This article describes how to bring the fraraty to 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 common denominator of some kind of shot will be natural numberwhich without a balance is divided into all the denominators of these fractions.

A number of natural numbers are infinite, and therefore, according to definition, each set ordinary fractions It has an infinite set of common denominators. 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. Find the 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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The material of this article explains how to find the smallest common denominator and how to bring a fraction to a 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 are disassembled.

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, a common denominator of a certain set of ordinary fractions is any natural number that is divided into all denominators 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 fractions.
• 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.

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.

The smallest common denominator (nos) of these non-interconnected fractions is the smallest common multiple (NOC) denominators of these fractions. ( see the topic "Finding the smallest total multiple":

To bring the fraction for the smallest common denominator, it is necessary: \u200b\u200b1) to find the smallest common multiple denominators of these fractions, it will be the smallest common denominator. 2) Find an additional factor for each fraction for which to share a new denominator to the denominator of each fraction. 3) multiply the numerator and denominator of each fraction on its additional factor.

Examples. Create the following fractions to the smallest general denominator.

We find the smallest general multiple denominators: NOC (5; 4) \u003d 20, since 20 is the smaller one that is divided into 5 and by 4. Find for the 1st fraction. Additional multiplier 4 (20 : 5 \u003d 4). For the 2nd fraction, the additional factor is 5 (20 : 4 \u003d 5). Multiply the numerator and denominator of the 1st fraction on 4, and the numerator and denominator of the 2nd fraction on 5. We led these fractions to the smallest general denominator ( 20 ).

The smallest general denominator of these fractions is the number 8, since 8 is divided into 4 and itself. An additional multiplier to the 1st fraction will not (or it can be said that it is equal to one), to the 2nd fraction Additional factor is 2 (8 : 4 \u003d 2). We multiply the numerator and denominator of the 2nd fraction on 2. We led these fractions to the smallest general denominator ( 8 ).

These fractions are not disracitable.

Sperate the 1st fraction on 4, and the 2nd fraction will reduce on 2. ( see examples of reducing ordinary fractions: Sitemap → 5.4.2. Examples of reducing ordinary fractions). Find Nok (16 ; 20)=2 4 · 5=16· 5 \u003d 80. An additional factor for the 1st fraction is 5 (80 : 16 \u003d 5). An additional factor for the 2nd fraction is 4 (80 : 20 \u003d 4). Multiply the numerator and denominator of the 1st fraction on 5, and the numerator and denominator of the 2nd fraction on 4. We led these fractions to the smallest general denominator ( 80 ).

We find the smallest common denominator nose (5 ; 6 and 15) \u003d NOC (5 ; 6 and 15) \u003d 30. An additional factor to the 1st fraction is 6 (30 : 5 \u003d 6), an additional factor of the 2nd fraction is 5 (30 : 6 \u003d 5), an additional factor to the 3rd fraction is 2 (30 : 15 \u003d 2). We multiply the numerator and denominator of the 1st fraction on 6, the numerator and denominator of the 2nd fraction on 5, the numerator and denominator of the 3rd fraction on 2. We led these fractions to the smallest general denominator ( 30 ).