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 bringing three or more fractions to a common denominator are analyzed.
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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.
It follows from the stated definition that this set of fractions has infinitely many common denominators, since there are an infinite number of 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 is the smallest number of all the common denominators of these fractions.
It remains to deal with the question of how to find the least common divisor.
Since is the least positive common divisor of a given set of numbers, 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.
Lesson topic: Reducing fractions to a common denominatorGoals:
educational: to form the ability to bring fractions to the lowest common denominator and find an additional factor in more complex cases; to form the ability to translate ordinary fractions into decimals;
developing: develop logical thinking, memory,students' computing skills
Educational: educate cognitive interest to the subject
During the classes
II. Verbal counting
1. Find the greatest common divisor and least common multiple of the numbers: 10 and 12; 12 and 8; 15 and 9; 6 and 4; 6 and 8; 12 and 15; 12 and 10; 16 and 20; 11 and 7.
2. Two tourists left the same point at the same time in different directions. The speed of the first tourist is 6 km/h, the speed of the second is 7 km/h. How far apart will they be in 3 hours?
3. The pump fills the pool in 48 minutes. What part of the pool will the pump fill in 1 minute?
4. There are five sons in the family, each of them has one sister. How many children are in the family? (6 children.)
III . Lesson topic message
- In the last lesson, we brought fractions to a new denominator. Today we will find a common denominator for several fractions and find out what is the smallest common denominator of fractions.
IV. Learning new material
1. Any 2 fractions can be reduced to the same denominator, or, in other words, to a common denominator.
- Find some common denominators of fractions. Name their lowest common denominator.
The common denominator of fractions can be any common multiple of their denominators. .
At the same time, as a rule, they try to choose the smallest common denominator (LCD) - then calculations with fractions turn out to be easier. The least common denominator is equal to the least common multiple of the denominators of the given fractions.
2. Let's look at examples of how to find NOZ of fractions.
1) Let's reduce the fractions 7/21 and 2/7 to a common denominator.
- What is special about the numbers 21 and 7? (21 is evenly divisible by 7.)
(Reasoning leads the teacher.)
- The larger denominator - the number 21 - is divisible by the smaller denominator 7, therefore, it can be taken as a common denominator of these fractions. This common denominator is the smallest possible.
This means that you only need to reduce the fraction 2/7 to the denominator 21. To do this, we will find an additional factor: 21: 7 = 3.
- What conclusion can be drawn? (If one denominator of a fraction is divisible by another, then NOZ will have a larger denominator.)
2) Let's reduce the fractions 3/4 and 2/5 to a common denominator.
- What can you say about the numbers 4 and 5? (Numbers are relatively prime.) The common denominator of these fractions must be divisible by both 4 and 5, i.e. be their common multiple. There are infinitely many common multiples of 4 and 5: 20, 40, 60, 80, etc. The smallest multiple of 20 is the product of 4 and 5.
So, you need to bring each of the fractions to the denominator 20:
- What conclusion can be drawn? (If the denominators of fractions are mutually prime numbers, then the least common denominator is their product.)
V. Physical education
VI. Working on a task
VII. Consolidation of the studied material
1. No. 279 p. 45 (oral). Work in pairs.
Someone from a couple answers the teacher.
- Why can't the fraction 3/5 be reduced to a denominator of 36? (36 is not a multiple of 5.)
2. No. 283 (а-е) p. 46 (with a detailed commentary at the blackboard and in notebooks, a) b) write down the decision in detail, then pronounce it all orally, write down only fractions with a new denominator).
Solution:
Additional multipliers: 24:6 = 4, 24:8 = 3.
Additional multipliers: 45:9 = 5, 45:15 = 3.
3. Name the numbers that:
a) more than 4/7, but less than 5/7; b) more than 1/6, but less than 2/6; c) more than 5/8, but less than 3/4.
- What needs to be done to complete the task? (Bring fractions to a new denominator.)
4. No. 281 p. 46 (c) (one student on the back of the board, the rest in notebooks, self-examination).
Solution:
VIII. Independent work
Option I
1. Bring fractions to a new denominator 24:
2. Bring the fraction 3/5 to a new denominator: 15; 25; 40; 55; 250; 300.
Option II
1. Bring fractions to a new denominator 48:
2. Bring the fraction 4/7 to a new denominator: 14; 28; 49; 70; 210; 350.
3. Express in hundredths of a fraction:
Option III (for more advanced students)
1. Bring fractions to a new denominator 84:
2. Bring the fraction 5/8 to a new denominator: 16; 24; 56; 80; 240; 3200.
3. Express in hundredths of a fraction:
IX. Consolidation of the studied material
1. No. 290 p. 47 (oral). Work in pairs.
- What was used in the solution? (The main property of a fraction.)
- Formulate the main property of a fraction.
(Answer: a) x = 3, b) x = 5, c) x = 5, d) x = 7.)
2. No. 289 (c, d) p. 47 (independently, mutual verification).
- What is the greatest common divisor of the numerator and denominator?
X. Summing up the lesson
- What number can be the common denominator of two fractions?
- How to bring fractions to the lowest common denominator?
- What is the rule for reducing fractions to a common denominator?
Homework:
Bringing fractions to a common denominator
The fractions I have the same denominator. They say they have common denominator 25. Fractions and have different denominators, but they can be reduced to a common denominator using the basic property of fractions. To do this, we find a number that is divisible by 8 and 3, for example, 24. We bring the fractions to the denominator 24, for this we multiply the numerator and denominator of the fraction by additional multiplier 3. An additional factor is usually written on the left above the numerator:
Multiply the numerator and denominator of the fraction by an additional factor of 8:
We bring the fractions to a common denominator. Most often, fractions lead to the least common denominator, which is the least common multiple of the denominators of the given fractions. Since LCM (8, 12) = 24, then the fractions can be reduced to the denominator 24. Let's find additional factors of fractions: 24:8 = 3, 24:12 = 2. Then
You can bring several fractions to a common denominator.
Example. We bring 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:
Fraction Comparison
On fig. 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 segment AD is greater than the length of segment AC, i.e., the fraction is greater than the fraction
Of the two fractions with a common denominator, the one with the larger numerator is larger, i.e.
For example, or
To compare any two fractions, they are reduced to a common denominator, and then the rule for comparing fractions with a common denominator is applied.
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 there be three fractions. Let's bring them to a common denominator. Let after that they will have the form Since the first fraction is less
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 its 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.
I originally wanted to include the common denominator methods in the "Adding and Subtracting Fractions" paragraph. But there was so much information, and its importance is so great (after all, not only numerical fractions have common denominators), that it is better to study this issue separately.
So let's say we have two fractions with different denominators. And we want to make sure that the denominators become the same. The main property of a fraction comes to the rescue, which, let me remind you, sounds like this:
A fraction does not change if its numerator and denominator are multiplied by the same non-zero number.
Thus, if you choose the factors correctly, the denominators of the fractions will be equal - this process is called reduction to a common denominator. And the desired numbers, "leveling" the denominators, are called additional factors.
Why do you need to bring fractions to a common denominator? Here are just a few reasons:
- Addition and subtraction of fractions with different denominators. There is no other way to perform this operation;
- Fraction comparison. Sometimes reduction to a common denominator greatly simplifies this task;
- Solving problems on shares and percentages. Percentages are, in fact, ordinary expressions that contain fractions.
There are many ways to find numbers that make the denominators equal when multiplied. We will consider only three of them - in order of increasing complexity and, in a sense, efficiency.
Multiplication "criss-cross"
The simplest and most reliable way, which is guaranteed to equalize the denominators. We will act "ahead": we multiply the first fraction by the denominator of the second fraction, and the second by the denominator of the first. As a result, the denominators of both fractions will become equal to the product of the original denominators. Take a look:
As additional factors, consider the denominators of neighboring fractions. We get:
Yes, it's that simple. If you are just starting to learn fractions, it is better to work with this method - this way you will insure yourself against many mistakes and are guaranteed to get the result.
The only drawback of this method is that you have to count a lot, because the denominators are multiplied "ahead", and as a result, very large numbers can be obtained. That's the price of reliability.
Common divisor method
This technique helps to greatly reduce the calculations, but, unfortunately, it is rarely used. The method is as follows:
- Look at the denominators before you go "thru" (i.e., "criss-cross"). Perhaps one of them (the one that is larger) is divisible by the other.
- The number resulting from such a division will be an additional factor for a fraction with a smaller denominator.
- At the same time, a fraction with a large denominator does not need to be multiplied by anything at all - this is the savings. At the same time, the probability of error is sharply reduced.
A task. Find expression values:
Note that 84: 21 = 4; 72:12 = 6. Since in both cases one denominator is divisible by the other without a remainder, we apply the method common factors. We have:
Note that the second fraction was not multiplied by anything at all. In fact, we have cut the amount of calculations in half!
By the way, I took the fractions in this example for a reason. If you're interested, try counting them using the criss-cross method. After the reduction, the answers will be the same, but there will be much more work.
This is the strength of the method of common divisors, but, again, it can only be applied when one of the denominators is divided by the other without a remainder. Which happens quite rarely.
Least common multiple method
When we reduce fractions to a common denominator, we are essentially trying to find a number that is divisible by each of the denominators. Then we bring the denominators of both fractions to this number.
There are a lot of such numbers, and the smallest of them will not necessarily equal the direct product of the denominators of the original fractions, as is assumed in the "cross-wise" method.
For example, for denominators 8 and 12, the number 24 is quite suitable, since 24: 8 = 3; 24:12 = 2. This number is much less than the product 8 12 = 96 .
The smallest number that is divisible by each of the denominators is called their least common multiple (LCM).
Notation: The least common multiple of a and b is denoted by LCM(a ; b ) . For example, LCM(16; 24) = 48 ; LCM(8; 12) = 24 .
If you manage to find such a number, the total amount of calculations will be minimal. Look at the examples:
A task. Find expression values:
Note that 234 = 117 2; 351 = 117 3 . Factors 2 and 3 are coprime (have no common divisors except 1), and factor 117 is common. Therefore LCM(234; 351) = 117 2 3 = 702.
Similarly, 15 = 5 3; 20 = 5 4 . Factors 3 and 4 are relatively prime, and factor 5 is common. Therefore LCM(15; 20) = 5 3 4 = 60.
Now let's bring the fractions to common denominators:
Note how useful the factorization of the original denominators turned out to be:
- Having found the same factors, we immediately reached the least common multiple, which, generally speaking, is a non-trivial problem;
- From the resulting expansion, you can find out which factors are “missing” for each of the fractions. For example, 234 3 \u003d 702, therefore, for the first fraction, the additional factor is 3.
To appreciate how much of a win the least common multiple method gives, try calculating the same examples using the criss-cross method. Of course, without a calculator. I think after that comments will be redundant.
Do not think that such complex fractions will not be in real examples. They meet all the time, and the above tasks are not the limit!
The only problem is how to find this NOC. Sometimes everything is found in a few seconds, literally “by eye”, but in general this is a complex computational problem that requires separate consideration. Here we will not touch on this.
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 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. Let's break them down into prime 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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