The result of reduction to a common denominator. Bringing fractions to a common denominator

This article explains, how to find the lowest common denominator and how to convert fractions common denominator . First, the definitions of the common denominator of fractions and the least common denominator are given, and it is also shown how to find the common denominator of fractions. The following is a rule for reducing fractions to a common denominator and examples of the application of this rule are considered. In conclusion, examples of reduction of three and more fractions to a common denominator.

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What is called reducing fractions to a common denominator?

Now we can say what it is to bring fractions to a common denominator. Bringing fractions to a common denominator is the multiplication of the numerators and denominators of given fractions by such additional factors that the result is fractions with the same denominators.

Common denominator, definition, examples

Now it's time to define the common denominator of fractions.

In other words, the common denominator of some set ordinary fractions is any natural number, which is divisible by all the denominators of the given fractions.

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

Determining the common denominator of fractions allows you to find the common denominators of given fractions. Let, for example, given fractions 1/4 and 5/6, their denominators are 4 and 6, respectively. The positive common multiples of 4 and 6 are the numbers 12, 24, 36, 48, ... Any of these numbers is the common denominator of the fractions 1/4 and 5/6.

To consolidate the material, consider the solution of the following example.

Example.

Is it possible to reduce the fractions 2/3, 23/6 and 7/12 to a common denominator of 150?

Solution.

To answer this question, we need to find out if the number 150 is a common multiple of the denominators 3, 6 and 12. To do this, check if 150 is evenly divisible by each of these numbers (if necessary, see the rules and examples of division of natural numbers, as well as the rules and examples of division of natural numbers with a remainder): 150:3=50 , 150:6=25 , 150: 12=12 (rest. 6) .

So, 150 is not divisible by 12, so 150 is not a common multiple of 3, 6, and 12. Therefore, the number 150 cannot be a common denominator of the original fractions.

Answer:

It is forbidden.

The lowest common denominator, how to find it?

In the set of numbers that are common denominators of these fractions, there is the smallest natural number, which is called the least common denominator. Let us formulate the definition of the least common denominator of these fractions.

Definition.

Lowest common denominator- this is smallest number, of all the common denominators of the given fractions.

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

Since is the least positive common divisor this set numbers, then the LCM of the denominators of these fractions is the least common denominator of these fractions.

Thus, finding the least common denominator of fractions is reduced to the denominators of these fractions. Let's take a look at an example solution.

Example.

Find the least common denominator of 3/10 and 277/28.

Solution.

The denominators of these fractions are 10 and 28. The desired least common denominator is found as the LCM of the numbers 10 and 28. In our case, it's easy: since 10=2 5 and 28=2 2 7 , then LCM(15, 28)=2 2 5 7=140 .

Answer:

140 .

How to bring fractions to a common denominator? Rule, examples, solutions

Common fractions usually lead to the lowest common denominator. Now we will write down a rule that explains how to reduce fractions to the lowest common denominator.

The rule for reducing fractions to the lowest common denominator consists of three steps:

First, find the least common denominator of the fractions.
Second, for each fraction, an additional factor is calculated, for which the lowest common denominator is divided by the denominator of each fraction.
Thirdly, the numerator and denominator of each fraction is multiplied by its additional factor.

Let's apply the stated rule to the solution of the following example.

Example.

Reduce the fractions 5/14 and 7/18 to the lowest common denominator.

Solution.

Let's perform all the steps of the algorithm for reducing fractions to the smallest common denominator.

First, we find the least common denominator, which is equal to the least common multiple of the numbers 14 and 18. Since 14=2 7 and 18=2 3 3 , then LCM(14, 18)=2 3 3 7=126 .

Now we calculate additional factors with the help of which the fractions 5/14 and 7/18 will be reduced to the denominator 126. For the fraction 5/14 the additional factor is 126:14=9 , and for the fraction 7/18 the additional factor is 126:18=7 .

It remains to multiply the numerators and denominators of the fractions 5/14 and 7/18 by additional factors of 9 and 7, respectively. We have and .

So, reduction of fractions 5/14 and 7/18 to the smallest common denominator is completed. The result was fractions 45/126 and 49/126.

In this lesson, we will look at reducing fractions to a common denominator and solve problems on this topic. Let us define the concept of a common denominator and an additional factor, recall the mutual prime numbers. Let's define the concept of the least common denominator (LCD) and solve a number of problems to find it.

Topic: Adding and subtracting fractions with different denominators

Lesson: Reducing fractions to a common denominator

Repetition. Basic property of a fraction.

If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then a fraction equal to it will be obtained.

For example, the numerator and denominator of a fraction can be divided by 2. We get a fraction. This operation is called fraction reduction. Can be done and inverse transformation, multiplying the numerator and denominator of the fraction by 2. In this case, we say that we have reduced the fraction to a new denominator. The number 2 is called an additional factor.

Conclusion. A fraction can be reduced to any denominator that is a multiple of the denominator of the given fraction. In order to bring a fraction to a new denominator, its numerator and denominator are multiplied by an additional factor.

1. Bring the fraction to the denominator 35.

The number 35 is a multiple of 7, that is, 35 is divisible by 7 without a remainder. So this transformation is possible. Let's find an additional factor. To do this, we divide 35 by 7. We get 5. We multiply the numerator and denominator of the original fraction by 5.

2. Bring the fraction to the denominator 18.

Let's find an additional factor. To do this, we divide the new denominator by the original one. We get 3. We multiply the numerator and denominator of this fraction by 3.

3. Bring the fraction to the denominator 60.

By dividing 60 by 15, we get an additional multiplier. It is equal to 4. Let's multiply the numerator and denominator by 4.

4. Bring the fraction to the denominator 24

In simple cases, reduction to a new denominator is performed in the mind. It is customary to only indicate an additional factor behind the bracket a little to the right and above the original fraction.

A fraction can be reduced to a denominator of 15 and a fraction can be reduced to a denominator of 15. Fractions have a common denominator of 15.

The common denominator of fractions can be any common multiple of their denominators. For simplicity, fractions are reduced to the lowest common denominator. It is equal to the least common multiple of the denominators of the given fractions.

Example. Reduce to the least common denominator of the fraction and .

First, find the least common multiple of the denominators of these fractions. This number is 12. Let's find an additional factor for the first and second fractions. To do this, we divide 12 by 4 and by 6. Three is an additional factor for the first fraction, and two for the second. We bring the fractions to the denominator 12.

We reduced the fractions to a common denominator, that is, we found fractions that are equal to them and have the same denominator.

Rule. To bring fractions to the lowest common denominator,

First, find the least common multiple of the denominators of these fractions, which will be their least common denominator;

Secondly, divide the least common denominator by the denominators of these fractions, that is, find an additional factor for each fraction.

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

a) Reduce the fractions and to a common denominator.

The lowest common denominator is 12. The additional factor for the first fraction is 4, for the second - 3. We bring the fractions to the denominator 24.

b) Reduce the fractions and to a common denominator.

The lowest common denominator is 45. Dividing 45 by 9 by 15, we get 5 and 3, respectively. We bring the fractions to the denominator 45.

c) Reduce the fractions and to a common denominator.

The common denominator is 24. The additional factors are 2 and 3, respectively.

Sometimes it is difficult to verbally find the least common multiple for the denominators of given fractions. Then the common denominator and additional factors are found by expanding into prime factors.

Reduce to a common denominator of the fraction and .

Let's decompose the numbers 60 and 168 into prime factors. Let's write out the expansion of the number 60 and add the missing factors 2 and 7 from the second expansion. Multiply 60 by 14 and get a common denominator of 840. The additional factor for the first fraction is 14. The additional factor for the second fraction is 5. Let's reduce the fractions to a common denominator of 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 6th grade. - Gymnasium, 2006.

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

4. Rurukin A.N., Chaikovsky I.V. Tasks for the course of mathematics grade 5-6. - ZSH MEPhI, 2011.

5. Rurukin A.N., Sochilov S.V., Chaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - ZSH MEPhI, 2011.

6. Shevrin L.N., Gein A.G., Koryakov I.O. etc. Mathematics: Interlocutor textbook for grades 5-6 high school. Library of the teacher of mathematics. - Enlightenment, 1989.

You can download the books specified in clause 1.2. this lesson.

Homework

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

Homework: No. 297, No. 298, No. 300.

Other tasks: #270, #290

This article explains how to reduce fractions to a common denominator and how to find the smallest common denominator. Definitions are given, a rule for reducing fractions to a common denominator is given, and practical examples are considered.

What is reducing a fraction to a common denominator?

Ordinary fractions consist of a numerator - the upper part, and a denominator - the lower part. If fractions have the same denominator, they are said to have a common denominator. For example, fractions 11 14 , 17 14 , 9 14 have the same denominator 14 . In other words, they are reduced to a common denominator.

If fractions have different denominators, then they can always be reduced to a common denominator with the help of simple actions. To do this, you need to multiply the numerator and denominator by certain additional factors.

Obviously, the fractions 4 5 and 3 4 are not reduced to a common denominator. To do this, you need to use additional factors 5 and 4 to bring them to a denominator of 20. How exactly to do this? Multiply the numerator and denominator of 45 by 4, and multiply the numerator and denominator of 34 by 5. Instead of fractions 4 5 and 3 4 we get 16 20 and 15 20 respectively.

Bringing fractions to a common denominator

Reducing fractions to a common denominator is the multiplication of the numerators and denominators of fractions by factors such that the result is identical fractions with the same denominator.

Common denominator: definition, examples

What is a common denominator?

Common denominator

The common denominator of fractions is any positive number, which is a common multiple of all given fractions.

In other words, the common denominator of some set of fractions will be such a natural number that is divisible without a remainder by all the denominators of these fractions.

The set of natural numbers is infinite, and therefore, by definition, every set of common fractions has an infinite number of common denominators. In other words, there are infinitely many common multiples for all denominators of the original set of fractions.

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

Consider an example.

Example 1. Common denominator

Can di fractions 1 3, 21 6, 5 12 be reduced to a common denominator, which is equal to 150?

To find out if this is the case, you need to check if 150 is a common multiple of the denominators of the fractions, that is, for the numbers 3, 6, 12. In other words, the number 150 must be divisible by 3, 6, 12 without a remainder. Let's check:

150 ÷ 3 = 50 , 150 ÷ 6 = 25 , 150 ÷ 12 = 12 , 5

This means that 150 is not a common denominator of the indicated fractions.

Lowest common denominator

The smallest natural number from the set of common denominators of some set of fractions is called the least common denominator.

Lowest common denominator

The least common denominator of fractions is the smallest number among all the common denominators of those fractions.

The least common divisor of a given set of numbers is the least common multiple (LCM). The LCM of all denominators of fractions is the least common denominator of those fractions.

How to find the lowest common denominator? Finding it comes down to finding the least common multiple of fractions. Let's look at an example:

Example 2: Find the lowest common denominator

We need to find the smallest common denominator for the fractions 1 10 and 127 28 .

We are looking for the LCM of numbers 10 and 28. We decompose them into simple factors and get:

10 \u003d 2 5 28 \u003d 2 2 7 N O K (15, 28) \u003d 2 2 5 7 \u003d 140

How to bring fractions to the lowest common denominator

There is a rule that explains how to reduce fractions to a common denominator. The rule consists of three points.

The rule for reducing fractions to a common denominator

Find the smallest common denominator of fractions.
For each fraction, find an additional factor. To find the multiplier, you need to divide the least common denominator by the denominator of each fraction.
Multiply the numerator and denominator by the found additional factor.

Consider the application of this rule on a specific example.

Example 3. Reducing fractions to a common denominator

There are fractions 3 14 and 5 18. Let's bring them to the lowest common denominator.

As a rule, we first find the LCM of the denominators of the fractions.

14 \u003d 2 7 18 \u003d 2 3 3 N O K (14, 18) \u003d 2 3 3 7 \u003d 126

We calculate additional factors for each fraction. For 3 14 the additional factor is 126 ÷ 14 = 9 , and for the fraction 5 18 the additional factor is 126 ÷ 18 = 7 .

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

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

Bringing Multiple Fractions to the Least Common Denominator

According to the considered rule, not only pairs of fractions, but also more of them can be reduced to a common denominator.

Let's take another example.

Example 4. Reducing fractions to a common denominator

Bring the fractions 3 2 , 5 6 , 3 8 and 17 18 to the lowest common denominator.

Calculate the LCM of the denominators. Find the LCM of three or more numbers:

N O C (2, 6) = 6 N O C (6, 8) = 24 N O C (24, 18) = 72 N O C (2, 6, 8, 18) = 72

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

We multiply the fractions by additional factors and go to the lowest common denominator:

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

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