The material of this article explains how to find the smallest common denominator and how to bring the fraraty to common denominator . First, the definitions of the overall denominator fractions and the smallest common denominator are given, and also shown how to find a common denominator. The following is a rule of defending to a common denominator and addressed examples of applying this rule. In conclusion, examples of bringing three and more fractions to the general denominator.

Navigating page.

What is called bringing fractions to a common denominator?

Now we can say that such a fraction to a common denominator. Bringing fractions to a common denominator - This is multiplying the numerals and denominators of these fractions on such additional factors, which result is a fraction with the same denominatives.

General denominator, definition, examples

Now it's time to give the definition of a common denominator fraction.

In other words, the general denominator of some set ordinary fractions is any natural numberwhich is divided into all denominers of these fractions.

From the voiced definition it follows that this set of fractions has infinitely many common denominators, since there is an infinite set of common multiple of all denominators of the original set of fractions.

The definition of the total denominator fraction allows you to find common denominators of these fractions. Let, for example, are given fractions 1/4 and 5/6, their denominators are equal to 4 and 6, respectively. The positive common multiple numbers 4 and 6 are numbers 12, 24, 36, 48, ... any of these numbers is a common denominator of 1/4 and 5/6 fractions.

To secure the material, consider the decision of the next example.

Example.

Is it possible to lead 5/3, 23/6 and 7/12 to the total denominator 150?

Decision.

For an answer to the question, we need to find out whether the number 150 is a total multiple denominator 3, 6 and 12. To do this, check whether 150 is aimed at each of these numbers (if necessary, see the rules and examples of dividing natural numbers, as well as rules and examples of dividing natural numbers with the residue): 150: 3 \u003d 50, 150: 6 \u003d 25, 150: 12 \u003d 12 (OST. 6).

So, 150 is not divisible to 12, therefore, 150 is not a common multiple numbers 3, 6 and 12. Consequently, the number 150 cannot be a common denominator of the initial fractions.

Answer:

It is impossible.

The smallest common denominator, how to find it?

In a set of numbers that are common denominators of these fractions, there is a smallest natural number, which is called the smallest common denominator. We formulate the definition of the smallest overall denominator of these fractions.

Definition.

The smallest common denominator - This is the smallest number, of all common denominators of these fractions.

It remains to deal with the question of how to find the smallest common divider.

Since it is the smallest positive common divider of this set of numbers, the NOC of the data denominators of the frains is the smallest common denominator of these fractions.

Thus, finding the smallest common denominator fractions is reduced to the denominators of these fractions. We will analyze the solution of the example.

Example.

Find the smallest overall denominator of the fractions 3/10 and 277/28.

Decision.

Data denominants of fractions are equal to 10 and 28. The desired smallest overall denominator is like NOC numbers 10 and 28. In our case, it is easy: since 10 \u003d 2 · 5, a 28 \u003d 2 · 2 · 7, then NOK (15, 28) \u003d 2 · 2 · 5 · 7 \u003d 140.

Answer:

140 .

How to bring a fraction for a common denominator? Rule examples solutions

Usually ordinary fractions lead to the smallest common denominator. Now we will write up the rule that explains how to bring the fraction for the smallest general denominator.

Rule of bringing fractions to the smallest general denominator Consists of three steps:

• First, there is a smallest common denominator fraction.
• Secondly, for each fraction, an additional factor is calculated, for which the smallest common denominator is divided into the denominator of each fraction.
• Thirdly, the numerator and the denominator of each fraction is multiplied by its additional factor.

Apply the rule of the rule to solve the following example.

Example.

Put fractions 5/14 and 7/18 to the smallest general denominator.

Decision.

Perform all the steps of the algorithm to bring the fractions to the smallest general denominator.

At first we find the smallest common denominator, which is equal to the smallest general multiple numbers 14 and 18. Since 14 \u003d 2 · 7 and 18 \u003d 2 · 3 · 3, then NOC (14, 18) \u003d 2 · 3 · 3 · 7 \u003d 126.

Now we calculate additional multipliers with which the fractions 5/14 and 7/18 will be shown to the denominator 126. For the fraction 5/14, the additional factor is 126: 14 \u003d 9, and for the fraction 7/18, the additional factor is 126: 18 \u003d 7.

It remains to multiply the numerals and denominators of the fractions 5/14 and 7/18 on additional faults 9 and 7, respectively. We have I. .

So, bringing fractions 5/14 and 7/18 to the smallest general denominator completed. As a result, it turned out the fractions 45/126 and 49/126.

Schema of bringing to a common denominator

1. It is necessary to determine which will be the smallest common multiple for denominators of fractions. If you are dealing with a mixed or integer, then you must first turn into a fraction, and only then define the smallest common multiple. To turn an integer into a fraction, you need to write this number in the numerator itself, and in the denominator - one. For example, the number 5 in the form of a fraction will look like this: 5/1. To turn a mixed number to fraction, you need to multiply an integer to the denominator and add a numerator to it. Example: 8 whole and 3/5 in the form of fractions \u003d 8x5 + 3/5 \u003d 43/5.
2. After that, it is necessary to find an additional factor, which is determined by the division of the nose on the denominator of each fraction.
3. The last step is the multiplication of the fraction on an additional factor.

It is important to remember that bringing to the general denominator is needed not only for addition or subtraction. To compare several fractions with different denominators, it is also necessary to first lead each of them to the general denominator.

Bringing fractions to a common denominator

In order to understand how to bring a fraction to the general denominator, it is necessary to understand some of the properties of fractions. So, an important propertyUsed to bring to nose, is the equality of fractions. In other words, if the numerator and denominator of the fraction is multiplied by the number, then the result is a fraction equal to the previous one. As an example, we give the following example. In order to bring the fraction 5/9 and 5/6 to the smallest common denominator, you need to perform the following actions:

1. First we find the smallest general multiple denominator. In this case, for numbers 9 and 6 NOCs will be equal to 18.
2. We define additional faults for each of the fractions. This is done as follows. We divide the NOC to the denominator of each fraction, as a result we obtain 18: 9 \u003d 2, and 18: 6 \u003d 3. These numbers will be additional multipliers.
3. We give two fractions to nos. Multiplying the fraction to the number, you need to multiply and the numerator, and the denominator. Fraction 5/9 can be multiplied by an additional factor 2, resulting in a fraction equal to this, 10/18. The same makes the same with the second fraction: 5/6 I multiply at 3, as a result of which we get 15/18.

As we see from the example above, both fractions were shown to the smallest common denominator. To finally sort out how to find a common denominator, it is necessary to master another defendance property. It lies in the fact that the numerator and denominator of the fraction can be reduced by the same number called a common divider. For example, a 12/30 fraction can be reduced to 2/5 if it is divided into a common divider - the number 6.

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 I. reliable waywhich is guaranteed to level the denominators. 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 strength of the method common divisorsBut, 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 numberwhich 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.

In this lesson, we will consider bringing fractions to a common denominator and solve the task on this topic. Let us define the concept of a common denominator and an additional factor, remember the mutual simple numbers. We give the definition of the concept of the smallest common denominator (Nos) and solve a number of tasks for its finding.

Topic: Addition and subtraction of fractions with different denominators

Lesson: bringing fractions to a common denominator

Reiteration. The main property of the fraction.

If the numerator and denominator of the fraction are multiplied or divided into one and the same natural number, then the fraction equal to it.

For example, the numerator and denominator of the fraction can be divided into 2. We will get a fraction. This operation is called the cutting of the fraction. You can also perform the reverse transformation, multiplying the numerator and denominator of the fraction on 2. In this case, it is said that we have led to a new denominator. The number 2 is called an additional factor.

Output.The fraction can be brought to any denominator to a multiple denominator of this fraction. In order to lead to a new denominator, its numerator and denominator multiplies to an additional factor.

1. Give a fraction to the denominator 35.

The number is 35 times 7, that is, 35 is divided into 7 without a residue. So this conversion is possible. Find an additional factor. To do this, we divide 35 to 7. We obtain 5. Multiply on 5 numer and denominator of the original fraction.

2. Give a fraction to the denominator 18.

Find an additional factor. To do this, we divide the new denominator to the original. We obtain 3. Multiply by 3 numerator and denominator of this fraction.

3. Give a fraction to the denominator 60.

Dividing 60 to 15, we obtain an additional factor. It is equal to 4. Multiply the numerator and denominator on 4.

4. Give a fraction to the denominator 24

In simple cases, bringing to a new denominator is performed in the mind. It is only applied to specify an additional factor behind a bracket of a little right and above the original fraction.

The fraction can be brought to the denominator 15 and the fraction can be brought to the denominator 15. The fractions and the overall denominator 15.

A common denominator can be any common multiple of their denominator. For simplicity, fractions lead to the smallest common denominator. It is equal to the smallest total multiple denominator denominations.

Example. Lead to the smallest overall denominator the fractions and.

We will find the smallest common multiple denominator denominators. This is a number 12. We find an additional factor for the first and for the second fraction. For this, 12 divide by 4 and by 6. Three are an additional factor for the first fraction, and two for the second. We give the fractions to the denominator 12.

We led a fraction and to a common denominator, that is, we found the fractions equal to them, who have the same denominator.

Rule. To bring a fraction for the smallest general denominator, it is necessary

First, find the smallest general multiple denominator of these fractions, it will be their smallest common denominator;

Secondly, divide the smallest common denominator to the data denominators of the fractions, i.e., to find for each fraction an additional multiplier.

Thirdly, multiply the numerator and the denominator of each fraction on its additional factor.

a) lead to a common denomoter and.

The smallest overall denominator is 12. An additional factor for the first fraction is 4, for the second - 3. Give the fraction to the denominator 24.

b) lead to a common denomoter and.

The smallest overall denominator is 45. Selecting 45 to 9 to 15, we obtain, respectively, 5 and 3. give the fractions to the denominator 45.

c) lead to a common denomoter and.

Common denominator - 24. Additional multipliers, respectively, - 2 and 3.

Sometimes it is difficult to choose orally the smallest total multiple for denominators of these fractions. Then the general denominator and additional multipliers are found using decomposition into simple multipliers.

Lead to a general denomoter and.

Spreads the numbers 60 and 168 to simple multipliers. We repel the decomposition of the number 60 and add missing multipliers 2 and 7 from the second decomposition. Multiply 60 by 14 and we obtain a common denominator 840. An additional factor for the first fraction is 14. An additional factor for the second fraction - 5. We give the fractions to the total denominator 840.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M.: Mnemozina, 2012.

2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics Grade 6. - Gymnasium, 2006.

3. Depima I.Ya., Vilenkin N.Ya. Behind the pages of the textbook of mathematics. - Enlightenment, 1989.

4. Rurukin A.N., Tchaikovsky I.V. Tasks at the rate of mathematics 5-6 class. - Zh MEPI, 2011.

5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. Manual for students of the 6th grade of the correspondence school of MEPI. - Zh MEPI, 2011.

6. Chevrine L.N., Gain A.G., Koryakov I.O. and others. Mathematics: Tutorial - Interlocutor for 5-6 classes high School. Library of Mathematics Teacher. - Enlightenment, 1989.

You can download the books specified in paragraph 1.2. This lesson.

Homework

Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M.: Mnemozina, 2012. (Reference See 1.2)

Homework: №297, №298, №300.

Other tasks: №270, №290