How to divide common fractions correctly. Actions with fractions
Last time we learned how to add and subtract fractions (see the lesson "Adding and subtracting fractions"). The most difficult moment in those actions was bringing the fractions to a common denominator.
Now it's time to deal with multiplication and division. The good news is that these operations are even easier to perform than addition and subtraction. To begin with, consider the simplest case when there are two positive fractions without a dedicated integer part.
To multiply two fractions, you must separately multiply their numerators and denominators. The first number will be the numerator of the new fraction, and the second will be the denominator.
To divide two fractions, you need to multiply the first fraction by the “inverted” second.
Designation:
From the definition it follows that the division of fractions is reduced to multiplication. To "flip" a fraction, just swap the numerator and denominator. Therefore, the entire lesson we will consider mainly multiplication.
As a result of multiplication, a cancellable fraction can arise (and often does arise) - it, of course, must be canceled. If, after all the contractions, the fraction turned out to be incorrect, the whole part should be selected in it. But what will definitely not happen with multiplication is reduction to a common denominator: no criss-cross methods, largest factors and least common multiples.
By definition, we have:
Multiplication of whole fractions and negative fractions
If there is an integer part in the fractions, they must be converted into incorrect ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the range of multiplication or even removed according to the following rules:
- Plus and minus gives a minus;
- Two negatives make an affirmative.
Until now, these rules were met only when adding and subtracting negative fractions, when it was required to get rid of the whole part. For production, they can be generalized to "burn" several cons at once:
- Cross out the minuses in pairs until they completely disappear. In an extreme case, one minus can survive - the one for which there was no pair;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out, since it did not find a pair, we move it outside the multiplication limits. You get a negative fraction.
A task. Find the meaning of the expression:
We convert all fractions into incorrect ones, and then we move the minuses out of the multiplication limits. We multiply what is left according to the usual rules. We get:
Let me remind you once again that the minus in front of a fraction with a highlighted integer part refers specifically to the entire fraction, and not just to its integer part (this applies to the last two examples).
Also pay attention to negative numbers: when multiplying, they are enclosed in parentheses. This is done in order to separate the minuses from the multiplication signs and make the whole notation more accurate.
Reducing fractions on the fly
Multiplication is a very time consuming operation. The numbers here are quite large, and to simplify the task, you can try to reduce the fraction even more before multiplication... Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be canceled using the basic property of a fraction. Take a look at examples:
A task. Find the meaning of the expression:
By definition, we have:
In all examples, the numbers that have undergone reduction and what is left of them are marked in red.
Please note: in the first case, the multipliers have been reduced completely. In their place, there are only a few that, generally speaking, can be omitted. In the second example, the complete reduction was not achieved, but the total amount of computation still decreased.
However, under no circumstances use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers there that you just want to reduce. Here, take a look:
You can't do that!
The error occurs due to the fact that when adding in the numerator of the fraction appears the sum, and not the product of numbers. Therefore, it is impossible to apply the basic property of a fraction, since this property deals precisely with the multiplication of numbers.
There is simply no other reason for reducing fractions, so the correct solution to the previous problem looks like this:
Correct solution:
As you can see, the correct answer turned out to be not so pretty. In general, be careful.
Multiplication and division of fractions.
Attention!
There are additional
materials in Special Section 555.
For those who are "not very ..."
And for those who "very much ...")
This operation is much nicer than addition-subtraction! Because it's easier. Let me remind you: to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). I.e:
For instance:
Everything is extremely simple... And please don't look for a common denominator! Don't need him here ...
To divide a fraction into a fraction, you need to flip second(this is important!) fraction and multiply them, i.e .:
For instance:
If you come across multiplication or division with integers and fractions - that's okay. As with addition, we make a fraction with one in the denominator out of an integer - and go! For instance:
In high school, you often have to deal with three-story (or even four-story!) Fractions. For instance:
How to bring this fraction to a decent look? It's very simple! Use two-point division:
But don't forget the division order! Unlike multiplication, this is very important here! Of course, 4: 2, or 2: 4, we will not confuse. But in a 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 determines the order of division? Or brackets, or (as here) the length of horizontal bars. Develop an eye. And if there are no brackets or dashes, like:
then we divide-multiply in order, from left to right!
And another very simple and important trick. In actions with degrees, oh, how useful he is! Divide the unit by any fraction, for example, by 13/15:
The fraction has turned over! And it always does. When dividing 1 by any fraction, the result is the same fraction, only inverted.
That's all for fractions. The thing is quite simple, but it gives more than enough errors. Take note of the practical advice, and there will be fewer (mistakes)!
Practical advice:
1. The most important thing when working with fractional expressions is accuracy and care! These are not general words, not good wishes! This is a dire necessity! Do all calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in the draft than to mess it up in the mind.
2. In examples with different types of fractions - go to ordinary fractions.
3. All fractions are reduced to stop.
4. Multi-storey fractional expressions are reduced to ordinary ones using division through two points (watch the division order!).
5. Divide the unit into a fraction mentally, simply by turning the fraction over.
Here are the tasks that you must definitely solve. Answers are given after all tasks. Use the materials on this topic and practical advice. Consider how many examples you were able to solve correctly. The first time! No calculator! And make the right conclusions ...
Remember - the correct answer is received from the second (especially the third) time - does not count! This is a harsh life.
So, we solve in exam mode ! This is already preparation for the exam, by the way. We solve the example, check, solve the next one. We decided everything - checked again from the first to the last. But only then look at the answers.
Calculate:
Have you solved it?
We are looking for answers that match yours. I deliberately wrote them down in a mess, away from temptation, so to speak ... Here they are, the answers, separated by semicolons.
0; 17/22; 3/4; 2/5; 1; 25.
And now we draw conclusions. If everything worked out, I'm glad for you! Basic calculations with fractions are 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 solved Problems.
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.
Division is. In this article we will talk about division of ordinary fractions... First, we will give a rule for dividing ordinary fractions and consider examples of dividing fractions. Next, we will focus on dividing an ordinary fraction by a natural number and numbers by a fraction. Finally, consider how the division of an ordinary fraction by a mixed number is carried out.
Page navigation.
Dividing a fraction by a fraction
It is known that division is the inverse of multiplication (see the relationship between division and multiplication). That is, division involves finding an unknown factor when the product and another factor are known. The same sense of division is retained in the division of ordinary fractions.
Let's look at examples of division of ordinary fractions.
Note that we should not forget about the reduction of fractions and about the separation of the whole part from the improper fraction.
Division of an ordinary fraction by a natural number
We will immediately give the rule for dividing an ordinary fraction by a natural number: to divide the fraction a / b by a natural number n, you need to leave the numerator unchanged, and multiply the denominator by n, that is,.
This division rule directly follows from the division rule for ordinary fractions. Indeed, representing a natural number as a fraction leads to the following equalities 
.
Consider an example of dividing a fraction by a number.
Example.
Divide 16/45 by the natural number 12.
Decision.
By the rule of dividing a fraction by a number, we have 
... Let's execute the reduction:. This completes the division.
Answer:
.
Division of a natural number by an ordinary fraction
The rule for dividing fractions is similar to the rule for dividing a natural number by an ordinary fraction: to divide a natural number n by an ordinary fraction a / b, you need to multiply the number n by the number that is the reciprocal of a / b.
According to the voiced rule, and the rule for multiplying a natural number by an ordinary fraction allows you to rewrite it in the form.
Let's look at an example.
Example.
Divide the natural number 25 by the fraction 15/28.
Decision.
Let's go from division to multiplication, we have 
... After cutting and isolating the whole part, we get.
Answer:
.
Division of an ordinary fraction by a mixed number
Division of an ordinary fraction by a mixed number easily reduced to division of ordinary fractions. To do this, it is enough to carry out
) and the denominator by the denominator (we get the denominator of the product).
The formula for multiplying fractions:
For instance:
Before you start multiplying the numerators and denominators, you need to check for the possibility of reducing the fraction. If you can reduce the fraction, then it will be easier for you to make further calculations.
Division of an ordinary fraction into a fraction.
Division of fractions with a natural number.
It's not as scary as it sounds. As in the case of addition, we convert an integer to a fraction with one in the denominator. For instance:
Multiplication of mixed fractions.
The rules for multiplying fractions (mixed):
- converting mixed fractions to incorrect ones;
- we multiply the numerators and denominators of fractions;
- we reduce the fraction;
- if you got an incorrect fraction, then convert the incorrect fraction to a mixed one.
Note! To multiply a mixed fraction by another mixed fraction, you first need to bring them to the form of improper fractions, and then multiply according to the rule of multiplying ordinary fractions.
The second way to multiply a fraction by a natural number.
It may be more convenient to use the second way of multiplying an ordinary fraction by a number.
Note! To multiply a fraction by a natural number, you need to divide the denominator of the fraction by this number, and leave the numerator unchanged.
From the above example, it is clear that this option is more convenient to use when the denominator of the fraction is divisible without a remainder by a natural number.
Multi-storey fractions.
In high school, three-story (or more) fractions are often found. Example:
To bring such a fraction to the usual form, division through 2 points is used:
Note!When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Note, eg:
When dividing one by any fraction, the result will be the same fraction, only inverted:
Practical tips for multiplying and dividing fractions:
1. The most important thing in working with fractional expressions is accuracy and care. Do all calculations carefully and accurately, with concentration and clarity. It is better to write a few extra lines in a draft than to get confused in the calculations in your head.
2. In tasks with different types of fractions - go to the form of ordinary fractions.
3. We reduce all fractions until it becomes impossible to reduce.
4. Multi-storey fractional expressions are reduced to ordinary ones, using division through 2 points.
5. Divide the unit into a fraction mentally, simply by turning the fraction over.
A fraction is one or more parts of a whole, which is usually taken as one (1). As with natural numbers, all basic arithmetic operations (addition, subtraction, division, multiplication) can be performed with fractions, for this you need to know the features of working with fractions and distinguish between their types. There are several types of fractions: decimal and ordinary, or simple. Each type of fractions has its own specifics, but having thoroughly figured out once how to handle them, you can solve any examples with fractions, since you will know the basic principles of performing arithmetic calculations with fractions. Let's look at examples of how to divide a fraction by an integer using different types of fractions.
How to divide a prime fraction by a natural number?Ordinary or simple are fractions written in the form of such a ratio of numbers, in which the dividend (numerator) is indicated at the top of the fraction, and the divisor (denominator) of the fraction is indicated below. How do you divide such a fraction by an integer? Let's look at an example! Let's say we want to divide 8/12 by 2.
To do this, we must perform a number of actions:
Thus, if we are faced with the task of dividing a fraction by an integer, the solution scheme will look something like this: 
Similarly, you can divide any ordinary (simple) fraction by an integer.
How do I divide a decimal by an integer?
A decimal fraction is a fraction that is obtained by dividing one into ten, a thousand, and so on. Decimal arithmetic is fairly straightforward.
Let's look at an example of how to divide a fraction by an integer. Let's say we need to divide the decimal fraction 0.925 by the natural number 5.
Summing up, we will focus on two main points that are important when performing the operation of dividing decimal fractions by an integer: - to divide a decimal fraction by a natural number, column division is used;
- the comma is placed in the quotient when the division of the integer part of the dividend is completed.
