How to solve the fraction of fractions to the general denominator. Bringing fractions to a new denominator - rule and examples

• Addition and subtraction of fractions with the same denominators
• Addition and subtraction of fractions with different denominators
• Concept of NOK.
• Bringing fractions to one denominator
• How to fold an integer and fraction

1 Addition and subtraction of fractions with the same denominators

To fold the fractions with the same denominators, it is necessary to fold their numerals, and the denominator leave the same, for example:

To subtract fractions with the same denominators, it is necessary from the numerator of the first fraction to deduct the numerator of the second fraction, and the denominator leave the same, for example:

To fold the mixed fractions, it is necessary to separately add their whole parts, and then folded their fractional parts, and record the result mixed fraction,

Example 1:

Example 2:

If the fraction of fractional parts turned out to be improper fraction, separated from it the whole part and add it to the whole part, for example:

2 Addition and subtraction of fractions with different denominators.

In order to fold or subtract fractions with different denominators, you must first lead them to one denominator, and then act as indicated at the beginning of this article. The overall denominator of several fractions is the NOC (the smallest common one). For the numerator of each fraction there are additional factors by dividing the NOC to the denominator of this fraction. We will look at the example later, after you figure it out what kind of NOK.

3 The smallest total multiple (NOK)

The smallest total multiple of two numbers (NOC) is the smallest natural number that is divided into both of these numbers without a residue. Sometimes the NOK can be selected orally, but more often, especially when working with large numbers, it is necessary to find NOC in writing, using the following algorithm:

In order to find the NOC of several numbers, you need:

1. Decompose these numbers for simple factors
2. Take the biggest decomposition, and write these numbers in the form of a work
3. To highlight in other expansions of the number that are not found in the largest decomposition (or there are fewer times in it), and add them to the work.
4. Multiply all the numbers in the work, it will be the NOC.

For example, we find NOC numbers 28 and 21:

4 Bringing fractions to one denominator

Let's return to the addition of fractions with different denominators.

When we give a fraction to the same denominator equal to the NOC of both denominators, we must multiply the number of these fractions on additional multipliers. It is possible to find them, dividing the NOC to the denominator of the corresponding fraction, for example:

Thus, in order to bring the fraction to one indicator, you must first find the NOC (that is, the smallest number that is divided into both denominator) of the denominators of these fractions, then put additional faults to the details of fractions. You can find them by dividing the general denominator (NOC) to the denominator of the corresponding fraction. Then you need to multiply the numerator of each fraction on an additional factor, and the denominator put the NOC.

5 How to fold an integer and fraction

In order to fold an integer and fraction, you just need to add this number before the fraction, the mixed fraction will be mixed, for example:

If we fold an integer and mixed fraction, we add this number to the whole part of the fraction, for example:

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Addition and subtraction of fractions with the same denominators.

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• If the result is incorrect fraction, you need to translate it into a mixed number.
• If the fraction can be reduced, be sure to reduce it, otherwise the wrong answer will be counted.

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Initially, I wanted to include the methods of bringing to a general denominator in paragraph "Addition and subtraction of fractions". But there was so much information, and its importance is so great (after all, general denominators are not only in numerical fractions), which is better to study this question separately.

So, let us have two fractions with different denominators. And we want to make the denominators become the same. The main property of the fraction comes to the rescue, which, remind, sounds as follows:

The fraction will not change if its numerator and denominator multiply the same number other than zero.

Thus, if you correctly select multipliers, the denominators in the frains are equal - this process is called bringing to a common denominator. And the artificial numbers, "leveling" denominants are called additional factories.

Why do you need to give a fraction to a common denominator? Here are just a few reasons:

1. Addition and subtraction of fractions with different denominators. In a different way, this operation is not fulfilled;
2. Comparison of fractions. Sometimes bringing to a common denominator greatly simplifies this task;
3. Solving tasks for shares and interest. Interest ratios are essentially ordinary expressions that contain fractions.

There are many ways to find numbers, when multiplying by which the denominators will become equal. We will consider only three of them - in order of increasing complexity and, in a sense, efficiency.

Multiplication of "Cross-Low"

The easiest and most reliable way that guarantees the denominators is guaranteed. We will act "across": we multiply the first fraction to the signator of the second fraction, and the second - to the denominator first. As a result, the denominators of both fractions will become equal to the product of the initial denominators. Take a look:

As an additional factors, consider the denominators of neighboring fractions. We get:

Yes, so everything is simple. If you are just starting to study the fraction, it is better to work exactly this method - so you are intensifying yourself from a variety of errors and guaranteed to get the result.

The only drawback of this method is to count a lot, because the denominers are multiplying, and as a result, very large numbers can get. Such is the payment of reliability.

Method of common divisors

This technique helps a lot to reduce the calculations, but, unfortunately, it is rarely applied. The method is as follows:

1. Before acting "Stroke" (i.e., by the cross-cross-time method), take a look at the denominators. Perhaps one of them (one that is more) is divided into another.
2. The number obtained as a result of this division will be an additional factor for a fraction with a smaller denominator.
3. At the same time, the fraction with a large denominator does not need to multiply anything - this is saving. At the same time, the probability of error sharply decreases.

A task. Find the values \u200b\u200bof expressions:

Note that 84: 21 \u003d 4; 72: 12 \u003d 6. Since in both cases one denominator is divided without a residue to another, we use the method of general factors. We have:

Note that the second fraction in general did not multiply anywhere. In fact, we have reduced the volume of calculations twice!

By the way, the fraction in this example I took it not by chance. If it is interesting, try to count them by the "Cross-crossing" method. After cutting, the answers will turn out the same, but the work will be much more.

This is the force of the method of common divisors, but I repeat, it is possible to apply it only when one of the denominators is divided into another without a residue. What happens quite rarely.

Method of the smallest total multiple

When we bring a fraction to a common denominator, we are essentially trying to find such a number that is divided into each of the denominators. Then lead to this number the denominators of both fractions.

There are a lot of such numbers, and the smallest of them will not necessarily be equal to the direct product of the denominators of the initial fractions, as it is assumed in the "Cross-crossroad" method.

For example, for denominators 8 and 12, the number 24 is quite suitable, since 24: 8 \u003d 3; 24: 12 \u003d 2. This number is much less than the work of 8 · 12 \u003d 96.

The smallest number that is divided into each of the denominators is called their smallest common multiple (NOC).

Designation: The smallest general multiple numbers A and B is denoted by NOC (A; B). For example, NOC (16; 24) \u003d 48; NOC (8; 12) \u003d 24.

If you manage to find such a number, the final amount of calculations will be minimal. Look at the examples:

A task. Find the values \u200b\u200bof expressions:

Note that 234 \u003d 117 · 2; 351 \u003d 117 · 3. Multiplers 2 and 3 are mutually simple (do not have common divisors, except 1), and the multiplier 117 is common. Therefore, NOK (234; 351) \u003d 117 · 2 · 3 \u003d 702.

Similarly, 15 \u003d 5 · 3; 20 \u003d 5 · 4. Multiplers 3 and 4 are mutually simple, and multiplier 5 is common. Therefore, NOK (15; 20) \u003d 5 · 3 · 4 \u003d 60.

Now we will give the fractions for general denominators:

Please note how good it was to decompose the initial denominator for factors:

1. Finding the same multipliers, we immediately went to the smallest common pain, which, generally speaking, is a nontrivial task;
2. From the resulting decomposition, you can find out which factors "not enough" each of the frains. For example, 234 · 3 \u003d 702, therefore, for the first fraction, the additional factor is 3.

To evaluate how the tremendous winnings give the least common multiple method, try to calculate the same examples by the method of the Cross. Of course, without a calculator. I think after that comments will be superfluous.

Do not think that there will be no such difficult fractions in these examples. They are constantly meeting, and the above tasks are not the limit!

The only problem is how to find this church. Sometimes everything is in a few seconds, literally "on the eye", but in general it is a complex computational task that requires separate consideration. Here we will not touch it.

How to bring algebraic (rational) fractions to a common denominator?

1) If there are polynomials in the denominators, you need to try one of the known methods.

2) the smallest common denominator (nos) consists of all multipliers taken in high degree.

The smallest common denominator for numbers is orally looking for as the smallest number, which is divided into other numbers.

3) To find an additional factor to each fraction, you need a new denominator to divide on the old one.

4) The numerator and denominator of the initial fraction multiply on an additional factor.

Consider examples of bringing algebraic fractions to a common denominator.

To find a common denominator for numbers, choose a larger number and check whether it is divided into less. 15 to 9 is not divisible. Multiply 15 by 2 and check whether the obtained number is divided by 9. 30 to 9 is not divided. We multiply 15 by 3 and check whether the obtained number is divided into 9. 45 to 9 is divided, it means that the general denominator for numbers is 45.

The smallest common denominator consists of all multipliers taken to the greatest extent. Thus, the overall denominator of these fractions is 45 BC (letters are accepted in alphabetical order).

To find an additional multiplier to each fraction, you need a new denominator to divide on the old one. 45BC: (15B) \u003d 3C, 45BC: (9C) \u003d 5B. We multiply the numerator and denominator of each fraction on an additional multiplier:

First we are looking for a common denominator for numbers: 8 to 6 is not divided, 8 ∙ 2 \u003d 16 to 6 is not divided, 8 ∙ 3 \u003d 24 to 6 is divided. Each of the variables must be included in the total denominator once. From the degrees we take a degree with a great figure.

Thus, the overall denominator of these fractions is 24a³BC.

To find an additional multiplier to each fraction, you need a new denominator to divide on the old: 24a³BC: (6a³c) \u003d 4B, 24a³BC: (8a²bc) \u003d 3A.

An additional factor is multiplied by a numerator and denominator:

Multicominates in denominators of these frains need. In the denominator of the first fraction - a full square of the difference: x²-18x + 81 \u003d (x-9) ²; In the denominator, the second is the difference of squares: x²-81 \u003d (x-9) (x + 9):

The general denominator consists of all multipliers made to the greatest extent, that is, equal to (x-9) ² (x + 9). We find additional multipliers and multiply them on the numerator and the denominator of each fraction:

This article describes how to bring a fraction to a common denominator and how to find the smallest common denominator. Definitions are given, the result of bringing fractions to a common denominator and considered practical examples.

What is the resulting fraction for a common denominator?

Ordinary fractions consist of a numerator - the upper part, and the denominator - the bottom. If the fraraty has the same denominator, they say that they are shown to the general denominator. For example, fractions 11 14, 17 14, 9 14 have the same denominator 14. In other words, they are shown to the general denominator.

If the fractions have different denominators, they can always be brought to a common denominator using non-hard action. To do this, you need a numerator and a denominator to multiply by certain additional factors.

Obviously, fractions 4 5 and 3 4 are not given to a common denominator. To do this, you need to use additional faults 5 and 4 to lead them to the denominator 20. How exactly do it? Multiply the numerator and denominator of the fraction 4 5 to 4, and the numerator and denominator of the fraction 3 4 multiply on 5. Instead of fractions 4 5 and 3 4, we obtain, respectively, 16 20 and 15 20.

Bringing fractions to a common denominator

Bringing fractions to a common denominator is the multiplication of the number and denominators of fractions on such multipliers that the resultant fraction with the same denominator is obtained.

General denominator: Definition, examples

What is a common denominator?

Common denominator

The overall denominator of fractions is any positive number that is a common multiple of all these fractions.

In other words, the general denominator of some kind of fraction will be such a natural number that is divided without a residue to all denominators of these fractions.

A number of natural numbers are infinite, and therefore, according to the definition, each set of ordinary fractions has an infinite set of common denominants. In other words, there are infinitely many common multiple for all denominers of the original set of fractions.

A common denominator for several fractions is easy to find using the definition. Let there be fractions 1 6 and 3 5. The overall denominator will be any positive common multiple for numbers 6 and 5. Such positive common multiple are the numbers 30, 60, 90, 120, 150, 180, 210, and so on.

Consider an example.

Example 1. Common denominator

Can die frame 1 3, 21 6, 5 12 lead to a common denominator, which is equal to 150?

To find out if it is, it is necessary to check whether 150 is common for denominators of fractions, that is, for numbers 3, 6, 12. In other words, the number 150 must be divided into 3, 6, 12 without a residue. Check:

150 ÷ \u200b\u200b3 \u003d 50, 150 ÷ \u200b\u200b6 \u003d 25, 150 ÷ \u200b\u200b12 \u003d 12, 5

So, 150 is not a common denominator of the specified fractions.

The smallest common denominator

The smallest natural number of a variety of common denominators of some kind of fraction is called the smallest common denominator.

The smallest common denominator

The smallest overall denominator of fractions is the smallest number among all the general denominators of these frains.

The smallest common divisor of this set of numbers is the smallest common multiple (NOC). The NOC of all the denominators frains is the smallest common denominator of these frains.

How to find the smallest common denominator? His finding is reduced to finding the smallest common fragrance fractions. Turn to the example:

Example 2. Find the smallest common denominator

It is necessary to find the smallest common denominator for fractions 1 10 and 127 28.

We are looking for NOC numbers 10 and 28. Spread them on simple factors and get:

10 \u003d 2 · 5 28 \u003d 2 · 2 · 7 N o to (15, 28) \u003d 2 · 2 · 5 · 7 \u003d 140

How to bring a fraction to the smallest general denominator

There is a rule that explains how to lead a fraction for a common denominator. The rule consists of three points.

Rule of bringing fractions to a common denominator

1. Find the smallest overall denominator fractions.
2. For each fraction to find an additional multiplier. To find a multiplier, you need the smallest common denominator to divide the denominator of each fraction.
3. Multiply the numerator and denominator to the found additional factor.

Consider the application of this rule on a specific example.

Example 3. Bringing fractions to a common denominator

There are fractions 3 14 and 5 18. We give them to the smallest overall denominator.

According to the rule, we first find the NOC of the denominators of fractions.

14 \u003d 2 · 7 18 \u003d 2 · 3 · 3 N o to (14, 18) \u003d 2 · 3 · 3 · 7 \u003d 126

Calculate additional multipliers for each fraction. For 3 14, the additional factor is like 126 ÷ 14 \u003d 9, and for the fraction 5 18, the additional factor will be 126 ÷ 18 \u003d 7.

We multiply the numerator and denominator of fractions for additional factors and get:

3 · 9 14 · 9 \u003d 27 126, 5 · 7 18 · 7 \u003d 35 126.

Bringing several fractions to the smallest general denominator

Under the considered rule, not only a pair of fractions can be brought to the general denominator, but more than their number.

We give another example.

Example 4. Bringing fractions to a shared denominator

Create fractions 3 2, 5 6, 3 8 and 17 18 to the smallest general denominator.

Calculate the NOC of the denominators. We find NOC three and more numbers:

N about K (2, 6) \u003d 6 N o to (6, 8) \u003d 24 N o to (24, 18) \u003d 72 N o to (2, 6, 8, 18) \u003d 72

For 3 2, the additional factor is 72 ÷ 2 \u003d 36, for 5 6 the additional factor is 72 ÷ 6 \u003d 12, for 3 8, the additional factor is 72 ÷ 8 \u003d 9, finally, for 17 18, the additional factor is 72 ÷ 18 \u003d 4.

We multiply the fraction on additional factors and go to the smallest general denominator:

3 2 · 36 \u003d 108 72 5 6 · 12 \u003d 60 72 3 8 · 9 \u003d 27 72 17 18 · 4 \u003d 68 72

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How to bring a fraction to a common denominator

If ordinary fractions have the same denominators, they say that these the fractions are given to a common denominator..

Example 1.

For example, the fractions $ \\ FRAC (3) (18) $ and $ \\ FRAC (20) (18) $ have the same denominators. It is said that they have a total denominator $ 18 $. Crushing $ \\ FRAC (1) (29) $, $ \\ FRAC (7) (29) $ and $ \\ FRAC (100) (29) $ has also identical denominators. It is said that they have a total denominator $ 29 $.

If fractions are not the same, they can be reduced to a common denominator. To do this, multiply their numerals and denominators to certain additional multipliers.

Example 2.

How to cite two fractions $ \\ FRAC (6) (11) $ and $ \\ FRAC (2) (7) $ to a shared denominator.

Decision.

Multiply crushes $ \\ FRAC (6) (11) $ and $ \\ FRAC (2) (7) $ for additional multipliers $ 7 $ and $ 11, respectively and give them to a total denominator $ 77 $:

$ \\ FRAC (6 \\ CDOT 7) (11 \\ CDOT 7) \u003d \\ FRAC (42) (77) $

$ \\ FRAC (2 \\ CDOT 11) (7 \\ CDOT 11) \u003d \\ FRAC (22) (77) $

In this way, bringing fractions to a common denominator Called multiplication of the numerator and denominator of these fractions on additional factors, which as a result allow you to get a fraction with the same denominators.

Common denominator

Definition 1.

Any positive common multiple of all denominers of a certain set of fractions are called common denominator.

In other words, the overall denominator of the specified ordinary fractions is any natural number that can be divided into all denominators of the specified frains.

From the definition, an infinite set of common denominators of this set of fractions are followed.

Example 3.

Find common denominators of the fractions $ \\ FRAC (3) (7) $ and $ \\ FRAC (2) (13) $.

Decision.

These fractions have denominators equal to $ 7 $ and $ 13 $, respectively. Positive common multiple numbers $ 2 $ and $ 5 $ are $ 91, 182, 273, 364 $, etc.

Any of these numbers can be used as a common denominator of the fractions $ \\ FRAC (3) (7) $ and $ \\ FRAC (2) (13) $.

Example 4.

Determine whether it is possible to determine $ \\ FRAC (1) (2) $, $ \\ FRAC (16) (7) $ and $ \\ FRAC (11) (9) $ to bring $ 252 $ denominator.

Decision.

To determine how to bring a fraction to a common denominator $ 252 $, it is necessary to check whether the number is $ 252 $ shared multiple denominants $ 2, $ 7 and $ 9 $. To do this, we divide the number $ 252 $ for each of the denominators:

$ \\ FRAC (252) (2) \u003d 126, $ $ \\ FRAC (252) (7) \u003d 36 $, $ \\ FRAC (252) (9) \u003d 28 $.

The number of $ 252 $ is divided by all the denominators, i.e. It is a common multiple numbers $ 2, $ 7 and $ 9 $. It means that these fractions $ \\ FRAC (1) (2) $, $ \\ FRAC (16) (7) $ and $ \\ FRAC (11) (9) $ can be reduced to a total denominator $ 252 $.

Answer: You can.

The smallest common denominator

Definition 2.

Among all common denominators, the smallest natural numbers that are called the smallest common denominator.

Because NOC - the smallest positive common divisor of this set of numbers, the NOC of the denominators of the specified fractions is the smallest common denominator of these fractions.

Consequently, to find the smallest common denominator of fractions, you need to find the NOC of the denominators of these fractions.

Example 5.

The fractions of $ \\ FRAC (4) (15) $ and $ \\ FRAC (37) (18) $ are given. Find their smallest common denominator.

Decision.

Datamen of these fractions are equal to $ 15 $ and $ 18. We find the smallest overall denominator as NOC numbers $ 15 and $ 18. We use for this decomposition of numbers to simple multipliers:

$ 15 \u003d 3 \\ CDot $ 5, $ 18 \u003d 2 \\ CDOT 3 \\ CDOT $ 3

$ NOK (15, 18) \u003d 2 \\ CDOT 3 \\ CDOT 3 \\ CDOT 5 \u003d 90 $.

Answer: $ 90 $.

Rule of bringing fractions to the smallest general denominator

Most often, when solving problems of algebra, geometry, physics, etc. Adopted ordinary fractions to bring to the smallest common denominator, and not to any general denominator.

Algorithm:

1. Using the NOC of the denominators of the specified fractions to find the smallest common denominator.
2. 2. Add an additional factor for the specified fractions. To do this, the lowest total denominator needs to be divided into the denominator of each fraction. The resulting number and will be an additional factor of this fraction.
3. Multiply the numerator and denominator of each fraction to the found additional factor.

Example 6.

Find the smallest overall denominator of the fraction $ \\ FRAC (4) (16) $ and $ \\ FRAC (3) (22) $ and bring both fractions to it.

Decision.

We use the algorithm of bringing fractions to the smallest general denominator.

Calculate the smallest total multiple numbers $ 16 $ and $ 22 $:

Spread the denominators to simple multipliers: $ 16 \u003d 2 \\ CDOT 2 \\ CDOT 2 \\ CDOT 2 $, $ 22 \u003d 2 \\ CDOT 11 $.

$ NOK (16, 22) \u003d 2 \\ CDOT 2 \\ CDOT 2 \\ CDOT 2 \\ CDOT 11 \u003d $ 176.

Calculate additional multipliers for each fraction:

$ 176 \\ div 16 \u003d 11 $ - for the fraction $ \\ FRAC (4) (16) $;

$ 176 \\ div 22 \u003d $ 8 - for the fraction $ \\ FRAC (3) (22) $.

I multiply the numerals and denominators of the fractions $ \\ FRAC (4) (16) $ and $ \\ FRAC (3) (22) $ for additional multipliers of $ 11 $ and $ 8 $, respectively. We get:

$ \\ FRAC (4) (16) \u003d \\ FRAC (4 \\ CDOT 11) (16 \\ CDOT 11) \u003d \\ FRAC (44) (176) $

$ \\ FRAC (3) (22) \u003d \\ FRAC (3 \\ CDOT 8) (22 \\ CDOT 8) \u003d \\ FRAC (24) (176) $

Both fractions are shown to the smallest general denominator $ 176 $.

Answer: $ \\ FRAC (4) (16) \u003d \\ FRAC (44) (176) $, $ \\ FRAC (3) (22) \u003d \\ FRAC (24) (176) $.

Sometimes in order to find the smallest common denominator, you need to hold a number of time consuming calculations, which may not justify the goal of solving the problem. In this case, you can use the simplest way to reduce the fraction for a common denominator, which is a product of data denominators.