Ordinary and decimal fractions and actions over them

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

Why do you need fractions?

The world around us consists of whole objects. Therefore, there is no necessity for shares. 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 you divide it into two, you get 6 parts. It will be well divided into three. But the five will not be able to give a whole number of chocolate pieces.

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

What is a "fraction"?

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 from above (left) is called the numerator. What stands at the bottom (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.

What are the fractions?

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 in the 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.

Which subspecies have these types of fractions?

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

1. Correct. Its numerator is always less than the denominator.

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

3. Reducible / irreducible. It can be both correct and 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, cut it.

4. Mixed. A whole number is assigned to its usual correct (incorrect) fractional part. And it always stands on the left.

5. 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).

How to convert a decimal into an ordinary fraction?

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 in question.

How to convert decimals into ordinary fractions if their whole part is missing, that 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 above examples will have numbers 9/10, 5/100. And the latter can be reduced by 5. Therefore, the result for it should be written 1/20.

How to make an ordinary decimal fraction, if its whole 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 conduct the same operation with them. The first number appears 23/100, the second - 108/100000. The second value must again be reduced. In the answer, we obtain such mixed fractions: 5 23/100 and 13 27/25000.

How to convert an infinite decimal into an ordinary decimal fraction?

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

The only thing that is allowed to do with such a fraction 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: conversion to decimal will never give an initial value. That is, infinite non-periodic fractions into ordinary ones are not translated. You need to remember this.

How to write an infinite periodic fraction in the form of an ordinary?

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

Those who have 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 an infinite decimal in the form of an ordinary fraction 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: the period is written to the numerator, and the denominator is the number 9, which is repeated as many times as the digits contain 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 one 9. That is, the answer is a fraction of 5/9.

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

• Count the fractional digits 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. Deductible - it's the same without a period.

For example, 0.5 (8) - write a periodic decimal in the form of an 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 subtract 5. It turns out 53. The answer to the example would be to write 53/90.

How are ordinary fractions converted to decimals?

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 easily turns into 10, 100, etc. For example, the numbers 5, 20, 25. They multiply 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: a finite or a periodic decimal.

Actions with ordinary fractions

Addition and subtraction

Students learn about them before others. And first, the fractions have the same denominators, and then different. General rules can be reduced to such a plan.

1. Find the least common multiple of the denominators.

2. Write additional factors to all common fractions.

3. Multiply the numerators and denominators by their multipliers.

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

5. 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.

6. 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.

7. In the second - it is necessary to apply the rule of subtraction from a smaller number greater. That is, subtract the module from the subtracted module, and put the "-" sign in response.

8. Carefully look at the result of addition (subtraction). If an irregular fraction is obtained, then it is necessary to allocate the whole part. That is, to divide the numerator by the denominator.

Multiplication and division

For their execution, fractions do not need to lead to a common denominator. This simplifies the execution of actions. But they still have to follow the rules.

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

2. Multiply numerators.

3. Multiply the denominators.

4. If a contractible fraction is obtained, then it is supposed to be simplified again.

5. When dividing, we first need to replace the division by multiplication, and the divisor (the second fraction) by the reverse fraction (swap the numerator and denominator).

6. Then act like multiplication (starting with point 1).

7. In tasks where multiplying (dividing) by an integer, the latter is supposed to be written in the form of an irregular fraction. That is, with the denominator 1. Then act as described above.

Actions with decimals

Addition and subtraction

Of course, you can always convert a decimal into an ordinary fraction. And act according to the plan already described. But sometimes it is more convenient to act without this translation. Then the rules for their addition and subtraction will be exactly the same.

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

2. Write the fraction so that the comma is under the comma.

3. Add (subtract) as natural numbers.

4. Take the comma off.

Multiplication and division

It is important that you do not need to add zeros here. Fractions are supposed to be left as they are given in the example. And then go on according to plan.

1. To multiply, you need to write fractions one under another, not paying attention to commas.

2. Multiply as natural numbers.

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

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

5. To multiply the dividend by the same number.

6. Divide the decimal into a natural number.

7. Put a comma in the reply at the moment when the whole part is finished.

What if, in one example, there are both types of fractions?

Yes, in mathematics, there are often examples in which you need to perform actions on ordinary and decimal fractions. In such tasks, there are two possible solutions. It is necessary to weigh the numbers objectively and choose the optimal one.

The first way: to introduce ordinary decimal

It is suitable if finite fractions are obtained by fission or translation. 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 count them.

The second way: to write down decimals ordinary

This method is convenient, if in the part after the decimal there are 1-2 digits. If there are more of them, you can get a very large ordinary fraction and the decimal notations will allow you to count the task faster and easier. Therefore, you always need to soberly assess the task and choose the simplest method of solution.