Initially, I wanted to include common denominator methods in the Adding and Subtracting Fractions paragraph. But there was so much information, and its importance is so great (after all, common denominators are not only for numeric fractions) 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 basic property of a fraction comes to the rescue, which, recall, sounds like this:
The fraction will not change if its numerator and denominator are multiplied by the same nonzero number.
Thus, if you choose the right factors, the denominators of the fractions become equal - this process is called common denominator reduction. And the required numbers, "leveling" the denominators, are called additional factors.
Why do you even 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;
- Comparison of fractions. Sometimes converting to a common denominator makes this task much easier;
- Solving problems for shares and percentages. Percentages are, in fact, common expressions that contain fractions.
There are many ways to find numbers that, when multiplied by, make the denominators of fractions equal. We will consider only three of them - in order of increasing complexity and, in a sense, efficiency.
Cross-multiplication
The simplest and reliable way which is guaranteed to flatten the denominators. We will go 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 become equal to the product initial denominators. Take a look:
Consider the denominators of neighboring fractions as additional factors. We get:
Yes, it's that simple. If you are just starting to learn fractions, it is better to work with this particular 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 "right through", and the result can be very large numbers... This is the price to pay for reliability.
Common divisors method
This technique helps to greatly reduce the calculations, but, unfortunately, it is rarely used. The method is as follows:
- Before you go ahead (that is, the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided by the other.
- The number obtained as a result of such division will be an additional factor for the fraction with a lower denominator.
- In this case, 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.
Task. Find the values of the expressions:
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 of common factors. We have:
Note that the second fraction was never multiplied by anything at all. In fact, we have cut the amount of computation in half!
By the way, I took the fractions in this example for a reason. If you're curious, try counting them crosswise. After the reduction, the answers will be the same, but there will be much more work.
This is the strength of the method. common divisors, but, again, it can be applied only when one of the denominators is divided by the other without a remainder. Which is rare enough.
Least Common Multiple Method
When we bring 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 be equal to the direct product of the denominators of the original fractions, as it is assumed in the "criss-cross" method.
For example, for the 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 can find such a number, the total amount of computation will be minimal. Take a look at examples:
Task. Find the values of the expressions:
Note that 234 = 117 · 2; 351 = 117 3. The factors 2 and 3 are relatively prime (they have no common factors other than 1), and the factor 117 is common. Therefore, the LCM (234; 351) = 117 · 2 · 3 = 702.
Similarly, 15 = 5 · 3; 20 = 5 4. The factors 3 and 4 are relatively prime, and the factor 5 is common. Therefore, LCM (15; 20) = 5 3 4 = 60.
Now we bring the fractions to common denominators:
Note how helpful factoring the original denominators was:
- Having found the same factors, we immediately arrived at the least common multiple, which, generally speaking, is a nontrivial problem;
- From the resulting expansion, you can find out which factors are "missing" for each of the fractions. For example, 234 3 = 702, therefore, for the first fraction, the additional factor is 3.
To estimate how colossal gains the least common multiple method gives, try calculating the same examples using the criss-cross method. Without a calculator, of course. I think after that comments will be superfluous.
Do not think that such complex fractions will not be in the real examples. They meet all the time, and the above tasks are not the limit!
The only problem is how to find this very NOC. Sometimes everything is found in a few seconds, literally "by eye", but on the whole this is a complex computational problem that requires separate consideration. We will not touch on this here.
To bring fractions to the lowest common denominator, you must: 1) find the smallest common multiple of the denominators of these fractions, it will be the lowest common denominator. 2) find an additional factor for each of the fractions, for which the new denominator is divided 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.
Find the smallest common multiple of the denominators: LCM (5; 4) = 20, since 20 is the smallest number that can be 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 brought these fractions to the lowest common denominator ( 20 ).
The lowest common denominator of these fractions is 8, since 8 is divisible by 4 and by itself. There will be no additional factor for the 1st fraction (or we can say that it is equal to one), to the 2nd fraction the additional factor is 2 (8 : 4 = 2). We multiply the numerator and denominator of the 2nd fraction by 2. We brought these fractions to the lowest common denominator ( 8 ).
These fractions are not irreducible.
Reduce the 1st fraction by 4, and the 2nd fraction by 2. ( see examples for reduction common fractions: Sitemap → 5.4.2. Examples of reduction of common fractions). Find the LCM (16 ; 20)=2 4 · 5=16· 5 = 80. The additional factor 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 brought these fractions to the lowest common denominator ( 80 ).
Find the lowest common denominator of the NOZ (5 ; 6 and 15) = LCM (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 brought these fractions to the lowest common denominator ( 30 ).
Page 1 of 1 1
To solve examples with fractions, you need to be able to find the lowest common denominator. Below is a detailed instruction.
How to find the lowest common denominator - concept
Least Common Denominator (LCN) in simple words Is the minimum number that is divisible by the denominators of all fractions this example... In other words, it is called Least Common Multiple (LCM). NOZ is used only if the denominators of the fractions are different.
How to find the lowest common denominator - examples
Let's consider examples of finding the NOZ.
Calculate 3/5 + 2/15.
Solution (Workflow):
- We look at the denominators of the fractions, make sure that they are different and the expressions are reduced as much as possible.
- Find the smallest number that is divisible by both 5 and 15. This number will be 15. Thus, 3/5 + 2/15 =? / 15.
- We figured out the denominator. What will be in the numerator? An additional multiplier will help us figure this out. The additional factor is the number obtained by dividing the NOZ by the denominator of a particular fraction. For 3/5, the additional factor is 3, since 15/5 = 3. For the second fraction, the additional factor is 1, since 15/15 = 1.
- Having found out the additional factor, we multiply it by the numerators of the fractions and add the resulting values. 3/5 + 2/15 = (3 * 3 + 2 * 1) / 15 = (9 + 2) / 15 = 11/15.
Answer: 3/5 + 2/15 = 11/15.
If in the example not 2, but 3 or more fractions are added or subtracted, then the NOZ must be sought for as many fractions as given.
Calculate: 1/2 - 5/12 + 3/6
Solution (sequence of actions):
- Find the lowest common denominator. The minimum divisible by 2, 12 and 6 is 12.
- We get: 1/2 - 5/12 + 3/6 =? / 12.
- We are looking for additional factors. For 1/2 - 6; for 5/12 - 1; for 3/6 - 2.
- We multiply by the numerators and assign the corresponding signs: 1/2 - 5/12 + 3/6 = (1 * 6 - 5 * 1 + 2 * 3) / 12 = 7/12.
Answer: 1/2 - 5/12 + 3/6 = 7/12.
V real life we need to operate with ordinary fractions. However, to add or subtract fractions with different denominators, such as 2/3 and 5/7, we need to find a common denominator. By bringing the fractions to a common denominator, we can easily perform addition or subtraction operations.
Definition
Fractions are one of the most difficult topics in elementary arithmetic, and rational numbers are intimidating to schoolchildren who meet them for the first time. We are used to operating with numbers written in decimal format. It is much easier to add 0.71 and 0.44 outright than it is to add 5/7 and 4/9. Indeed, to sum the fractions, they must be brought to a common denominator. However, fractions represent the value of quantities much more accurately than their decimal equivalents, and in mathematics, the representation of series or ir rational numbers in the form of a fraction becomes a priority. Such a task is called "reducing an expression to a closed form."
If both the numerator and the denominator of the fraction are multiplied or divided by the same coefficient, then the value of the fraction will not change. This is one of the most important properties fractional numbers... For example, the fraction 3/4 in decimal is 0.75. If we multiply the numerator and denominator by 3, then we get the fraction 9/12, which also equals 0.75. Thanks to this property, we can multiply different fractions so that they all have the same denominator. How to do it?
Finding a common denominator
The lowest common denominator (LCM) is the lowest common multiple of all the denominators of an expression. We can find such a number in three ways.
Using the maximum denominator
This is one of the simplest but time-consuming methods of finding NOZs. First, we write out the largest number from the denominators of all fractions and check its divisibility by smaller numbers. If divisible, then the largest denominator is the NOZ.
If in the previous operation the numbers are divided with a remainder, then the largest of them must be multiplied by 2 and repeat the divisibility test. If it is divided without a remainder, then the new coefficient becomes NOZ.
If not, then the largest denominator is multiplied by 3, 4, 5, and so on, until the smallest common multiple of lower parts all fractions. In practice, it looks like this.
Let's say we have fractions 1/5, 1/8 and 1/20. Check 20 for divisibility of 5 and 8. 20 is not divisible by 8. Multiply 20 by 2. Check 40 for divisibility of 5 and 8. Numbers are divisible without remainder, therefore, NOZ (1/5, 1/8 and 1/20) = 40 , and fractions become 8/40, 5/40 and 2/40.
Sequential enumeration of multiples
The second way is a simple enumeration of multiples and selection of the smallest one. To find multiples, we multiply the number by 2, 3, 4, and so on, so the number of multiples goes to infinity. You can limit this sequence by the limit, which is the product of the given numbers. For example, for numbers 12 and 20, the LCM is found as follows:
- write out numbers that are multiples of 12 - 24, 48, 60, 72, 84, 96, 108, 120;
- write out numbers that are multiples of 20 - 40, 60, 80, 100, 120;
- determine common multiples - 60, 120;
- choose the smallest of them - 60.
So 1/12 and 1/20 have a common denominator of 60, and the fractions convert to 5/60 and 3/60.
Prime factorization
This method of finding the LCM is the most relevant. This method implies the decomposition of all numbers from the lower parts of fractions into indivisible factors. After that, a number is compiled that contains the factors of all denominators. In practice, it works like this. Find the LCM for the same pair 12 and 20:
- factor 12 - 2 × 2 × 3;
- lay out 20 - 2 × 2 × 5;
- we combine the factors in such a way that they contain the numbers and 12, and 20 - 2 × 2 × 3 × 5;
- we multiply the indivisibles and get the result - 60.
In the third paragraph, we combine the multipliers without repetitions, that is, two twos are enough to form 12 in combination with a three and 20 with a five.
Our calculator allows you to determine the NOZ for an arbitrary number of fractions written in both ordinary and decimal forms. To search for NOZ, you just need to enter values separated by tabs or commas, after which the program will calculate the common denominator and display the converted fractions.
Real life example
Adding fractions
Suppose that in an arithmetic problem we need to add five fractions:
0,75 + 1/5 + 0,875 + 1/4 + 1/20
A manual solution would be done in the following way... First, we need to represent numbers in one notation form:
- 0,75 = 75/100 = 3/4;
- 0,875 = 875/1000 = 35/40 = 7/8.
Now we have a number of ordinary fractions that need to be reduced to the same denominator:
3/4 + 1/5 + 7/8 + 1/4 + 1/20
Since we have 5 terms, the easiest way is to use the method of finding the NOZ by the largest number. Check 20 for divisibility by the rest of the numbers. 20 is not divisible by 8 without a remainder. We multiply 20 by 2, check 40 for divisibility - all numbers divide 40 completely. This is our common denominator. Now, to sum the rational numbers, we need to determine additional factors for each fraction, which are defined as the ratio of the LCM to the denominator. Additional factors will look like this:
- 40/4 = 10;
- 40/5 = 8;
- 40/8 = 5;
- 40/4 = 10;
- 40/20 = 2.
Now we multiply the numerator and denominator of the fractions by the corresponding additional factors:
30/40 + 8/40 + 35/40 + 10/40 + 2/40
For such an expression, we can easily determine the amount equal to 85/40 or 2 integers and 1/8. These are cumbersome calculations, so you can simply enter the problem data into the calculator form and get the answer right away.
Conclusion
Fraction arithmetic - not too much convenient thing, because in order to find the answer, you have to carry out a lot of intermediate calculations. Use our online calculator to reduce fractions to a common denominator and solve school problems quickly.
Most operations with algebraic fractions, such as addition and subtraction, require preliminary reduction of these fractions to the same denominator. Such denominators are also often referred to as “common denominator”. In this topic, we will consider the definition of the concepts "common denominator algebraic fractions"And" the least common denominator of algebraic fractions (LCF) ", consider the points of the algorithm for finding the common denominator and solve several problems on the topic.
Yandex.RTB R-A-339285-1
Common denominator of algebraic fractions
If we talk about ordinary fractions, then the common denominator is a number that is divisible by any of the denominators of the original fractions. For ordinary fractions 1 2 and 5 9 36 can be a common denominator, since it is divisible by 2 and 9 without remainder.
The common denominator of algebraic fractions is defined in a similar way, only polynomials are used instead of numbers, since they are the ones that are in the numerators and denominators of an algebraic fraction.
Definition 1
Common denominator of an algebraic fraction Is a polynomial that is divisible by the denominator of any of the fractions.
In connection with the peculiarities of algebraic fractions, which will be discussed below, we will often deal with common denominators presented in the form of a product, and not in the form of a standard polynomial.
Example 1
The polynomial written as a product 3 x 2 (x + 1), corresponds to the polynomial standard view 3 x 3 + 3 x 2... This polynomial can be the common denominator of algebraic fractions 2 x, - 3 x y x 2 and y + 3 x + 1, due to the fact that it is divisible by x, on x 2 and on x + 1... Information about the divisibility of polynomials is in the corresponding topic of our resource.
Least Common Denominator (LCN)
For given algebraic fractions, the number of common denominators can be infinite.
Example 2
Take the fractions 1 2 x and x + 1 x 2 + 3 as an example. Their common denominator is 2 x (x 2 + 3) like - 2 x (x 2 + 3) like x (x 2 + 3) like 6, 4 x (x 2 + 3) (y + y 4) like - 31 x 5 (x 2 + 3) 3, etc.
When solving problems, you can make your work easier by using a common denominator, which among the whole set of denominators has the simplest form. This denominator is often referred to as the lowest common denominator.
Definition 2
Least common denominator of algebraic fractions Is the common denominator of algebraic fractions, which has the simplest form.
By the way, the term “lowest common denominator” is not generally accepted, therefore it is better to limit ourselves to the term “common denominator”. And that's why.
Earlier, we focused your attention on the phrase “the denominator of the most simple kind". The main meaning of this phrase is as follows: any other common denominator of the data in the condition of the problem of algebraic fractions should be divided without a remainder by the denominator of the simplest form. In this case, in the product, which is the common denominator of fractions, you can use various numerical coefficients.
Example 3
Take the fractions 1 2 x and x + 1 x 2 + 3. We have already found out that it will be easiest for us to work with a common denominator of the form 2 x (x 2 + 3). Also, the common denominator for these two fractions can be x (x 2 + 3) which does not contain a numeric factor. The question is which of these two common denominators is the lowest common denominator of the fractions. There is no unequivocal answer, therefore it is more correct to speak simply of a common denominator, and to take into work the option with which it will be most convenient to work. So, we can use such common denominators as x 2 (x 2 + 3) (y + y 4) or - 15 x 5 (x 2 + 3) 3 that have more complex view but it can be more difficult to deal with them.
Finding the common denominator of algebraic fractions: an algorithm of actions
Suppose we have several algebraic fractions for which we need to find a common denominator. To solve this problem, we can use the following algorithm of actions. First, we need to factor the denominators of the original fractions. Then we compose a work, in which we successively include:
- all factors from the denominator of the first fraction together with powers;
- all the factors present in the denominator of the second fraction, but which are not in the written work or their degree is not enough;
- all missing factors from the denominator of the third fraction, and so on.
The resulting product will be the common denominator of algebraic fractions.
As the multipliers of the product, we can take all the denominators of the fractions given in the problem statement. However, the multiplier that we get in the end will be far from the NOZ in meaning and its use will be irrational.
Example 4
Find the common denominator of the fractions 1 x 2 y, 5 x + 1, and y - 3 x 5 y.
Solution
In this case, we do not need to factor out the denominators of the original fractions. Therefore, we will begin to apply the algorithm by compiling a work.
From the denominator of the first fraction, we take the factor x 2 y, from the denominator of the second fraction the factor x + 1... We get the work x 2 y (x + 1).
The denominator of the third fraction can give us a multiplier x 5 y, however, in the work we compiled earlier, there are already factors x 2 and y... Therefore, we add more x 5 - 2 = x 3... We get the work x 2 y (x + 1) x 3 which can be reduced to the form x 5 y (x + 1)... This will be our NOZ of algebraic fractions.
Answer: x 5 y (x + 1).
Now we will consider examples of problems when the denominators of algebraic fractions have integer numerical factors. In such cases, we also act according to the algorithm, having previously decomposed integer numerical factors into prime factors.
Example 5
Find the common denominator of the fractions 1 12 x and 1 90 x 2.
Solution
Expanding the numbers in the denominators of fractions into prime factors, we get 1 2 2 · 3 · x and 1 2 · 3 2 · 5 · x 2. We can now move on to drawing up a common denominator. To do this, from the denominator of the first fraction, we take the product 2 2 3 x and add factors 3, 5 and x from the denominator of the second fraction. We get 2 2 3 x 3 5 x = 180 x 2... This is our common denominator.
Answer: 180 x 2.
If you look closely at the results of the two analyzed examples, you will notice that the common denominators of fractions contain all the factors present in the expansions of the denominators, and if a certain factor is present in several denominators, then it is taken with the largest available exponent. And if there are integer coefficients in the denominators, then in the common denominator there is a numerical factor equal to the least common multiple of these numerical coefficients.
Example 6
The denominators of both algebraic fractions 1 12 x and 1 90 x 2 have a factor x... In the second case, the factor x is squared. To compile a common denominator, we need to take this factor to the greatest extent, i.e. x 2... There are no other multipliers with variables. Integer numeric coefficients of the original fractions 12 and 90 , and their least common multiple is 180 ... It turns out that the desired common denominator has the form 180 x 2.
Now we can write another algorithm for finding the common factor of algebraic fractions. For this we:
- we decompose the denominators of all fractions into factors;
- compose the product of all alphabetic factors (if there is a factor in several expansions, we take the option with the highest exponent);
- add the LCM of the numerical expansion coefficients to the resulting product.
The above algorithms are equivalent, so any of them can be used in solving problems. It's important to pay attention to detail.
There are times when the common factors in the denominators of fractions may not be noticeable behind the numerical coefficients. Here it is advisable to first take out the numerical coefficients at the highest powers of the variables outside the brackets in each of the factors in the denominator.
Example 7
What is the common denominator of the fractions 3 5 - x and 5 - x · y 2 2 · x - 10.
Solution
In the first case, the minus one must be taken out of the brackets. We get 3 - x - 5. Multiply the numerator and denominator by - 1 in order to get rid of the minus in the denominator: - 3 x - 5.
In the second case, we put two out of the parenthesis. This allows us to get the fraction 5 - x · y 2 2 · x - 5.
Obviously, the common denominator of these algebraic fractions - 3 x - 5 and 5 - x y 2 2 x - 5 is 2 (x - 5).
Answer:2 (x - 5).
Fractions data in the problem statement can have fractional coefficients. In these cases, you must first get rid of the fractional coefficients by multiplying the numerator and denominator by some number.
Example 8
Simplify the algebraic fractions 1 2 x + 1 1 14 x 2 + 1 7 and - 2 2 3 x 2 + 1 1 3 and find their common denominator.
Solution
Let's get rid of fractional coefficients by multiplying the numerator and denominator in the first case by 14, in the second case by 3. We get:
1 2 x + 1 1 14 x 2 + 1 7 = 14 1 2 x + 1 14 1 14 x 2 + 1 7 = 7 x + 1 x 2 + 2 and - 2 2 3 x 2 + 1 1 3 = 3 - 2 3 2 3 x 2 + 4 3 = - 6 2 x 2 + 4 = - 6 2 x 2 + 2.
After the transformations, it becomes clear that the common denominator is 2 (x 2 + 2).
Answer: 2 (x 2 + 2).
If you notice an error in the text, please select it and press Ctrl + Enter