To bring fractions to the least common denominator, you must: 1) find the least common multiple of the denominators of these fractions, it will be the least common denominator. 2) find an additional factor for each of the fractions, for which we divide the new denominator by the denominator of each fraction. 3) multiply the numerator and denominator of each fraction by its additional factor.
Examples. Reduce the following fractions to the lowest common denominator.
We find the least common multiple of the denominators: LCM(5; 4) = 20, since 20 is the smallest number that is divisible by both 5 and 4. We find for the 1st fraction an additional factor 4 (20 : 5=4). For the 2nd fraction, the additional multiplier is 5 (20 : 4=5). We multiply the numerator and denominator of the 1st fraction by 4, and the numerator and denominator of the 2nd fraction by 5. We reduced these fractions to the lowest common denominator ( 20 ).
The lowest common denominator of these fractions is 8, since 8 is divisible by 4 and itself. There will be no additional multiplier to the 1st fraction (or we can say that it is equal to one), to the 2nd fraction the additional multiplier is 2 (8 : 4=2). We multiply the numerator and denominator of the 2nd fraction by 2. We reduced these fractions to the lowest common denominator ( 8 ).
These fractions are not irreducible.
We reduce the 1st fraction by 4, and we reduce the 2nd fraction by 2. ( see examples on the reduction of ordinary fractions: Sitemap → 5.4.2. Examples of reduction of ordinary fractions). Find LCM(16 ; 20)=2 4 · 5=16· 5=80. The additional multiplier for the 1st fraction is 5 (80 : 16=5). The additional multiplier for the 2nd fraction is 4 (80 : 20=4). We multiply the numerator and denominator of the 1st fraction by 5, and the numerator and denominator of the 2nd fraction by 4. We reduced these fractions to the lowest common denominator ( 80 ).
Find the least common denominator of the NOC(5 ; 6 and 15) = LCM(5 ; 6 and 15)=30. The additional multiplier to the 1st fraction is 6 (30 : 5=6), the additional multiplier to the 2nd fraction is 5 (30 : 6=5), the additional multiplier to the 3rd fraction is 2 (30 : 15=2). We multiply the numerator and denominator of the 1st fraction by 6, the numerator and denominator of the 2nd fraction by 5, the numerator and denominator of the 3rd fraction by 2. We reduced these fractions to the lowest common denominator ( 30 ).
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Scheme of reduction to a common denominator
- It is necessary to determine what will be the least common multiple for the denominators of fractions. If you are dealing with a mixed or integer number, then you must first turn it into a fraction, and only then determine the least common multiple. To turn an integer into a fraction, you need to write the number itself in the numerator, and one in the denominator. For example, the number 5 as a fraction would look like this: 5/1. To turn a mixed number into a fraction, you need to multiply the whole number by the denominator and add the numerator to it. Example: 8 integers and 3/5 as a fraction = 8x5+3/5 = 43/5.
- After that, it is necessary to find an additional factor, which is determined by dividing the NOZ by the denominator of each fraction.
- The last step is to multiply the fraction by an additional factor.
It is important to remember that reduction to a common denominator is needed not only for addition or subtraction. To compare several fractions with different denominators, it is also necessary to first reduce each of them to a common denominator.
Bringing fractions to a common denominator
In order to understand how to reduce a fraction to a common denominator, it is necessary to understand some properties of fractions. So, an important property used to reduce to NOZ is the equality of fractions. In other words, if the numerator and denominator of a fraction are multiplied by a number, then the result is a fraction equal to the previous one. Let's take the following example as an example. In order to reduce the fractions 5/9 and 5/6 to the lowest common denominator, you need to do the following:
- First, find the least common multiple of the denominators. In this case, for the numbers 9 and 6, the NOC will be 18.
- We determine additional factors for each of the fractions. This is done in the following way. We divide the LCM by the denominator of each of the fractions, as a result we get 18: 9 \u003d 2, and 18: 6 \u003d 3. These numbers will be additional factors.
- We bring two fractions to NOZ. When multiplying a fraction by a number, you need to multiply both the numerator and the denominator. The fraction 5/9 can be multiplied by an additional factor of 2, resulting in a fraction equal to the given one - 10/18. We do the same with the second fraction: multiply 5/6 by 3, resulting in 15/18.
As you can see from the example above, both fractions have been reduced to the lowest common denominator. To finally understand how to find a common denominator, you need to master one more property of fractions. It lies in the fact that the numerator and denominator of a fraction can be reduced by the same number, which is called the common divisor. For example, the fraction 12/30 can be reduced to 2/5 if it is divided by a common divisor - the number 6.
In this lesson, we will look at reducing fractions to a common denominator and solve problems on this topic. Let's give a definition of the concept of a common denominator and an additional factor, remember about coprime 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. You can also perform the reverse transformation by 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 factoring 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. and others. Mathematics: A textbook-interlocutor for grades 5-6 of 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 the fraction 4 5 by 4, and multiply the numerator and denominator of the fraction 3 4 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 a fraction is any positive number that is a common multiple of all the 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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