Problems of reducing fractions to a common denominator. Reducing fractions to the lowest common denominator, rule, examples, solutions

Reducing fractions to a common denominator

Fractions I have the same denominators. They say they have common denominator 25. Fractions have different denominators, but they can be reduced to a common denominator using the basic property of fractions. To do this, we will find a number that is divisible by 8 and 3, for example, 24. Let’s bring the fractions to the denominator 24, to do this we multiply the numerator and denominator of the fraction by additional multiplier 3. The additional factor is usually written to the left above the numerator:

Multiply the numerator and denominator of the fraction by an additional factor of 8:

Let's bring the fractions to a common denominator. Most often, fractions are reduced to a lowest common denominator, which is the smallest common multiple of the denominators of the given fractions. Since LCM (8, 12) = 24, then the fractions can be reduced to a denominator of 24. Let’s find additional factors of the fractions: 24:8 = 3, 24:12 = 2. Then

Several fractions can be reduced to a common denominator.

Example. Let's bring the fractions to a common denominator. Since 25 = 5 2, 10 = 2 5, 6 = 2 3, then LCM (25, 10, 6) = 2 3 5 2 = 150.

Let's find additional factors of fractions and bring them to the denominator 150:

Comparison of fractions

In Fig. Figure 4.7 shows a segment AB of length 1. It is divided into 7 equal parts. Segment AC has length , and segment AD has length .

The length of the segment AD is greater than the length of the segment AC, i.e. the fraction is greater than the fraction

Of two fractions with a common denominator, the one with the larger numerator is greater, i.e.

For example, or

To compare any two fractions, reduce them to a common denominator and then apply the rule for comparing fractions with a common denominator.

Example. Compare fractions

Solution. LCM (8, 14) = 56. Then Since 21 > 20, then

If the first fraction is less than the second, and the second is less than the third, then the first is less than the third.

Proof. Let three fractions be given. Let's bring them to a common denominator. Let them then look like Since the first fraction is smaller

second, then r< s. Так как вторая дробь меньше третьей, то s < t. Из полученных неравенств для натуральных чисел следует, что r < t, тогда первая дробь меньше третьей.

The fraction is called correct, if its numerator is less than its denominator.

The fraction is called wrong, if its numerator is greater than or equal to the denominator.

For example, fractions are proper and fractions are improper.

A proper fraction is less than 1, and an improper fraction is greater than or equal to 1.

The least common denominator (LCD) of these irreducible fractions is the least common multiple (LCM) of the denominators of these fractions. ( see the topic "Finding the least common multiple":

To reduce fractions to the least common denominator, you need to: 1) find the least common multiple of the denominators of the given fractions, it will be the least common denominator. 2) find an additional factor for each fraction by dividing 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 their 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. Find for the 1st fraction an additional factor 4 (20 : 5=4). For the 2nd fraction the additional factor 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 have reduced these fractions to the lowest common denominator ( 20 ).

The lowest common denominator of these fractions is the number 8, since 8 is divisible by 4 and itself. There will be no additional factor for the 1st fraction (or we can say that it is equal to one), for the 2nd fraction the additional factor is 2 (8 : 4=2). We multiply the numerator and denominator of the 2nd fraction by 2. We have reduced these fractions to the lowest common denominator ( 8 ).

These fractions are not irreducible.

Let's reduce the 1st fraction by 4, and reduce the 2nd fraction by 2. ( see examples on reducing ordinary fractions: Sitemap → 5.4.2. Examples of reducing common fractions). Find the LOC(16 ; 20)=2 4 · 5=16· 5=80. The additional multiplier for the 1st fraction is 5 (80 : 16=5). The additional factor 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 have reduced these fractions to the lowest common denominator ( 80 ).

We find the lowest common denominator NCD(5 ; 6 and 15)=NOK(5 ; 6 and 15)=30. The additional factor to the 1st fraction is 6 (30 : 5=6), the additional factor to the 2nd fraction is 5 (30 : 6=5), the additional factor 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 have reduced these fractions to the lowest common denominator ( 30 ).

This article explains how to reduce fractions to a common denominator and how to find the lowest common denominator. Definitions are given, the 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 be reduced to a common denominator. For example, the 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 using simple steps. To do this, you need to multiply the numerator and denominator by certain additional factors.

It is obvious that the fractions 4 5 and 3 4 are not reduced to a common denominator. To do this, you need to use additional factors of 5 and 4 to bring them to the 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 the fractions 4 5 and 3 4, we get 16 20 and 15 20, respectively.

Reducing fractions to a common denominator

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

Common denominator: definition, examples

What is the common denominator?

Common denominator

The common denominator of a fraction is any positive number that is a common multiple of all given fractions.

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

The series 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 of all the 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 the numbers 30, 60, 90, 120, 150, 180, 210, and so on.

Let's look at an example.

Example 1. Common denominator

Can the fractions 1 3, 21 6, 5 12 be brought to a common denominator, which is 150?

To find out if this is the case, you need to check whether 150 is a common multiple of the denominators of 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 the common denominator of these fractions.

Lowest common denominator

The smallest natural number among the many common denominators of a set of fractions is called the least common denominator.

Lowest common denominator

The least common denominator of a fraction 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 the fractions. Let's look at an example:

Example 2: Find the lowest common denominator

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

We are looking for the LCM of the numbers 10 and 28. Let's factor them into simple factors and get:

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

How to reduce fractions to 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 lowest common denominator of fractions.
Find an additional factor for each fraction. To find the factor, divide the lowest common denominator by the denominator of each fraction.
Multiply the numerator and denominator by the additional factor found.

Let's consider the application of this rule using a specific example.

Example 3: Reducing fractions to a common denominator

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

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

14 = 2 7 18 = 2 3 3 N O K (14, 18) = 2 3 3 7 = 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 the fractions by additional factors and get:

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

Reducing multiple fractions to their lowest common denominator

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

Let's give another example.

Example 4: Reducing fractions to a common denominator

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

Let's calculate the LCM of the denominators. Find the LCM of three or more numbers:

NOK (2, 6) = 6 NOK (6, 8) = 24 NOK (24, 18) = 72 NOK (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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In this lesson we will look at reducing fractions to a common denominator and solve problems on this topic. Let's define the concept of a common denominator and an additional factor, and remember about relatively prime numbers. Let's define the concept of the lowest 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. The main property of a fraction.

If the numerator and denominator of a fraction are multiplied or divided by the same natural number, you get an equal fraction.

For example, the numerator and denominator of a fraction can be divided by 2. We get the fraction. This operation is called fraction reduction. You can also perform the inverse 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. To bring a fraction to a new denominator, its numerator and denominator are multiplied by an additional factor.

1. Reduce the fraction to the denominator 35.

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

2. Reduce the fraction to denominator 18.

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

3. Reduce the fraction to a denominator of 60.

Dividing 60 by 15 gives an additional factor. It is equal to 4. Multiply the numerator and denominator by 4.

4. Reduce the fraction to the denominator 24

In simple cases, reduction to a new denominator is performed mentally. It is only customary to indicate the additional factor behind a bracket slightly 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 also 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 their lowest common denominator. It is equal to the least common multiple of the denominators of the given fractions.

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

First, let's 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, divide 12 by 4 and 6. Three is an additional factor for the first fraction, and two is for the second. Let's bring the fractions to the denominator 12.

We brought the fractions to a common denominator, that is, we found equal fractions that have the same denominator.

Rule. To reduce fractions to their lowest common denominator, you must

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

Secondly, divide the lowest common denominator by the denominators of these fractions, i.e. find an additional factor for each fraction.

Third, 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 reduce 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 gives 5 and 3, respectively. We reduce the fractions to the denominator 45.

c) Reduce the fractions and to a common denominator.

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

Sometimes it can be difficult to verbally find the least common multiple of the denominators of given fractions. Then the common denominator and additional factors are found using prime factorization.

Reduce the fractions and to a common denominator.

Let's factor 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. Let's 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 bring the fractions to a common denominator of 840.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M.: Mnemosyne, 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., Tchaikovsky I.V. Assignments for the mathematics course for grades 5-6. - ZSh MEPhI, 2011.

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

6. Shevrin L.N., Gein A.G., Koryakov I.O. and others. Mathematics: Textbook-interlocutor for 5-6 grades of secondary school. Math teacher's library. - Enlightenment, 1989.

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

Homework

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

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

Other tasks: No. 270, No. 290