In this material we will look at how to correctly convert fractions to a new denominator, what an additional factor is and how to find it. After this, we will formulate the basic rule for reducing fractions to new denominators and illustrate it with examples of problems.

The concept of reducing a fraction to another denominator

Let us recall the basic property of a fraction. According to him, an 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 number). In other words, everything common 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 reduce a fraction to another 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 will be a fraction equal to the original one.

Let's illustrate this with an example.

Example 1

Convert the fraction 11 25 to the new denominator.

Solution

Let's take an arbitrary natural number 4 and multiply both sides of the original fraction by it. We count: 11 · 4 = 44 and 25 · 4 = 100. The result is a fraction of 44 100.

All calculations can be written in this form: 11 25 = 11 4 25 4 = 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 new fraction. So, for a b the denominator can only contain numbers b m that are multiples of b. Review the basic concepts of division—multiples and divisors. If the number is not a multiple of b, but it cannot be a divisor of the new fraction. Let us illustrate our idea with an example of solving a 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 in the denominator of the new fraction (i.e. 54 can be divided by 9). This means that such a reduction is possible. But we cannot divide 21 by 9, so this 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 represents a natural number by which both sides of a fraction are multiplied to bring it to a new denominator.

Those. when we do this with a fraction, we take an additional factor for it. For example, to convert the fraction 7 10 to the form 21 30, we need an additional factor of 3. And you can get the fraction 15 40 from 3 8 using the multiplier 5.

Accordingly, if we know the denominator to which a fraction needs to be reduced, then we can calculate an additional factor for it. Let's figure out how to do this.

We have a fraction a b that can be reduced to a certain denominator c; Let's 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 conditions of the problem b · m = c. Let's remember how multiplication and division are related to each other. This connection will prompt 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 the additional factor, we need to divide the required denominator by the original one.

Example 3

Find the additional factor with which the fraction 17 4 was reduced to the denominator 124.

Solution

Using the rule above, we simply divide 124 by the denominator of the original fraction, four.

We count: 124: 4 = 31.

This type of calculation is often required when converting fractions to common denominator.

The rule for reducing fractions to the specified denominator

Let's move on to defining the basic rule with which you can reduce fractions to the specified denominator. So,

Definition 2

To reduce a fraction to the specified denominator you need:

1. determine an additional factor;
2. multiply both the numerator and denominator of the original fraction by it.

How to apply this rule in practice? Let's give an example of solving the problem.

Example 4

Reduce the fraction 7 16 to the denominator 336.

Solution

Let's start by calculating the additional multiplier. Divide: 336: 16 = 21.

We multiply the resulting answer by both parts of the original fraction: 7 16 = 7 · 21 16 · 21 = 147 336. So we brought the original fraction to the desired denominator 336.

Answer: 7 16 = 147 336.

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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 ).

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I originally wanted to include common denominator techniques in the Adding and Subtracting Fractions section. But there turned out to be 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 basic property of a fraction comes to the rescue, which, let me remind you, sounds like this:

A fraction will not change if its numerator and denominator are multiplied by the same number other than zero.

Thus, if you choose the factors correctly, the denominators of the fractions will become equal - this process is called reduction to a common denominator. And the required numbers, “evening out” the denominators, are called additional factors.

Why do we need to reduce fractions to a common denominator? Here are just a few reasons:

1. Adding and subtracting fractions with different denominators. There is no other way to perform this operation;
2. Comparing fractions. Sometimes reduction to a common denominator greatly simplifies this task;
3. Solving problems involving fractions and percentages. Percentages are essentially ordinary expressions that contain fractions.

There are many ways to find numbers that, when multiplied by them, will make the denominators of fractions equal. We will consider only three of them - in order of increasing complexity and, in a sense, effectiveness.

Criss-cross multiplication

The simplest and reliable way, which is guaranteed to equalize the denominators. We will act “in a headlong manner”: 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 study fractions, it is better to work using this method - this way you will insure yourself against many mistakes and are guaranteed to get the result.

The only drawback this method- you have to count a lot, because the denominators are multiplied “throughout”, and the result can be very big numbers. This is the price to pay for reliability.

Common Divisor Method

This technique helps to significantly reduce calculations, but, unfortunately, it is used quite rarely. The method is as follows:

1. Before you go straight ahead (i.e., using the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided into the other.
2. The number resulting from this division will be an additional factor for the fraction with a smaller denominator.
3. In this case, a fraction with a large denominator does not need to be multiplied by anything at all - this is where the savings lie. At the same time, the probability of error is sharply reduced.

Task. Find the meanings of the expressions:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divided 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 cut the amount of computation in half!

By the way, I didn’t take the fractions in this example by chance. If you're interested, try counting them using the criss-cross method. After reduction, the answers will be the same, but there will be much more work.

This is the power of the common divisors method, but, again, it can only be used when one of the denominators is divisible 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 denominator. 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 is assumed in the “criss-cross” 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:

Task. Find the meanings of the expressions:

Note that 234 = 117 2; 351 = 117 3. Factors 2 and 3 are coprime (have no common factors other than 1), and factor 117 is common. Therefore LCM(234, 351) = 117 2 3 = 702.

Likewise, 15 = 5 3; 20 = 5 · 4. Factors 3 and 4 are coprime, and factor 5 is common. Therefore LCM(15, 20) = 5 3 4 = 60.

Now let's bring the fractions to common denominators:

Notice how useful it was to factorize the original denominators:

1. Having discovered identical factors, we immediately arrived at the least common multiple, which, generally speaking, is a non-trivial problem;
2. From the resulting expansion you can find out which factors are “missing” in each fraction. For example, 234 · 3 = 702, therefore, for the first fraction the additional factor is 3.

To appreciate how much of a difference the least common multiple method makes, try calculating these same examples using the criss-cross method. Of course, without a calculator. I think after this comments will be unnecessary.

Don't think that there won't be such complex fractions 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 can be found in a few seconds, literally “by eye,” but in general this is a complex computational task that requires separate consideration. We won't touch on that here.

This article explains how to find the lowest common denominator And how to reduce fractions to a common denominator. First, the definitions of common denominator of fractions and least common denominator are given, and it is shown how to find the common denominator of fractions. Below 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 discussed.

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What is called reducing fractions to a common denominator?

Now we can say what it is to reduce fractions to a common denominator. Reducing 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 same denominators.

Common denominator, definition, examples

Now it's time to define the common denominator of fractions.

In other words, the common denominator of a certain set of ordinary fractions is any natural number that is divisible by all the denominators of these fractions.

From the stated definition it follows that a given set of fractions has infinitely many common denominators, since there is 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 the fractions 1/4 and 5/6, their denominators are 4 and 6, respectively. Positive common multiples of the numbers 4 and 6 are the numbers 12, 24, 36, 48, ... Any of these numbers is a common denominator of the fractions 1/4 and 5/6.

To consolidate the material, consider the solution to the following example.

Example.

Can the fractions 2/3, 23/6 and 7/12 be reduced to a common denominator of 150?

Solution.

To answer the question we need to find out whether the number 150 is a common multiple of the denominators 3, 6 and 12. To do this, let’s check whether 150 is divisible by each of these numbers (if necessary, see the rules and examples of dividing natural numbers, as well as the rules and examples of dividing natural numbers with a remainder): 150:3=50, 150:6=25, 150: 12=12 (remaining 6) .

So, 150 is not evenly divisible by 12, therefore 150 is not a common multiple of 3, 6, and 12. Therefore, the number 150 cannot be the common denominator of the original fractions.

Answer:

It is forbidden.

Lowest common denominator, how to find it?

In the set of numbers that are common denominators of given fractions, there is a smallest natural number, which is called the least common denominator. Let us formulate the definition of the lowest common denominator of these fractions.

Definition.

Lowest common denominator- This smallest number, from all common denominators of these fractions.

It remains to deal with the question of how to find the least common divisor.

Since is the smallest positive common divisor of a given set of numbers, then the LCM of the denominators of the given fractions is the least common denominator of the given fractions.

Thus, finding the lowest common denominator of fractions comes down to the denominators of those fractions. Let's look at the solution to the example.

Example.

Find the lowest common denominator of the fractions 3/10 and 277/28.

Solution.

The denominators of these fractions are 10 and 28. The desired lowest 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 reduce fractions to a common denominator? Rule, examples, solutions

Common fractions usually result in a lowest common denominator. We will now write down a rule that explains how to reduce fractions to their lowest common denominator.

Rule for reducing fractions to lowest common denominator consists of three steps:

• First, find the lowest common denominator of the fractions.
• Second, an additional factor is calculated for each fraction by dividing the lowest common denominator by the denominator of each fraction.
• Third, the numerator and denominator of each fraction are multiplied by its additional factor.

Let us apply the stated rule to solve the following example.

Example.

Reduce the fractions 5/14 and 7/18 to their lowest common denominator.

Solution.

Let's perform all the steps of the algorithm for reducing fractions to the lowest 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 9 and 7, respectively. We have and .

So, reducing the fractions 5/14 and 7/18 to the lowest common denominator is complete. The resulting fractions were 45/126 and 49/126.

When adding and subtracting algebraic fractions with different denominators, the fractions first lead to common denominator. This means that they find one denominator that is divided by the original denominator of each algebraic fraction included in the given expression.

As you know, if the numerator and denominator of a fraction are multiplied (or divided) by the same number other than zero, the value of the fraction will not change. This is the main property of a fraction. Therefore, when fractions are reduced to a common denominator, they essentially multiply the original denominator of each fraction by the missing factor to obtain a common denominator. In this case, you need to multiply the numerator of the fraction by this factor (it is different for each fraction).

For example, given the following sum of algebraic fractions:

It is required to simplify the expression, that is, add two algebraic fractions. To do this, first of all, you need to bring the fraction terms to a common denominator. The first step is to find a monomial that is divisible by both 3x and 2y. In this case, it is desirable that it be the smallest, that is, find the least common multiple (LCM) for 3x and 2y.

For numerical coefficients and variables, the LCM is searched separately. LCM(3, 2) = 6, and LCM(x, y) = xy. Next, the found values ​​are multiplied: 6xy.

Now we need to determine by what factor we need to multiply 3x to get 6xy:
6xy ÷ 3x = 2y

This means that when reducing the first algebraic fraction to a common denominator, its numerator must be multiplied by 2y (the denominator has already been multiplied when reducing to a common denominator). The multiplier for the numerator of the second fraction is looked for similarly. It will be equal to 3x.

Thus we get:

Then you can act as with fractions with identical denominators: add up the numerators, and write one common denominator:

After transformations, a simplified expression is obtained, which is one algebraic fraction, which is the sum of two original ones:

Algebraic fractions in the original expression may contain denominators that are polynomials rather than monomials (as in the example above). In this case, before searching for a common denominator, you should factor the denominators (if possible). Next, the common denominator is collected from different factors. If the multiplier is in several original denominators, then it is taken once. If the multiplier has different powers in the original denominators, then it is taken with the larger one. For example:

Here the polynomial a 2 – b 2 can be represented as the product (a – b)(a + b). The factor 2a – 2b is expanded as 2(a – b). Thus, the common denominator will be 2(a – b)(a + b).