change the value of the whole number.   Example: In the problem 3 ÷ 1.2, our whole number is 3. Since 1.2 has one place to the right of the decimal point, rewrite 3 as 3.0, so it also has one place after the decimal. Now our problem is 3.0 ÷ 1.2. Warning: do not add zeros to the left of the decimal point! 3 is the same as 3.0 or 3.00, but it is not the same as 30 or 300. In division problems, you're allowed to move the decimal points, but only if you move them by the same amount for each number. This lets you turn the problem into whole numbers.  Example: To turn 3.0 ÷ 1.2 into whole numbers, move the decimal points one space to the right. 3.0 becomes 30, and 1.2 becomes 12. Now our problem is 30 ÷ 12. . Put the dividend (usually the larger number) under the long division symbol. Write the divisor outside it. Now you have an ordinary long division problem using whole numbers. If you want a reminder of how to do long division, read the next section.


Запишите задачу с корнем, как задачу деления столбиком. Сгруппируйте цифры по парам. Найдите наибольшее число, квадрат которого меньше или равен первой «группе». Удвойте только что написанное число, затем опустите его под корень и отнимите. Заполните пробел. Продолжайте использовать «пустые» числа, чтобы найти ответ. Передвиньте точку десятичной дроби вперед от изначального «делимого» числа.

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Though it can be a little time-consuming, it's possible to solve for the square roots of difficult imperfect squares without a calculator. To do this, we'll use a solving method (or algorithm) that's similar — but not exactly the same — as basic long division.  Start by writing out your square root problem in the same from as a long division problem. For example, let's say that we want to find the square root of 6.45, which is definitely not a convenient perfect square. First, we'd write an ordinary radical symbol (√), then we'd write our number underneath it. Next, we'd make a line above our number so that it's in a little "box" — just like in long division. When we're done, we should have a long-tailed "√" symbol with 6.45 written under it. We'll be writing numbers above our problem, so be sure to leave space. To start solving your problem, group the digits of the number under the radical sign into pairs, starting at the decimal point. You may want to make small marks (like dots, slashes, commas, etc.) between your pairs to keep track of them. In our example, we would divide 6.45 into pairs like this: 6-.45-00. Note that there is a "leftover" digit on the left — this is OK. Start with the first number or pair on the left. Pick the biggest number with a square that is less than or equal to the "group". For instance, if the group was 37, you would pick 6, because 62 = 36 < 37 but 72 = 49 > 37. Write this number above the first group. This is the first digit of your answer. In our example, the first group in 6-.45-00 is 6. The biggest number that is less than or equal to 6 when squared is 2 — 22 = 4. Write a "2" above the 6 under the radical. Take the first digit of your answer (the number you just found) and double it. Write this underneath your first group and subtract to find the difference. Drop the next pair of numbers down beside the answer. Finally, write the last digit of the double of the first digit of your answer to the left and leave a space next to it. In our example, we would start by taking the...


Запишите задачу. Вычтите подобные члены. Запишите окончательный ответ.

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Let's say you're working with the following problem: 3x2 - 5x + 2y - z - (2x2 + 2x + y). Write the first set of terms above the second. When you're working with variables, you can only add or subtract terms that have the same variable and which are written to the same degree. This means that you can subtract 4x2 from 7x2, for example, but not 4x from 4y. This means that you can break the problem down like this:  3x2 - 2x2 = x2  -5x - 2x = -7x 2y - y = y -z - 0 = -z Now that you've subtracted all of the like terms you can subtract, all you can do is state your final answer, which will contain each of the terms that you subtracted. Here's the final answer: 3x2 - 5x + 2y - z - (2x2 + 2x + y) = x2 - 7x + y - z


Запишите задачу. Преобразуйте целое число в дробь со знаменателем, равным знаменателю вычитаемой дроби. Перепишите задачу. Вычтите числители двух дробей. Запишите результат вычитания числителей над прежним знаменателем, чтобы получить окончательный ответ.

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Let's say you're working with the following problem: 5 - 3/4. Write it down. You'll want to convert the number 5 into a fraction with the denominator of 4 in order to subtract the two numbers. So, you can first think of 5 as a fraction that is really 5/1. Then, you can multiply both the top and bottom of the new fraction by 4 in order to create two fractions with the same denominator. So, 5/1 x 4/4 = 20/4. This fraction is really equal to 5, but it allows you to subtract two fractions. The new problem can be written like this: 20/4 - 3/4. Now, you can simply subtract 3 from 20 to get the final answer. 20 - 3 = 17, so 17 is your new numerator. You can keep the denominator as it is. Your final answer is 17/4. If you'd like to state it as a mixed number, simply divide 17 by 4 to get 4, with 1 left over as the remainder. This will make your final answer 17/4 equal to 4 1/4.


Запишите задачу. Разложите меньшее число на десятки и единицы. Умножьте большее число на десятки меньшего числа. Умножьте большее число на единицы меньшего числа. Сложите два полученных результата.

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Let's say you're multiplying 325 times 12. Write it down. One number should be right next to the other, not below it. Keep 325 and split up 12 into 10 and 2. The 1 is in the tens digit, so you should add a 0 afterward to keep its place, and since the 2 is in the ones place, you can just write down 2. Now, multiply 325 times 10. All you have to do is add a zero to the end to equal 3250. Now, just multiply 325 by 2. You can eyeball it and see that the answer is 650, since 300 times 2 is 600 and 25 times 2 is 50. Add up 600 and 50 to equal 650. Now, just add up 3250 and 650. You can do this using the good old fashioned addition method. Just write 3250 over 650 and do all the work. You'll get 3,900. Really, this is similar to doing the standard long multiplication, but splitting up a number into ones and tens allows you to do a bit more of the math in your head and to avoid multiplying and carrying too much. Either method will yield the same results, and it all depends on which one works more quickly for you.


Запишите задачу. Сложите все значения (переменные и числа). Вычтите сумму из 180°. Найдите х. Проверьте ответ.

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Sometimes, instead being lucky enough to know the measurements of two of the angles of a triangle, you'll only be given a few variables, or some variables and an angle measurement. Let's say you're working with this problem: Find the measurements of angle "x" of the triangle whose measurements are "x," "2x," and 24. First, just write it down. It's the same principle that you would follow if you did know the measurements of the two angles. Simply add up the measurements of the angles, combining the variables. So, x + 2x + 24° = 3x + 24°. Now, subtract these measurements from 180° to get closer to solving the problem. Make sure you set the equation equal to 0. Here's what it would look like:  180° - (3x + 24°) = 0 180° - 3x - 24° = 0 156° - 3x = 0 Now, just put the variables on one side of the equation and the numbers on the other side. You'll get 156° = 3x. Now, divide both sides of the equation by 3 to get x = 52°. This means that the measurement of the third angle of the triangle is 52°. The other angle, 2x, is 2 x 52°, or 104°. If you want to make sure that this is a valid triangle, just add up the three angle measurements to make sure that they add up to 180°. That's 52° + 104° + 24° = 180°. You're all done.


Запишите задачу. Целое число умножьте на числитель. Полученный результат разделите на знаменатель.

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If the problem you are working with is a word problem, first you need to write out all of the numbers involved in the calculation. If you were given the numbers written out already, you can skip this step.  For example: What is one-third of seven? When you see “of” written between two numbers in a word problem, think of it as meaning multiply. Thus, one-third of seven actually means one-third multiplied by seven. Writing out the problem gives you (1/3) x 7. When working with whole numbers, you will only multiply the number by the numerator (the top number) of the fraction. The denominator remains the same throughout the multiplication process. For example: (1/3) x 7 = 7/3. Divide the product by the denominator (the bottom number) of the fraction. At this stage, the fraction might be an improper fraction, meaning that the numerator is larger than the denominator, or it will just need to be reduced. For example: After multiplying, the fraction is 7/3. Three doesn’t divide evenly into seven so you will have a remainder. Three goes into seven twice with one leftover; therefore, the final answer is the mixed number 21/3.


Запишите закон Рауля. Определите, какое вещество будет растворителем, а какое – растворенным веществом. Найдите температуру раствора, так как она будет влиять на давление его пара. Найдите давление паров растворителя. Найдите мольную долю растворителя. Теперь подставьте данные и найденные значения величин в уравнение Рауля, приведенное в начале этого раздела (Pраствор = PрастворительXрастворитель).

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In real life, it's rare to work with a single pure liquid — usually, we deal with liquids that are mixtures of several different component substances. Some of the most common of these mixtures are created by dissolving a small amount of a certain chemical called a solute in a large amount of a chemical called a solvent to create a solution. In these cases, it's useful to know an equation called Raoult's Law (named for physicist François-Marie Raoult), which looks like this: Psolution=PsolventXsolvent. In this formula, the variables refer to;   Psolution: The vapor pressure of the entire solution (all of the component parts combined)  Psolvent: The vapor pressure of the solvent  Xsolvent: The mole fraction of the solvent. Don't worry if you don't know terms like "mole fraction" — we'll explain these in the next few steps. Before you calculate the vapor pressure of a mixed liquid, you need to identify the substances with which you are working. As a reminder, a solution is formed when a solute is dissolved in a solvent — the chemical that dissolves is always the solute and the chemical that does the dissolving is always the solvent.  Let's work through a simple example in this section to illustrate the concepts we're discussing. For our example, let's say that we want to find the vapor pressure of simple syrup. Traditionally, simple syrup is one part sugar dissolved in one part water, so we'll say that sugar is our solute and water is our solvent.  Note that the chemical formula for sucrose (table sugar) is C12H22O11. This will be important soon. As we saw in the Clausius-Clapeyron section above, a liquid's temperature will affect its vapor pressure. In general, the higher the temperature, the greater the vapor pressure — as the temperature increases, more of the liquid will evaporate and form vapor, increasing the pressure in the container. In our example, let's say that the simple syrup's current temperature is 298 K ( about 25 C). Chemical reference materials usually have vapor pressure values for many common...


Запишите и отрепетируйте. Продумайте вступление. Говорите спокойно, придавая значение каждому слову. Составьте список положительных моментов. Оставьте разговор открытым.

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You should approach this conversation almost like you are preparing to give an important speech. You want to write out how you want the conversation to go. Visualize it going well. Practice saying all your reasons out loud to yourself in the mirror or to a friend. You want to get your parent's full attention as you convince them to help you in switching schools.  Say something like, “Hey, Mom and Dad! Can we all sit down at the table together? I have something I’d like to talk about with you, and I’d really love to hear what you think.” You want to let your parents know that this is important to you and that you appreciate them listening to your thoughts. Even if your parents don’t respond at first, you don’t want to come across as whiney because this will make it less likely that your parents will support you in switching schools. At the same time, be honest.  You want them to know how much staying in your current situation will hurt you. Make your statements sincere and to-the-point.   If you’re being bullied, don’t be too embarrassed to show them how much it’s affecting your performance at school and how much it hurts you. Say something like, “There is a group of kids in my class that writes mean notes to me every day and steal things out of my desk. They call me names, and it makes me sad. I’ve asked them to stop, and I’ve talked to the teacher, but they still do it behind her back. I have a hard time enjoying school or focusing because I can’t stop thinking about it.” If you think you need a school with more academic attention say something like, “I’ve been having a hard time finishing my work in school because I don’t understand it. There are so many kids in my class that the teacher usually doesn’t have time to help me.” Or, if you want more of a challenge, say, “I get all A’s at school because the work is too easy. I finish all my work first, and I end up just sitting there in class. My teacher doesn’t have time to make special assignments for me.” These are the reasons that changing would improve...


Запишите известные величины. Запишите формулу теоремы косинусов. Найдите значения a{\displaystyle a}, b{\displaystyle b} и c{\displaystyle c}. Перемножьте числа. Возведите c{\displaystyle c} в квадрат. Сложите квадраты a{\displaystyle a} и b{\displaystyle b}. Изолируйте косинус неизвестного угла. Вычислите арккосинус.

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To find the missing angle of a triangle using the cosine rule, you need to know the length of all three sides of the triangle. For example, you might have triangle RST. Side SR is 8 cm long. Side ST is 10 cm long. Side RT is 12 cm long. What is the measurement of angle S? The formula is c2=a2+b2−2abcosC{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos {C}}. In this formula, cosC{\displaystyle \cos {C}} equals the cosine of the angle you are trying to find. The variable c{\displaystyle c} equals the side opposite the missing angle. The variables a{\displaystyle a} and b{\displaystyle b} are the lengths of the other two sides. Plug these values into the formula. For example, since side RT is opposite the missing angle, angle S, side RT will equal c{\displaystyle c} in the formula. The other two side lengths will be a{\displaystyle a} and b{\displaystyle b}. It doesn’t matter which side is which variable. So, your formula should look like this: 122=82+102−2(8)(10)cosC{\displaystyle 12^{2}=8^{2}+10^{2}-2(8)(10)\cos {C}}. You are multiplying 2ab{\displaystyle 2ab} times the cosine of the missing angle, which you don’t know yet. So, the variable should remain. For example, 122=82+102−160cosC{\displaystyle 12^{2}=8^{2}+10^{2}-160\cos {C}}. Remember that to square a number, you multiply the number by itself. For example, 144=82+102−160cosC{\displaystyle 144=8^{2}+10^{2}-160\cos {C}} Make sure you square each number first, and then add them together. For example:144=64+100−160cosC{\displaystyle 144=64+100-160\cos {C}}144=164−160cosC{\displaystyle 144=164-160\cos {C}} To do this, subtract the sum of a2{\displaystyle a^{2}} and b2{\displaystyle b^{2}} from both sides of the equation. Then, divide each side of the equation by the coefficient of the missing angle’s cosine. For example, to isolate the cosine of the missing angle, subtract 164 from both sides of the equation, then divide each side by -160:144−164=164−164−160cosC{\displaystyle 144-164=164-164-160\cos {C}}−20=−160cosC{\displaystyle -20=-160\cos...


Запишите исходную дробь. Запишите множители и числителя, и знаменателя. Вычислите наибольший общий делитель (НОД) и разделите на него дробь.

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Leave a large space to the right hand side of your paper -- you'll need it to write out the factors. Keep them in separate lists. It may be easiest if the lists line up on top of each other. Start with 1 and work your way up, listing them in pairs.  For example, if your fraction is 24/60, start off with 24.You'd write: 24 -- 1, 2, 3, 4, 6, 8, 12, 24  Then, move onto 60.You'd write: 60 -- 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 This may be referred to as the GCF in your textbook. What's the biggest number that goes into both the numerator and the denominator? Whatever it is, divide both numbers by that. For our example, the biggest number that is a factor of both numbers is 12. Therefore, we divide 24 by 12 and 60 by 12, leaving us with 2/5 -- our reduced fraction!


Запишите каноническое уравнение гиперболы. Приравняйте каноническое уравнение к нулю (а не к единице). Разложите на множители новое уравнение. Приравняйте каждый двучлен (то есть выражение внутри каждой пары скобок) к нулю и вычислите «y». Выполните описанные действия с гиперболой, уравнение которой отличается от канонического.

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We'll start with a simple example: a hyperbola with the center of its origin. For these hyperbolas, the standard form of the equation is x2/a2 - y2/b2 = 1 for hyperbolas that extend right and left, or y2/b2 - x2/a2 = 1 for hyperbolas that extend up and down. Remember, x and y are variables, while a and b are constants (ordinary numbers).   Example 1: x2/9 - y2/16 = 1  Some textbooks and teachers switch the position of a and b in these equations. Follow the equation closely so you understand what's going on. If you just memorize the equations you won't be prepared when you see a different notation. This new equation represents both asymptotes, though it will take a little more work to separate them.  Example 1: x2/9 - y2/16 = 0 Factor the left hand side of the equation into two products. Refresh your memory on factoring a quadratic if you need to, or follow along while we continue Example 1:  We'll end up with an equation in the form (__ ± __)(__ ± __) = 0.  The first two terms need to multiply together to make x2/9, so take the square root and write it in those spaces: (x/3 ± __)(x/3 ± __) = 0  Similarly, take the square root of y2/16 and place it in the two remaining spaces: (x/3 ± y/4)(x/3 ± y/4) = 0  Since there are no other terms, write one plus sign and one minus sign so the other terms cancel when multiplied: (x/3 + y/4)(x/3 - y/4) = 0 To get the equations for the asymptotes, separate the two factors and solve in terms of y.   Example 1: Since (x/3 + y/4)(x/3 - y/4) = 0, we know x/3 + y/4 = 0 and x/3 - y/4 = 0  Rewrite x/3 + y/4 = 0 → y/4 = - x/3 → y = - 4x/3  Rewrite x/3 - y/4 = 0 → - y/4 = - x/3 → y = 4x/3 We've just found the asymptotes for a hyperbola centered at the origin. A hyperbola centered at (h,k) has an equation in the form (x - h)2/a2 - (y - k)2/b2 = 1, or in the form (y - k)2/b2 - (x - h)2/a2 = 1. You can solve these with exactly the same factoring method described above. Just leave the (x - h) and (y - k) terms intact until the last step.   Example 2: (x - 3)2/4 - (y + 1)2/25 = 1  Set...


Запишите кошку на прием к ветеринару. Обратите внимание на поведенческие симптомы слепоты. Осмотрите глаза кошки на предмет слепоты.

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If you suspect that your cat has gone blind or is losing her sight, schedule an appointment with your cat's veterinarian right away. Some types of blindness are reversible if they are treated right away. For example, if your cat is suffering from blindness caused by hypertension, a daily medication may restore her sight and prevent further health problems as well. Make sure that your cat gets regular check-ups with a veterinarian even if she seems healthy. Early diagnosis and treatment of a problem can prevent your cat from going blind. Sometimes a cat will go blind over a period of days, weeks, or months. After the cat has become mostly or totally blind, you may begin to notice some strange behavior. Observe your cat to determine if she might be blind. Some of the behaviors that blind cats display include:  Seeming confused about her surroundings Bumping into furniture and other objects Missing jumps or landings that she used to manage with ease Seeming reluctant to move around or venture outdoors Rubbing or squinting her eyes If your cat's behavior has you suspicious that she might have gone blind, inspect her eyes to see if they look different than usual. Even if your cat is not showing signs of blindness, you should inspect your cat's eyes regularly to check for early symptoms of blindness. Some common symptoms of blindness in cats include:  Eyes don't respond to light, pupils are dilated Eyes look cloudy, inflamed, or discolored


Запишите краткий план, чтобы запомнить ключевые моменты. Сделайте карточки, если нужно запомнить цитаты. Создайте схему главных идей сочинения, если вы лучше запоминаете визуальные информацию.

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Write an outline of the essay's main ideas, concepts, and arguments. Make sure you only include the most important information, and organize it in the correct order. You might memorize the outline instead of the entire essay. If you need to recall quotes from literary or academic essays, write the quotes on flash cards. Memorize the quotes 1 at a time. Make sure you also learn the author, year, and any other information that may be tested. Sketch out a flowchart or map of the main ideas of the essay. Put the thesis in the center of the chart, and draw lines out to its supporting ideas.  When you need to remember the essay, you can redraw the chart to help you remember all the different pieces you need to recall. You can also draw images in your chart or sketch out the main events of the essay in comic form.


Запишите лекцию на диктофон и прослушивайте запись дома или в пути. Делайте пометки в тетради во время лекции. Перечитывайте конспекты каждый день. Перенесите пометки, сделанные на занятии, в отдельную тетрадь для конспектов. Повторите пройденный за неделю материал в выходные. Организуйте свои пометки. Используйте карточки. Учитесь с помощью гаджетов.

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Ask your instructor's permission before recording any part of your class sessions. With his or her permission, use an audio recording device during class. If you're using a digital recorder, convert the file into an mp3 and listen to your lecture while you're commuting or working out in the morning. Write down important ideas, concepts, names, and dates, rather than trying to take down every single word your instructor is saying. This should be done immediately after class, if possible. If you cannot study right after class has ended, studying as soon as possible that day is crucial, as most in-class information is forgotten after 24 hours.  Read through each line of notes slowly and carefully. Ask your instructor about anything that doesn't make sense or is unclear to you. This will allow you to compile the most vital information in one place, and can help you make more sense of the notes you took in class. But don't just copy the material mindlessly! Rephrasing class notes in your own words will also help you understand the material beyond simply re-stating what you've been told. This will help reinforce the things you learned that week, and may help you contextualize each day's lesson better within the framework of the entire week's lesson plan. Color-coding your notes by lesson or topic may be helpful, or try using a series of folders to create an orderly system. Try different organization methods until you find one that works right for you. This may include organizing handouts separate from notes, or organizing everything by date, by chapter, or by topic. Flashcards can help you memorize important names, dates, places, events, and concepts. They can be used for nearly every single subject taught in school.  Choose the most important names, dates, concepts, etc. Write the name on one side and the definition on the back. For math formulae, write the equation on one side and the solution on the back. Quiz yourself. Once you've mastered giving the definition or solution based on the front of the card, try...


Запишите линейное уравнение. Перепишите уравнение в виде линейного уравнения. Сравните угловые коэффициенты.

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The formula of a line in slope-intercept form is y = mx + b, where m is the slope, b is the y-intercept, and x and y are variables that represent coordinates on the line; generally, you will see them remain as x and y in the equation. In this form, you can easily determine the slope of the line as the variable "m". For example. Rewrite 4y - 12x = 20 and y = 3x -1. The equation 4y - 12x = 20 needs to be rewritten with algebra while y = 3x -1 is already in slope-intercept form and does not need to be rearranged. Oftentimes, the formula of the line you are given will not be in slope-intercept form. It only takes a little math and rearranging of variables to get it into slope-intercept.  For example: Rewrite line 4y-12x=20 into slope-intercept form. Add 12x to both sides of the equation: 4y – 12x + 12x = 20 + 12x Divide each side by 4 to get y on its own: 4y/4 = 12x/4 +20/4 Slope-intercept form: y = 3x + 5. Remember, when two lines are parallel to each other, they will have the exact same slope. Using the equation y = mx + b where m is the slope of the line, you can identify and compare the slopes of two lines.  In our example, the first line has an equation of y = 3x + 5, therefore it’s slope is 3. The other line has an equation of y = 3x – 1 which also has a slope of 3. Since the slopes are identical, these two lines are parallel. Note that if these equations had the same y-intercept, they would be the same line instead of parallel.


Запишите людей, которых вы хотите включить в генеалогическое дерево. Заполните пустые места, проведя некоторые исследования. Решите, какую информацию вы хотите выдвинуть на первый план. Решите, включать ли фотографии.

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A family tree begins with you, and branches out from there. Start by writing down the names of people in your immediate family, then move to your parents' generation. Make sure you don't leave anyone out! Your family tree will be an important piece of family history, so take time to create an accurate diagram.  Write down your name, your siblings' names, and your parents' names. Write down your grandparents' names, your aunts' and uncles' names and your cousins' names. Write down your great grandparents' names and your great aunts' and uncles' names. Many people stop there, but you can include as many generations as you want. It might start getting difficult to fill in names once you go back a few generations. In an effort to make sure everyone gets included and all the names are accurate, conduct some research  to double check yourself. This is a great opportunity to learn more about  your family history.  Talk to older members of your family to find out more information. Find out the names of your grandparents' siblings, their spouses, their children. Ask around to find out as much as you can. If you're lucky, you'll get the opportunity to hear some surprising family stories and secrets, too. Do online research using a genealogy tool. There are many online tools that let you simply enter your name and your parents' name, then do all the research for you. These sites usually give you a limited amount of information free, then require payment for more in-depth research. If you're serious about mapping your ancestry, this is a great way to collect information. In addition to including the first and last names of your family, you might want to list their birth dates, death dates (if applicable), marriage dates, and so on. Having these dates on the family tree will make it even more informative as a historical document for your family. In addition to dates, you could consider including your relatives' birthplaces or hometowns. If you have access to photographs of your ancestors, you could include a small...


Запишите матрицу размерностью 3 x 3. Выберите строку или столбец матрицы. Зачеркните строку или столбец с первым элементом. Найдите определитель матрицы 2 x 2. Умножьте полученный ответ на выбранный элемент матрицы M. Определите знак полученного результата. Повторите все вышеописанные действия со вторым элементом опорной строки (или столбца). Повторите с третьим элементом. Сложите полученные результаты.

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We'll start with a 3 x 3 matrix A, and try to find its determinant |A|. Here's the general matrix notation we'll be using, and our example matrix: M=(a11a12a13a21a22a23a31a32a33)=(153247462){\displaystyle M={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}={\begin{pmatrix}1&5&3\\2&4&7\\4&6&2\end{pmatrix}}} This will be your reference row or column. You'll get the same answer no matter which one you choose. For now, just pick the first row. Later, we'll give some advice on how to choose the easiest option to calculate. Let's choose the first row of our example matrix A. Circle the 1 5 3. In general terms, circle a11 a12 a13. Look at the row or column you circled and select the first element. Draw a line through its row and column. You should be left with four numbers. We'll treat these as a 2 x 2 matrix.  In our example, our reference row is 1 5 3. The first element is in row 1 and column 1. Cross out all of row 1 and column 1. Write the remaining elements as a 2 x 2 matrix:   1  5 3 2  4 1 4  6 2 Remember, the matrix (abcd){\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}} has a determinant of ad - bc. You may have learned this by drawing an X across the 2 x 2 matrix. Multiply the two numbers connected by the \ of the X. Then subtract the product of the two numbers connected by the /. Use this formula to calculate the determinate of the matrix you just found.  In our example, the determinant of the matrix (4762){\displaystyle {\begin{pmatrix}4&7\\6&2\end{pmatrix}}} = 4 * 2 - 7 * 6 = -34. This determinant is called the minor of the element we chose in our original matrix. In this case, we just found the minor of a11. Remember, you selected an element from your reference row (or column) when you decided which row and column to cross out. Multiply this element by the determinant you just calculated for the 2x2 matrix. In our example, we selected a11, which had a value of 1. Multiply this by -34 (the determinant of the 2x2) to get 1*-34 = -34. Next, you'll multiply...


Запишите множители числителя и множители знаменателя. Вычислите наибольший общий делитель (НОД) числителя и знаменателя. Разделите числитель и знаменатель на наибольший общий делитель. Проверьте полученный результат.

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Factors are numbers that you multiply together to get another number. For example, 3 and 4 are both factors of 12, because you can multiply them together to get 12. To list the factors of a number, you simply have to list all of the numbers that can be multiplied to get that number, and which therefore can be evenly divided into that number. List the factors of that number from smallest to largest, not forgetting to include 1 or that number. For example, this is how you would list the factors of the numerator and denominator of the fraction, 24/32:  24: 1, 2, 3, 4, 6, 8, 12, 24. 32: 1, 2, 4, 8, 16, 32. The GCF is the highest number that divides evenly into two or more numbers. Once you've listed all of the factors of that number, all you have to do is find the largest number that is repeated in both lists.  24: 1, 2, 3, 4, 6, 8, 12, 24. 32: 1, 2, 4, 8, 16, 32. The GCF of 24 and 32 is 8, because 8 is the largest number that evenly divides into both 24 and 32. Now that you've found your GCF, all you have to do is to divide the numerator and denominator by that number to reduce your fraction to its lowest terms.  Here's how to do it:  24/8 = 3 32/8 = 4 The simplified fraction is 3/4. If you'd like to make sure that you've simplified the fraction correctly, you can simply multiply the new numerator and new denominator by the GCF to make sure that you can return to the original fraction. Here's how to do it:  3 * 8 = 24 4 * 8 = 32 You have returned to the original fraction, 24/32. You can also examine the fraction to make sure that it cannot be reduced any more. Since 3 is a prime number, it can only be divisible by 1 and itself, and four is not divisible by 3, so the fraction cannot be reduced further.


Запишите начальное и конечное значения. Найдите разницу между начальным и конечным значениями. Полученный результат разделите на начальное значение. Полученный результат умножьте на 100.

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For example, let's say your auto insurance premium just went up. Write down these values:  Your car insurance premium was $400 before the increase. This is the starting value. After the increase, it costs $450. This is the end value. Subtract the starting value from the end value to find the out how much it increased. We're still working with ordinary numbers at this point, not percentages. In our example, $450 - $400 = a $50 increase. A percentage is just a special kind of fraction. For example, "5% of doctors" is quick way to write "5 out of 100 doctors." By dividing the answer by the starting value, we turn it into a fraction that compares the two values. In our example, $50 / $400 = 0.125. This converts your last result into a percentage. The final answer to our example is 0.125 x 100 = 12.5% increase in auto insurance premiums.


Запишите начальное и конечное значения. Разделите конечное значение на начальное значение. Полученный результат умножьте на 100. Из полученного результата вычтите 100.

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Let's start with a new example. The world population went from 5,300,000,00 people in 1990 to 7,400,000,000 in 2015. There's a trick to these problems with many zeroes. Instead of counting the zeroes each step of the way, we can rewrite these as 5.3 billion and 7.4 billion. This will tell us how much bigger the end result is than the original.  7.4 billion ÷ 5.3 billion = about 1.4. We've rounded to two significant digits because that's how many there were in the original problem. This will tell you the percentage comparison between the two values. If the value increased (instead of decreasing), your answer should always be larger than 100. 1.4 x 100 = 140%. This means the world population in 2015 is 140% the size of the population in 1990. In this kind of problem, "100%" is the size of the starting value. By subtracting this from our answer, we're left with just the percentage size of the increase.  140% - 100% = a 40% increase in population. This works because starting value + increase = end value. Rearrange the equation and we get increase = end value - starting value.


Запишите начальную цену и процент скидки. Обратите внимание на процент скидки. Преобразуйте найденный процент в десятичную дробь. Умножьте начальную цену на десятичную дробь. Вычислите размер скидки.

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The original price is usually the big number that retailers will mark through or make especially large so you know how good of a “deal” you’re getting. Be sure to also pay attention to whether the discounted percentage applies to a single item or the total. This trick lets you perform 1 calculation instead of 2! The opposite of a percent is 100% minus the percent you are working with, i.e. the amount of the item you are still paying for. If you want to buy a shirt that is 30% off, the opposite of this is 70%, meaning you would have to pay 70% of the original price. To convert a percent into a decimal, divide by 100%, multiply times 0.01, or move the decimal two places to the left. In this example, 70%/100% = 70/100 = 7/10 = 0.7. If the shirt you want is $20, multiply $20 by 0.7. This comes to $14, meaning the shirt is now on sale for $14. Once you get home, it can be a good idea to check your receipts, not only to make sure the discounts were applied correctly but to see how much you saved altogether. Here, you can simply subtract the sale price from the original price ($20 - $14 = $6 saved)!


Запишите основную причину, по которой вы хотите перейти в другую школу. Отметьте дату, когда вы хотите перейти в новую школу. Выберите школу, в которую вы хотите перейти. Запишите положительные причины перехода в другую школу.

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Before you can make a good argument with your parents, you need to understand why you really want to change schools. You need to be able to state this reason clearly. Some reasons you might want to change schools include:   You’ve been dealing with  bullying, and you don’t think it’s going to get better or you don’t feel comfortable staying around those people. Before asking your parents make sure your certain you'd like to switch (writing pros and cons may help). If you know this school will cost money, show them how much you want to go to this school despite the cost. You feel lost in the crowd of a large school with large classes, and you’d like a smaller environment. You don’t think your school is helping you academically. You might need a more challenging school or a school where you could help more individualized help. There’s another school that has programs you’re really interested in, like a superior drama, music, art, band, or sports program. The social environment is not what you want, maybe you don't have a lot of friends or have different views than your peers. When presenting this reason, word it carefully so it doesn't give your parents the idea that you just want to party. Don't say it like the only thing in school is about having friends, either. Tell them you need a study buddy, and no one at your current school is willing to help you out. As you are writing down this reason, make sure it is important enough to switch. For example, if you just don’t like math, and your school gives you a lot of homework, that’s not a good reason to switch. Or, if your boyfriend or best friend goes to a different school, this is not necessarily a good enough reason to switch. This will affect how you approach your parents about your situations. If you give your parents exact dates that you want to change schools, it will make it easier for them to say yes and not put off letting you switch.  If you’re being bullied, you might want to make a mid-year switch. However, if you want to change to a school that will...