Trying to decide math problems with fractions, the student understands that the desire to solve these problems is not enough for him. Knowledge of calculations with fractional numbers is also required. In some problems, all initial data are given in the condition in fractional form. In others, some of them may be fractions, and some may be whole numbers. To make some calculations with these given values, you must first bring them to a single form, that is, convert integers to fractional ones, and then do the calculations. In general, the way to convert an integer to a fraction is very simple. To do this, write the given number itself in the numerator of the final fraction, and one in its denominator. That is, if you need to convert the number 12 into a fraction, then the resulting fraction will be 12/1.

Such modifications help to bring fractions to a common denominator. This is necessary in order to be able to subtract or add fractional numbers. When multiplying and dividing them, a common denominator is not required. You can consider an example of how to convert a number into a fraction and then add two fractional numbers. Suppose you need to add the number 12 and the fractional number 3/4. The first term (the number 12) is reduced to the form 12/1. However, its denominator is 1, while the second term is 4. For the subsequent addition of these two fractions, they must be reduced to a common denominator. Due to the fact that one of the numbers has a denominator equal to 1, this is generally easy to do. It is necessary to take the denominator of the second number and multiply by it both the numerator and the denominator of the first.

The result of multiplication will be: 12/1=48/4. If 48 is divided by 4, then 12 is obtained, which means that the fraction is reduced to the correct denominator. Thus, at the same time, you can understand how to translate a fraction into an integer. This only applies to improper fractions, because they have a larger numerator than a denominator. In this case, the numerator is divided by the denominator and, if there is no remainder, there will be an integer. With the remainder, the fraction remains a fraction, but with a highlighted whole part. Now regarding the reduction to a common denominator in the considered example. If the first term had the denominator equal to some other number than 1, the numerator and denominator of the first number would have to be multiplied by the denominator of the second, and the numerator and denominator of the second by the denominator of the first.

Both terms are reduced to their common denominator and are ready for addition. It turns out that in this problem you need to add two numbers: 48/4 and 3/4. When adding two fractions with the same denominator, you only need to sum their upper parts, that is, the numerators. The denominator of the sum will remain unchanged. In this example, it should be 48/4+3/4=(48+3) /4=51/4. This will be the result of the addition. But in mathematics it is customary to reduce improper fractions to proper ones. Above, it was considered how to turn a fraction into a number, but in this example, an integer will not be obtained from the fraction 51/4, since the number 51 is not divisible by the number 4 without a remainder. Therefore, you need to select the integer part of this fraction and its fractional part. The integer part will be the number that is obtained by dividing by an integer the first number less than 51.

That is, one that can be divided by 4 without a remainder. The first number in front of the number 51, which is completely divisible by 4, will be the number 48. Dividing 48 by 4, the number 12 is obtained. This means that the integer part of the required fraction will be 12. It remains only to find the fractional part of the number. The denominator of the fractional part remains the same, i.e. 4 in this case. To find the numerator of the fractional part, it is necessary to subtract from the original numerator the number that was divided by the denominator without a remainder. In this example, it is required to subtract the number 48 from the number 51. That is, the numerator of the fractional part is 3. The result of the addition will be 12 integers and 3/4. The same is true when subtracting fractions. Suppose you need to subtract the fractional number 3/4 from the integer 12. To do this, the integer 12 is converted into a fractional 12/1, and then reduced to a common denominator with the second number - 48/4.

When subtracting in the same way, the denominator of both fractions remains unchanged, and subtraction is carried out with their numerators. That is, the numerator of the second is subtracted from the numerator of the first fraction. AT this example it will be 48/4-3/4=(48-3) /4=45/4. And again it turned out to be an improper fraction, which must be reduced to the correct one. To select the integer part, the first number up to 45 is determined, which is divisible by 4 without a remainder. It will be 44. If the number 44 is divided by 4, you get 11. So the integer part of the final fraction is 11. In the fractional part, the denominator is also left unchanged, and the number that was divided by the denominator without a remainder is subtracted from the numerator of the original improper fraction. That is, it is necessary to subtract 44 from 45. So the numerator in the fractional part is 1 and 12-3/4=11 and 1/4.

If one integer and one fractional number is given, but its denominator is 10, then it is easier to convert the second number into a decimal fraction, and then perform calculations. For example, you need to add the integer 12 and the fractional number 3/10. If the number 3/10 is written as a decimal, it will be 0.3. Now it is much easier to add 0.3 to 12 and get 2.3 than to bring fractions to a common denominator, perform calculations, and then extract the integer and fractional parts from an improper fraction. Even the simplest problems with fractional numbers assume that the student (or student) knows how to convert an integer to a fraction. These rules are too simple and easy to remember. But with the help of them it is very easy to carry out calculations of fractional numbers.

Fractions

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

Fractions in high school are not very annoying. For the time being. Until you come across exponents with rational exponents and logarithms. And there…. You press, you press the calculator, and it shows all the full scoreboard of some numbers. You have to think with your head, like in the third grade.

Let's deal with fractions, finally! Well, how much can you get confused in them!? Moreover, it is all simple and logical. So, what are fractions?

Types of fractions. Transformations.

Fractions are of three types.

1. Common fractions , eg:

Sometimes, instead of a horizontal line, they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens ...), tell yourself the phrase with the expression: " Zzzzz remember! Zzzzz denominator - out zzzz u!" Look, everything will be remembered.)

A dash, which is horizontal, which is oblique, means division top number (numerator) to bottom number (denominator). And that's it! Instead of a dash, it is quite possible to put a division sign - two dots.

When the division is possible entirely, it must be done. So, instead of the fraction "32/8" it is much more pleasant to write the number "4". Those. 32 is simply divided by 8.

32/8 = 32: 8 = 4

I'm not talking about the fraction "4/1". Which is also just "4". And if it doesn’t divide completely, we leave it as a fraction. Sometimes you have to do the reverse. Make a fraction from a whole number. But more on that later.

2. Decimals , eg:

It is in this form that it will be necessary to write down the answers to tasks "B".

3. mixed numbers , eg:

Mixed numbers are practically not used in high school. In order to work with them, they must be converted to ordinary fractions. But you definitely need to know how to do it! And then such a number will come across in the puzzle and hang ... From scratch. But we remember this procedure! A little lower.

Most versatile common fractions. Let's start with them. By the way, if there are all sorts of logarithms, sines and other letters in the fraction, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!

Basic property of a fraction.

So let's go! First of all, I will surprise you. The whole variety of fraction transformations is provided by a single property! That's what it's called basic property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction will not change. Those:

It is clear that you can write further, until you are blue in the face. Do not let sines and logarithms confuse you, we will deal with them further. The main thing to understand is that all these various expressions are the same fraction . 2/3.

And we need it, all these transformations? And how! Now you will see for yourself. First, let's use the basic property of a fraction for fraction abbreviations. It would seem that the thing is elementary. We divide the numerator and denominator by the same number and that's it! It's impossible to go wrong! But... man is a creative being. You can make mistakes everywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.

How to reduce fractions correctly and quickly without doing unnecessary work can be found in special Section 555.

A normal student does not bother dividing the numerator and denominator by the same number (or expression)! He just crosses out everything the same from above and below! This is where it hides typical mistake, blooper if you want.

For example, you need to simplify the expression:

There is nothing to think about, we cross out the letter "a" from above and the deuce from below! We get:

Everything is correct. But really you shared the whole numerator and the whole denominator "a". If you are used to just cross out, then, in a hurry, you can cross out the "a" in the expression

and get again

Which would be categorically wrong. Because here the whole numerator on "a" already not shared! This fraction cannot be reduced. By the way, such an abbreviation is, um ... a serious challenge to the teacher. This is not forgiven! Remember? When reducing, it is necessary to divide the whole numerator and the whole denominator!

Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. And how to work with her now? Without a calculator? Multiply, say, add, square!? And if you are not too lazy, but carefully reduce by five, and even by five, and even ... while it is being reduced, in short. We get 3/8! Much nicer, right?

The basic property of a fraction allows you to convert ordinary fractions to decimals and vice versa without calculator! This is important for the exam, right?

How to convert fractions from one form to another.

It's easy with decimals. As it is heard, so it is written! Let's say 0.25. It's zero point, twenty-five hundredths. So we write: 25/100. We reduce (divide the numerator and denominator by 25), we get the usual fraction: 1/4. Everything. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.

What if integers are non-zero? Nothing wrong. Write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three whole, seventeen hundredths. We write 317 in the numerator, and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary Watson! From all of the above, a useful conclusion: any decimal fraction can be converted to a common fraction .

But the reverse conversion, ordinary to decimal, some cannot do without a calculator. And it is necessary! How will you write down the answer on the exam!? We carefully read and master this process.

What is a decimal fraction? She has in the denominator always is worth 10 or 100 or 1000 or 10000 and so on. If your usual fraction has such a denominator, there is no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. And if in the answer to the task of section "B" it turned out 1/2? What will we write in response? Decimals are required...

We remember basic property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. For anyone, by the way! Except zero, of course. Let's use this feature to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But, then the numerator must also be multiplied by 5. This is already maths demands! We get 1/2 \u003d 1x5 / 2x5 \u003d 5/10 \u003d 0.5. That's all.

However, all sorts of denominators come across. For example, the fraction 3/16 will fall. Try it, figure out what to multiply 16 by to get 100, or 1000... Doesn't work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide in a corner, on a piece of paper, as they taught in elementary grades. We get 0.1875.

And there are some very bad denominators. For example, the fraction 1/3 cannot be turned into a good decimal. Both on a calculator and on a piece of paper, we get 0.3333333 ... This means that 1/3 into an exact decimal fraction does not translate. Just like 1/7, 5/6 and so on. Many of them are untranslatable. Hence another useful conclusion. Not every common fraction converts to a decimal. !

By the way, this helpful information for self-test. In section "B" in response, you need to write down a decimal fraction. And you got, for example, 4/3. This fraction is not converted to decimal. This means that somewhere along the way you made a mistake! Come back, check the solution.

So, with ordinary and decimal fractions sorted out. It remains to deal with mixed numbers. To work with them, they all need to be converted to ordinary fractions. How to do it? You can catch a sixth grader and ask him. But not always a sixth grader will be at hand ... We will have to do it ourselves. It is not difficult. Multiply the denominator of the fractional part by the integer part and add the numerator of the fractional part. This will be the numerator of a common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but it's actually quite simple. Let's see an example.

Let in the problem you saw with horror the number:

Calmly, without panic, we understand. The whole part is 1. One. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of an ordinary fraction. That's all. It looks even simpler in mathematical notation:

Clearly? Then secure your success! Convert to common fractions. You should get 10/7, 7/2, 23/10 and 21/4.

The reverse operation - converting an improper fraction into a mixed number - is rarely required in high school. Well, if... And if you - not in high school - you can look into the special Section 555. In the same place, by the way, you will learn about improper fractions.

Well, almost everything. You remembered the types of fractions and understood How convert them from one type to another. The question remains: why do it? Where and when to apply this deep knowledge?

I answer. Any example suggests necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed into a bunch, we translate everything into ordinary fractions. It can always be done. Well, if something like 0.8 + 0.3 is written, then we think so, without any translation. Why do we need extra work? We choose the solution that is convenient us !

If the task is entirely decimals, but um... some evil ones, go to the ordinary ones, try it! Look, everything will be fine. For example, you have to square the number 0.125. Not so easy if you have not lost the habit of the calculator! Not only do you need to multiply the numbers in a column, but also think about where to insert the comma! It certainly doesn't work in my mind! And if you go to an ordinary fraction?

0.125 = 125/1000. We reduce by 5 (this is for starters). We get 25/200. Once again on 5. We get 5/40. Oh, it's shrinking! Back to 5! We get 1/8. Easily square (in your mind!) and get 1/64. Everything!

Let's summarize this lesson.

1. There are three types of fractions. Ordinary, decimal and mixed numbers.

2. Decimals and mixed numbers always can be converted to common fractions. Reverse Translation not always available.

3. The choice of the type of fractions for working with the task depends on this very task. In the presence of different types fractions in one task, the most reliable thing is to switch to ordinary fractions.

Now you can practice. First, convert these decimal fractions to ordinary ones:

3,8; 0,75; 0,15; 1,4; 0,725; 0,012

You should get answers like this (in a mess!):

On this we will finish. In this lesson, we brushed up on the key points on fractions. It happens, however, that there is nothing special to refresh ...) If someone has completely forgotten, or has not mastered it yet ... Those can go to a special Section 555. All the basics are detailed there. Many suddenly understand everything are starting. And they solve fractions on the fly).

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

Then press the buttons, and the task is completed. As a result, you will get either an integer or a decimal fraction. A decimal fraction can have a long remainder after . In this case, the fraction must be rounded to a certain digit you need using rounding (numbers up to 5 are rounded down, from 5 inclusive and more - up).

If the calculator is not at hand, but you will have to. Write the numerator of a fraction with a denominator, between them a little corner, meaning. For example, convert the fraction 10/6 to a number. To begin with, divide 10 by 6. It turns out 1. Write down the result in a corner. Multiply 1 by 6, you get 6. Subtract 6 from 10. You get a remainder of 4. The remainder must be divided by 6 again. Add 0 to 4, and divide 40 by 6. You get 6. Write 6 in the result, after the decimal point. Multiply 6 by 6. You get 36. Subtract 36 from 40. You get the remainder again 4. Then you can not continue, because it becomes obvious that the result will be the number 1.66 (6). Round the given fraction to the digit you need. For example, 1.67. This is the final result.

Related article

Sources:

• converting fractions to whole numbers

Fractions are needed to denote numbers that consist of one or more parts of the unit. The term "fraction" comes from the Latin fractura, which means "to crush, break". There are ordinary and decimal fractions. At the same time, in ordinary fractions, a unit can be divided into any number of parts, and in decimal fractions, this number must be a multiple of 10. Any fraction can be both ordinary and decimal.

You will need

• To calculate the result, you will need a calculator or a piece of paper and a pen.

Instruction

So, for starters, take common fraction and divide it into parts. For example, 2 1/8, in which 2 is an integer part, and 1/8 is a fraction. From it you can see that the number was divided by 8, but only one was taken. The part that was taken is the numerator, and the number of parts into which it is divided is the denominator.

note

Often there are fractions that cannot be fully converted to decimals. This is where rounding comes in handy. If you want to round to thousandths, then look at the fourth number after the decimal point. If it is less than 5, then write down in response, the first three digits after the decimal point without change, otherwise, one must be added to the last digit of the three. For example, 0.89643123 can be written as 0.896, but 0.89663123 can be written as 0.897.

Helpful advice

If you calculate the result manually, then before dividing the fraction, it is better to reduce it as much as possible, and also to select whole parts from it.

Sources:

• how to convert fractions

Fraction is one of the elements of formulas for the input of which in the word processor Word there is a Microsoft Equation tool. With it, you can enter any complex mathematical or physical formulas, equations and other elements that include special characters.

Instruction

To launch the Microsoft Equation tool, you need to go to the address: "Insert" -> "Object", in the dialog box that opens, on the first tab from the list, select Microsoft Equation and click "OK" or double-click on the selected item. After launching the editor, a toolbar will open in front of you and an input field will be displayed: a rectangle in a dotted one. The toolbar is divided into sections, each of which contains a set of action signs or expressions. When you click on one of the sections, a list of the tools in it will expand. From the list that opens, select the desired symbol and click on it. Once selected, the specified character will appear in a selected rectangle in the document.

The section that contains elements for writing fractions is located in the second line of the toolbar. When you hover your mouse cursor over it, you will see the tooltip "Fraction and Radical Patterns". Click a section once and expand the list. The drop-down menu has templates for horizontal and oblique fractions. Among the options that appear, you can choose the one that suits your task. Click on desired option. After clicking, in the input field that opened in the document, a fraction symbol and places for entering the numerator and denominator, framed by a dotted line, will appear. The default cursor is automatically placed in the field for entering the numerator. Enter the numerator. In addition to numbers, you can also enter symbols, letters, or action signs. They can be entered both from the keyboard and from the corresponding sections of the Microsoft Equation toolbar. After the numerator water, press the TAB key to move to the denominator. You can also go by clicking the mouse in the field for entering the denominator. Once written, click with the mouse pointer anywhere in the document, the toolbar will close, the fraction input will be completed. To edit , double-click on it with the left mouse button.

If, when you open the menu "Insert" -> "Object", you did not find the Microsoft Equation tool in the list, you need to install it. Run the installation disc, disc image, or Word distribution file. In the installer window that appears, select "Add or remove components. Adding or removing individual components" and click "Next". In the next window, check the item "Advanced application settings". Click next. In the next window, find the list item "Office Tools" and click on the plus sign on the left. In the expanded list, we are interested in the item "Formula Editor". Click on the icon next to "Formula Editor" and, in the menu that opens, click "Run from computer". After that, click "Update" and wait until the required component is installed.

A decimal has two parts separated by commas. The first part is an integer unit, the second part is tens (if the number after the decimal point is one), hundreds (two numbers after the decimal point, like two zeros in a hundred), thousandths, etc. Let's look at examples of decimals: 0, 2; 7, 54; 235.448; 5.1; 6.32; 0.5. These are all decimals. How do you convert a decimal to a common fraction?

Example one

We have a fraction, for example, 0.5. As mentioned above, it consists of two parts. The first number, 0, shows how many integer units the fraction has. In our case, they are not. The second number shows tens. The fraction even reads zero point five tenths. Decimal number convert to fraction now it will not be difficult, we write 5/10. If you see that the numbers have a common divisor, you can reduce the fraction. We have this number 5, dividing both parts of the fraction by 5, we get - 1/2.

Example two

Let's take a more complex fraction - 2.25. It is read like this - two whole and twenty-five hundredths. Pay attention - hundredths, since there are two numbers after the decimal point. Now you can convert to a common fraction. We write down - 2 25/100. The integer part is 2, the fractional part is 25/100. As in the first example, this part can be shortened. The common divisor for 25 and 100 is 25. Note that we always choose the greatest common divisor. Dividing both parts of the fraction by GCD, we got 1/4. So 2, 25 is 2 1/4.

Example three

And to consolidate the material, let's take the decimal fraction 4.112 - four whole and one hundred and twelve thousandths. Why thousandths, I think, is clear. Now we write down 4 112/1000. According to the algorithm, we find the GCD of the numbers 112 and 1000. In our case, this is the number 6. We get 4 14/125.

Output

1. We break the fraction into integer and fractional parts.
2. We look at how many digits after the decimal point. If one is tens, two is hundreds, three is thousandths, etc.
3. We write the fraction in the usual form.
4. We reduce the numerator and denominator of the fraction.
5. Write down the resulting fraction.
6. We perform a check, divide the upper part of the fraction by the lower one. If there is an integer part, add to the resulting decimal fraction. It turned out the original version - great, so you did everything right.

Using examples, I showed how you can convert a decimal fraction to an ordinary one. As you can see, it is very easy and simple to do this.

Materials on fractions and study sequentially. below for you detailed information with examples and explanations.

1. Mixed number into a common fraction.Let's write in general view number:

We remember a simple rule - we multiply the whole part by the denominator and add the numerator, that is:

Examples:

2. On the contrary, an ordinary fraction into a mixed number. *Of course, this can only be done with an improper fraction (when the numerator is greater than the denominator).

With “small” numbers, no action, in general, needs to be done, the result is “seen” immediately, for example, fractions:

*Details:

15:13 = 1 remainder 2

4:3 = 1 remainder 1

9:5 = 1 remainder 4

But if the numbers are more, then you can’t do without calculations. Everything is simple here - we divide the numerator by the denominator by a corner until the remainder is less than the divisor. Division scheme:

For example:

* The numerator is the dividend, the denominator is the divisor.

We get the integer part (incomplete quotient) and the remainder. We write down - an integer, then a fraction (there is a remainder in the numerator, and we leave the denominator the same):

3. We translate the decimal into an ordinary one.

Partially in the first paragraph, where we talked about decimal fractions, we have already touched on this. As we hear, so we write. For example - 0.3; 0.45; 0.008; 4.38; 10.00015

We have the first three fractions without an integer part. And the fourth and fifth have it, we will translate them into ordinary ones, we already know how to do this:

*We see that fractions can also be reduced, for example, 45/100 = 9/20, 38/100 = 19/50 and others, but we will not do this here. For the reduction, a separate paragraph awaits you below, where we will analyze everything in detail.

4. Ordinary translate into decimal.

It's not all that simple. For some fractions, you can immediately see and clearly what to do with it so that it becomes decimal, for example:

We use our wonderful basic property of a fraction - we multiply the numerator and denominator, respectively, by 5, 25, 2, 5, 4, 2, we get:

If there is an integer part, then nothing complicated either:

We multiply the fractional part, respectively, by 2, 25, 2 and 5, we get:

And there are those for which, without experience, it is impossible to determine that they can be converted into decimals, for example:

What numbers should you multiply the numerator and denominator by?

Here again, a proven method comes to the rescue - division by a corner, a universal method, you can always use it to convert an ordinary fraction to a decimal:

So you can always determine whether a fraction is converted to a decimal. The fact is that not every ordinary fraction can be converted to decimal, for example, such as 1/9, 3/7, 7/26 are not translated. And what then turns out for a fraction when dividing 1 by 9, 3 by 7, 5 by 11? I answer - infinite decimal (we talked about them in paragraph 1). Let's divide:

That's all! Good luck to you!

Sincerely, Alexander Krutitskikh.