The division formula of ordinary fractions. Multiplication and division of fractions
) And the denominator on the denominator (we get a denominator of the work).
Formula multiplication fractions:
For example:
Before proceeding with multiplication of numerals and denominators, it is necessary to check the possibility of cutting the fraction. If it turns out to shorten the fraction, then you will be easier to carry out calculations.
Division of ordinary fraction on the fraction.
Division fractions with the participation of a natural number.
It's not as scary as it seems. As in the case of adding, we translate an integer in the fraction with a unit in the denominator. For example:
Multiplying mixed fractions.
Rules of multiplication of fractions (mixed):
- we transform mixed fractions into the wrong;
- reduce the numerals and denominators of fractions;
- reducing the fraction;
- if you got the wrong fraction, we transform the wrong fraction into a mixed one.
Note! To multiply a mixed fraction on another mixed fraction, you need to begin to bring them to the mind of the wrong fractions, and then multiply by the rule of multiplication ordinary fractions.
The second method of multiplication of the fraction on a natural number.
It is more convenient to use the second way of multiplying an ordinary fraction for a number.
Note! For multiplication of fractions on natural number A denominator is needed to divide the number, and the numerator is left unchanged.
From the above, the example is clear that this option is more convenient for use when the denoter of the fraction is divided without a residue on a natural number.
Multi-storey fractions.
In high school classes, three-story (or more) fractions are found. Example:
To bring such a fraction to the usual mind, use division after 2 points:
Note!In dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Note, eg:
When dividing units on any fraction, the result will the same fraction, only inverted:
Practical tips when multiplying and dividing fractions:
1. The most important in working with fractional expressions is accuracy and attentiveness. All calculations do carefully and gently, concentrately and clearly. Better write down a few unnecessary lines in the drafts, than getting confused in the calculations in the mind.
2. In tasks with different types of fractions - go to the species of ordinary fractions.
3. All fractions reducing until it is impossible to cut.
4. Multi-storey fractional expressions are in the form of ordinary, using the division after 2 points.
5. Unit of fraction divide in mind, just turning the fraction.
Last time we learned to fold and deduct the fraction (see the lesson "Addition and subtraction of fractions"). The most difficult moment in the actions was to bring fractions to the general denominator.
Now it's time to deal with multiplication and division. Good news It is that these operations are performed even easier than addition and subtraction. To begin with, consider the simplest case when there are two positive fractions without a selected part.
To multiply two fractions, it is necessary to multiply their numerals and denominators. The first number will be the numerator of the new fraction, and the second is the denominator.
To split two fractions, you need to multiply the first fraction to the "inverted" second.
Designation:
From the definition it follows that the division of fractions is reduced to multiplication. To "flip" the fraction, it is enough to change the numerator and denominator in places. Therefore, we will consider the whole lesson mostly multiplying.
As a result of multiplication, it may occur (and often it really occurs) a shortage of fraction - it, of course, must be reduced. If after all the cuts, the fraction was incorrect, it should be allocated to the whole part. But what exactly will not be when multiplying, it is to bring to a common denominator: no methods of "cross-elder", the greatest multipliers and the smallest common multiples.
By definition, we have:
Multiplication of fractions with a whole part and negative fractions
If in the frauds there is a whole part, they must be translated into the wrong - and only then multiplied according to the schemes above.
If there is a minus in a denoter in a denoter or before it, it can be reached out of multiplication or completely removed according to the following rules:
- Plus, minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have met only when adding and subtracting negative fractions when it was required to get rid of the whole part. For the work, they can be generalized to "burn" several minuses at once:
- I draw out the minuses in pairs until they disappear completely. In extreme cases, one minus can survive - the one who did not find a couple;
- If there are no minuses, the operation is completed - you can proceed to multiplication. If the last minus does not cross out, since he did not find a couple, we endure it outside the multiplication. It turns out a negative fraction.
A task. Find the value of the expression:
All fractions are translated into the wrong, and then we endure the minuses outside the multiplication. What remains, multiply by the usual rules. We get:
Once again remind you that minus, which is before the fraction with the selected whole part, it is precisely the whole fraction, and not just to its whole part (this applies to the last two examples).
Also pay attention to negative numbers: When multiplying, they are in brackets. This is done in order to separate the minuses from the multiplication signs and make the entire record more accurate.
Reduction of fractions "On the fly"
Multiplication is a very laborious operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction more to multiplication. After all, essentially, the numerals and denominants of fractions are ordinary multipliers, and therefore they can be cut using the main property of the fraction. Take a look at the examples:
A task. Find the value of the expression:
By definition, we have:
In all examples, the numbers that were subjected to reduction were marked, and what remained from them.
Please note: in the first case, the multipliers decreased completely. There are few units in their place, which, generally speaking, you can not write. In the second example, it was not possible to achieve a complete reduction, but the total volume of computation was still decreased.
However, in no case do not use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you want to cut. Here, look:
So you can not do!
An error occurs due to the fact that when adding the fraction in the numerator, the amount appears, and not the product of numbers. Therefore, it is impossible to apply the main property of the fraction because in this property. we are talking It is about multiplication of numbers.
Other grounds for cutting fractions simply does not exist, so correct solution The previous task looks like this:
Correct solution:
As you can see, the correct answer was not so beautiful. In general, be careful.
Multiplication and division of fractions.
Attention!
This topic has additional
Materials in a special section 555.
For those who are strongly "not very ..."
And for those who are "very ...")
This operation is much more nicer addition-subtraction! Because it's easier. I remind you: To multiply the fraction on the fraction, you need to multiply the numerators (it will be the resultant) and the denominators (this will be the denominator). I.e:
For example:
Everything is extremely simple. And please do not look for a common denominator! Do not need him here ...
To divide the fraction for the fraction, you need to flip over second(This is important!) Fraction and multiply them, i.e.:
For example:
If multiplication or division with integers and fractions was caught - nothing terrible. As with the addition, we make a fraction with a unit in the denominator - and forward! For example:
In high schools, it is often necessary to deal with three-story (or even four-storey!) Droks. For example:
How to bring this fraction to a decent mind? Yes, very simple! Use division in two points:
But do not forget about the order of division! Unlike multiplication, it is very important here! Of course, 4: 2, or 2: 4 We are not confused. But in the three-story fraction it is easy to make a mistake. Note, for example:
In the first case (expression on the left):
In the second (expression on the right):
Do you feel the difference? 4 and 1/9!
And what is the order of division? Or brackets, or (as here) the length of horizontal lines. Develop the eye meter. And if there are no brackets, nor dash, like:
then divide-multiply in a few, left to right!
And a very simple and important technique. In actions with degrees, he oh, how can I come in handy! We divide the unit to any fraction, for example, by 13/15:
The fraction turned over! And it always happens. When dividing 1 to any fraction, as a result, we get the same fraction only inverted.
That's all the actions with fractions. The thing is quite simple, but the mistakes gives more than enough. Note practical adviceAnd their (errors) will be less!
Practical tips:
1. The most important thing when working with fractional expressions is accuracy and attentiveness! Is not general words, not good wishes! This is a harsh need! All calculations on the exam make as a full task, focusing and clearly. It is better to write two extra lines in the draft, than to accumulate when calculating the mind.
2. In the examples with different types of fractions - we turn to ordinary fractions.
3. All fractions cut until it stops.
4. Multi-storey fractional expressions are reduced to ordinary, using division in two points (follow the order of division!).
5. Unit of fraction divide in mind, just turning the fraction.
Here are the tasks you need to break. Answers are given after all tasks. Use the materials of this topic and practical advice. Count how many examples you could solve correctly. The first time! Without a calculator! And make faithful conclusions ...
Remember - the correct answer, the resulting from the second (even more - the third) times - not considered! Such is a harsh life.
So, we decide in the exam mode ! This is already prepared for the exam, by the way. We solve the example, check, solve the following. They decided everything - they checked again from the first to last. Only later We look at the answers.
Calculate:
Did you cut?
We are looking for answers that coincide with yours. I specifically recorded them in disarray, away from the temptation, so to speak ... So they are answered, the point with the comma is recorded.
0; 17/22; 3/4; 2/5; 1; 25.
And now we make conclusions. If everything happened - I am glad for you! Elementary calculations with fractions - not your problem! You can do more serious things. If not...
So you have one of two problems. Or both at once.) Lack of knowledge and (or) Inattention. But this resolved Problems.
If you like this site ...
By the way, I have another couple of interesting sites for you.)
It can be accessed in solving examples and find out your level. Testing with instant check. Learn - with interest!)
You can get acquainted with features and derivatives.
The fraction is one or more of a whole share for which one is usually accepted (1). As with natural numbers, with fractions you can perform all the main arithmetic action (addition, subtraction, division, multiplication), for this you need to know the features of working with fractions and distinguish their views. There are several types of fractions: decimal and ordinary, or simple. Its specifics have each type of fractions, but, thoroughly dealting once, how to contact them, you can solve any examples with fractions, because you will know the basic principles of performing arithmetic calculations with fractions. Consider on the examples how to divide the fraction for an integer using different types frains.
How to split simple fraction on a natural number?Ordinary or simple, the fractions recorded in the form of such a ratio of numbers, at which the end of the fraction is specified by the divisible (numerator), and below the divider (denominator) of the fraction. How to divide such a fraction for an integer? Consider on the example! Suppose we need to divide 8/12 to 2.
To do this, we must fulfill a number of actions:
Thus, if we facilitate the task to divide the fraction for an integer, the solution scheme will look something like this: 
Similarly, you can divide any ordinary (simple) fraction for an integer.
How to divide the decimal fraction for an integer?
The decimal fraction is such a fraction that is obtained due to dividing unit for ten, a thousand and so on. Arithmetic actions with decimal fractions are performed quite simple.
Consider on the example how to split the fraction for an integer. Suppose we need to share the decimal fraction of 0.925 per natural number 5.
Summing up, let's stop at two main points that are important when performing a division operation. decimal fractions For an integer: - for the separation of the decimal fraction on the natural number, division in the column is used;
- the comma is placed in private when the division of the whole part of the dividend is completed.
To solve various tasks from the course of mathematics, physics have to divide fractions. It is very easy to do, if you know certain rules for performing this mathematical action.
Before moving to the formulation of the rule, how to share the fractions, let's remember some mathematical terms:
- The upper part of the fraci is called the numerator, and the lower - denominator.
- When dividing numbers is called this: divisible: divider \u003d private
How to share fractions: simple fractions
To perform the division of two simple fractions, you should multiply dividically on the fraction, reverse divider. This fraction is differently called another inverted, because it is obtained as a result of replacing the numerator and denominator. For example:
3/77: 1/11 = 3 /77 * 11 /1 = 3/7
How to share fractions: mixed fractions
If we have to divide the mixed fraction, then here it is also quite simple and understandable. First we translate the mixed fraction into the usual incorrect fraction. To do this, we multiply the denominator of such a fraction on an integer and the numerator add to the obtained product. As a result, we received a new numerator of a mixed fraction, and the denominator will remain unchanged. Further division of fractions will be carried out in the same way as the division of simple frains. For example:
10 2/3: 4/15 = 32/3: 4/15 = 32/3 * 15 /4 = 40/1 = 40
How to divide the fraction
In order to divide the simple fraction to the number, the latter should be written in the form of a fraction (incorrect). It is very easy to do: at the site of the numerator, this number is written, and the denominator is such a fraction equal to one. Further division is performed in the usual way. Consider this on the example:
5/11: 7 = 5/11: 7/1 = 5/11 * 1/7 = 5/77
How to share decimal fractions
Often, an adult is having difficulty if necessary without the help of a calculator to divide an integer or decimal fraction for a decimal fraction.
So, to perform the division of decimal fractions, you need to simply cross the comma in the divider and stop paying attention to it. In Delim, the comma is needed to move right at exactly so much signs as it was in the fractional part of the divider, if necessary, adding zeros. And further produce ordinary division by an integer. To make it more clear, we give the following example.
