Already in primary school students facefractions. And then they appear in each topic. You can not forget the actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are simple, the main thing is to understand everything in order.

The world around us consists of whole objects. Therefore in shares of necessity there is no. But everyday life constantly pushes people to work with parts of things and things.

For example, chocolate consists of several lobules.Consider the situation when its tile is formed by twelve rectangles. If it is divided into two, then it turns out to be 6 parts. She will be well divided into three. But the five will not be able to give a whole number of chocolate slices.

By the way, these segments are already fractions. And their further division leads to the appearance of more complex numbers.

This number consists of parts of the unit.Externally it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written on top (left) is called the numerator. What stands from below (on the right) is the denominator.

In fact, a fractional line is a sign of division. That is, the numerator can be called divisible, and the denominator can be called a divisor.

In mathematics there are only two types:ordinary and decimal fractions. With the first students get acquainted in the primary classes, calling them just "fractions." The second learn in grade 5. It is then that these names appear.

Ordinary fractions are all those that are written inthe form of two numbers separated by a line. For example, 4/7. Decimal is the number in which the fractional part has a positional record and is separated from the whole by a comma. For example, 4.7. Students need to clearly understand that the two examples given are completely different numbers.

Each simple fraction can be written as a decimal. This statement is almost always true in the opposite direction. There are rules that allow you to write a decimal fraction with an ordinary fraction.

Start better in chronological order, as they are being studied. The first are ordinary fractions. Among them, there are 5 subspecies.

Correct. Its numerator is always less than the denominator.

Wrong. Its numerator is greater than or equal to the denominator.

Reducible / irreducible.It can be either correct or incorrect. Another important thing is whether the numerator with the denominator has common factors. If there are, then they are supposed to divide both parts of the fraction, that is, reduce it.

Mixed. To its usual correct (incorrect) fractional part, an integer is assigned. And it always stands on the left.

Compound. It is formed from two divided fractions. That is, it has three fractional features at once.

Decimal fractions have only two subspecies:

final, that is, one whose fractional part is bounded (has an end);

infinite - a number whose digits after the comma do not end (they can be written endlessly).

If this is a finite number, then an association based on the rule is applied - as I hear, so I write. That is, you need to read it correctly and write it down, but without a comma, but with a fractional line.

As a clue about the necessary denominator, you need to remember that it is always one and several zeros. The latter need to write as many as the digits in the fractional part of the number being considered.

How to convert decimals into ordinary fractions,if their whole part is absent, that is, is equal to zero? For example, 0.9 or 0.05. After applying this rule, it turns out that you need to write zero integers. But it is not specified. It remains to write down only fractional parts. For the first, the denominator will be 10, and the second will be 100. That is, the indicated examples will have numbers 9/10, 5/100. And the last one turns out to be cut by 5. Therefore, the result for it should be written 1/20.

How to make a decimal fraction,if its integer part is different from zero? For example, 5.23 or 13.00108. In both examples, the whole part is read and its value is written. In the first case, this is 5, in the second case, 13. Then we need to go to the fractional part. They are supposed to carry out the same operation with them. The first number appears 23/100, the second - 108/100000. The second value should be reduced again. In the answer, these mixed fractions are obtained: 5 23/100 and 13 27/25000.

If it is non-periodical, then such an operation will not be possible. This fact is related to the fact that each decimal fraction is always translated either to the final or to the periodic one.

The only thing that is allowed to do with suchfractions, is to round it. But then the decimal will be approximately equal to that infinite. It can already be turned into an ordinary one. But the reverse process: translating to decimal will never give an initial value. That is, infinite non-periodic fractions into ordinary fractions are not translated. You need to remember this.

In these numbers, after the comma,one or more digits that are repeated. They are called a period. For example, 0.3 (3). Here "3" in the period. They are classed as rational, since they can be converted into ordinary fractions.

Those who met with periodic fractions,it is known that they can be pure or mixed. In the first case, the period starts immediately from the comma. In the second - the fractional part begins with any numbers, and then the repetition begins.

The rule by which you want to write in the formordinary fraction of infinite decimal, will be different for the two types of numbers indicated. Pure periodic fractions to write ordinary are quite simple. As with the finite, they need to be transformed: in the numerator write the period, and the denominator will be the number 9, repeated as many times as the number contains the period.

For example, 0, (5). The whole part of the number is not, so immediately you need to start fractional. In the numerator write 5, and in the denominator of one 9. That is, the answer is a fraction of 5/9.

The rule about how to write down an ordinary decimal periodic fraction, which is mixed.

Count the digits of the fractional part before the period. They will indicate the number of zeros in the denominator.

Look at the length of the period. So much will have a denominator.

Write down the denominator: first nine, then zero.

To determine the numerator, you need to write down the difference of two numbers. Decrements will be all digits after the decimal point, along with the period. Subtractable - it's the same without a period.

For example, 0.5 (8) - write down the periodicdecimal fraction in the form of ordinary. In the fractional part, up to the period there is one figure. So zero will be one. In the period, too, only one figure is 8. That is, one is nine. That is, in the denominator it is necessary to write 90.

To determine the numerator from 58, you need to subtract 5. It turns out 53. The answer to the example would be to write 53/90.

The simplest version is a number whose denominator is 10, 100, and so on. Then the denominator is simply discarded, and a comma is placed between the fractional and the whole parts.

There are situations when the denominator is easyturns into 10, 100, etc. For example, the numbers 5, 20, 25. They are multiplied by 2, 5 and 4, respectively. Only multiply is assigned not only the denominator, but also the numerator by the same number.

For all other cases, a simple rule is useful: divide the numerator by the denominator. In this case, you can get two variants of answers: the final or periodic decimal.

*Addition and subtraction*

With them, students get acquainted before others. And first, the fractions have the same denominators, and then different. General rules can be reduced to such a plan.

Find the least common multiple of the denominators.

Write additional factors to all ordinary fractions.

Multiply the numerators and denominators by the factors that are specified for them.

Add (subtract) the numerators of fractions, and leave the common denominator unchanged.

If the numerator of the reduced is less than the subtrahend, then we need to find out whether we have a mixed number or a proper fraction.

In the first case, the whole part needs to occupy the unit. Add a denominator to the numerator of the fraction. And then perform the subtraction.

In the second, you must apply the rule of subtracting from a smaller number the greater That is, subtract the module from the subtracted module to subtract the decremented module, and in the answer put the “-” sign.

Carefully look at the result of addition (subtraction). If you get the wrong fraction, it is necessary to allocate the whole part. That is, divide the numerator by denominator.

*Multiplication and division*

To perform them, the fraction does not need to be reduced to a common denominator. This simplifies the execution of actions. But they still need to follow the rules.

When multiplying ordinary fractions, it is necessary to consider the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.

Multiply the numerators.

Multiply the denominators.

If it turns out reduced fraction, then it should be simplified again.

When dividing, you must first replace the division by multiplication, and the divisor (the second fraction) by the reciprocal fraction (swap the numerator and denominator).

Then act as in multiplication (starting from point 1).

In tasks where you need to multiply (divide) by an integer, the latter is supposed to be written in the form of an improper fraction. That is, with the denominator 1. Then act as described above.

*Addition and subtraction*

Of course, you can always turn the decimalin ordinary. And act on the already described plan. But sometimes it is more convenient to act without this translation. Then the rules for their addition and subtraction will be exactly the same.

Equalize the number of digits in the fractional part of the number, that is, after the comma. Assign in it the missing number of zeros.

Write fractions so that the comma is under the comma.

Add (subtract) as natural numbers.

Demolish comma.

*Multiplication and division*

It is important that there is no need to add zeros. The shot should be left as it is given in the example. And then go according to plan.

For multiplication, you need to write fractions one under the other, not paying attention to commas.

Multiply as natural numbers.

Put a comma in the answer, counting from the right end of the answer as many digits as they are in the fractional parts of both factors.

To divide, you must first convert the divider: make it a natural number. That is, multiply it by 10, 100, etc., depending on how many digits in the fractional part of the divider.

On the same number multiply the dividend.

Divide the decimal fraction by a natural number.

Put in the answer a comma at the moment when the division of the whole part ends.

Yes, in math there are often examples, inwhich need to perform operations on ordinary and decimal fractions. In such tasks there are two possible solutions. You need to objectively weigh the numbers and choose the best.

*First way: submit ordinary decimal*

It is suitable if, when dividing or translatingthe resulting fractions are obtained. If at least one number gives a periodic part, then this method is prohibited. Therefore, even if you do not like working with ordinary fractions, you will have to consider them.

*Second way: write decimal fractions ordinary*

This technique is convenient if in partafter the comma are 1-2 digits. If there are more, there can be a very large ordinary fraction and decimal records will make it possible to count the task faster and easier. Therefore, you always need to soberly evaluate the task and choose the easiest method of solution.

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