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.
This article explains, how to find the lowest common denominator and how to bring fractions to a 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 of 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 whether 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.
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 downside this method- you have to count a lot, because the denominators are multiplied "throughout", and as a result you can get very big numbers. 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 without a remainder by the other, we use the method of 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 . The factors 2 and 3 are coprime (have no common divisors except 1), and the 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.
In this material, we will analyze how to correctly bring fractions to a new denominator, what an additional factor is and how to find it. After that, we formulate the basic rule for reducing fractions to new denominators and illustrate it with examples of problems.
The concept of reducing a fraction to a different denominator
Recall the basic property of a fraction. According to him, the ordinary fraction a b (where a and b are any numbers) has an infinite number of fractions that are equal to it. Such fractions can be obtained by multiplying the numerator and denominator by the same number m (natural). In other words, all ordinary fractions can be replaced by others of the form a m b m . This is the reduction of the original value to a fraction with the desired denominator.
You can bring a fraction to a different denominator by multiplying its numerator and denominator by any natural number. The main condition is that the multiplier must be the same for both parts of the fraction. The result is a fraction equal to the original.
Let's illustrate this with an example.
Example 1
Convert the fraction 11 25 to a new denominator.
Solution
Take an arbitrary natural number 4 and multiply both parts of the original fraction by it. We consider: 11 4 \u003d 44 and 25 4 \u003d 100. The result is a fraction of 44,100.
All calculations can be written in this form: 11 25 \u003d 11 4 25 4 \u003d 44 100
It turns out that any fraction can be reduced to a huge number of different denominators. Instead of four, we could take another natural number and get another fraction equivalent to the original one.
But not any number can become the denominator of a new fraction. So, for a b the denominator can only contain numbers b · m that are multiples of b . Recall the basic concepts of division - multiples and divisors. If the number is not a multiple of b, but it cannot be a divisor of a new fraction. Let us explain our idea with an example of solving the problem.
Example 2
Calculate whether it is possible to reduce the fraction 5 9 to the denominators 54 and 21.
Solution
54 is a multiple of nine, which is the denominator of the new fraction (i.e. 54 can be divided by 9). Hence, such a reduction is possible. And we cannot divide 21 by 9, so such an action cannot be performed for this fraction.
The concept of an additional multiplier
Let us formulate what an additional factor is.
Definition 1
Additional multiplier is a natural number by which both parts of a fraction are multiplied to bring it to a new denominator.
Those. when we perform this action on a fraction, we take an additional multiplier for it. For example, to reduce the fraction 7 10 to the form 21 30, we need an additional factor 3 . And you can get a fraction 15 40 out of 3 8 using a multiplier 5.
Accordingly, if we know the denominator to which the fraction must be reduced, then we can calculate an additional factor for it. Let's figure out how to do it.
We have a fraction a b , which can be reduced to some denominator c ; calculate the additional factor m . We need to multiply the denominator of the original fraction by m. We get b · m , and according to the condition of the problem b · m = c . Recall how multiplication and division are related. This connection will lead us to the following conclusion: the additional factor is nothing more than the quotient of dividing c by b, in other words, m = c: b.
Thus, to find an additional factor, we need to divide the required denominator by the original one.
Example 3
Find the additional factor by which the fraction 17 4 was brought to the denominator 124 .
Solution
Using the rule above, we simply divide 124 by the denominator of the original fraction, four.
We consider: 124: 4 \u003d 31.
This type of calculation is often required when reducing fractions to a common denominator.
The rule for reducing fractions to a specified denominator
Let's move on to the definition of the basic rule, with which you can bring fractions to the specified denominator. So,
Definition 2
To bring a fraction to the specified denominator, you need:
- determine an additional multiplier;
- multiply by it both the numerator and the denominator of the original fraction.
How to apply this rule in practice? Let us give an example of solving the problem.
Example 4
Carry out the reduction of the fraction 7 16 to the denominator 336 .
Solution
Let's start by calculating the additional multiplier. Divide: 336: 16 = 21.
We multiply the received answer by both parts of the original fraction: 7 16 \u003d 7 21 16 21 \u003d 147 336. So we brought the original fraction to the desired denominator 336.
Answer: 7 16 = 147 336.
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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 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. 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