Bringing fractions to a common denominator. Reduction of fractions to the lowest common denominator, rule, examples, solutions

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 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. common divisors, but, I repeat, it can only be used if one of the denominators is divided by the other without a remainder. Which happens quite rarely.

Reduction of fractions to the lowest common denominator, rule, examples, solutions.

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 reduction of three and more fractions to a common denominator.

What is called reducing fractions to a common denominator?

If ordinary fractions have equal denominators, then these fractions are said to be reduced to a common denominator.

So the fractions 45/76 and 143/76 are reduced to a common denominator of 76, and the fractions 1/3, 3/3, 17/3 and 1000/3 are reduced to a common denominator of 3.

If the denominators of fractions are not equal, then such fractions can always be reduced to a common denominator by multiplying their numerator and denominator by certain additional factors.

For example, ordinary fractions 2/5 and 7/4 with the help of additional factors 4 and 5, respectively, are reduced to a common denominator of 20. Indeed, multiplying the numerator and denominator of the fraction 2/5 by 4, we get the fraction 8/20, and multiplying the numerator and denominator fractions 7/4 by 5, we come to a fraction 35/20 (see reduction of fractions to a new 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.

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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 above 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 12, 24, 36, 48, ... Any of these numbers is the common denominator of 1/4 and 5/6.

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

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

To answer the 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 dividing 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.

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

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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.

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

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

Thus, finding the least common denominator of fractions is reduced to finding the LCM of the denominators of these fractions.

Let's take a look at an example solution.

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

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 is easy to find the LCM by factoring the numbers into prime factors: since 10=2 5, and 28=2 2 7 , then LCM(15, 28)=2 2 5 7=140.

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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.

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

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 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.

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Reduction to the least common denominator of three or more fractions

The rule from the previous paragraph allows you to bring to the lowest common denominator not only two fractions, but also three fractions, and more of them.

Let's consider an example solution.

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

The least common denominator of these fractions is equal to the least common multiple of the numbers 2, 6, 8 and 18. To find the LCM(2, 6, 8, 18), we will use the information from the section on finding the LCM of three or more numbers.

We get LCM(2, 6)=6, LCM(6, 8)=24, and finally LCM(24, 18)=72, so LCM(2, 6, 8, 18)=72. So the lowest common denominator is 72.

Now we calculate additional factors. For fraction 3/2 the additional factor is 72:2=36, for fraction 5/6 it is 72:6=12, for fraction 3/8 the additional factor is 72:8=9, and for fraction 17/18 it is 72 :18=4.

Bringing fractions to a common denominator

There remains the last step in bringing the original fractions to the lowest common denominator: .

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Common denominator is any positive common multiple of all the denominators of the given fractions.

Lowest common denominator- this is smallest number, of all the common denominators of the given fractions.

Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics: textbook for 5 cells. educational institutions.
Vilenkin N.Ya. etc. Mathematics. Grade 6: textbook for educational institutions.

Common denominator of common fractions

If regular factions have same denominators, then these fractions have a common denominator. For example,

they have a common denominator.

Common denominator This is the number that is the denominator for two or more regular fractions.

Fractions with different denominators can be reduced to a common denominator.

Giving fractions to a common denominator

Giving fractions to a common denominator Is replacing these fractions with different denominators of the same fractions with the same denominators?

Fractions can simply be reduced to a common denominator or the lowest common denominator.

smallest common denominator This is the least common denominator of these fractions.

Common denominator of factions on the Internet

To give fractions the lowest common denominator you need:

If possible, perform fraction reduction.
Find the smallest common catalogs of these fractions. NOC will become their smallest common denominator.
Divide LCM by the denominators of these fractions. This measure finds an additional factor for each of these fractions. Additional coefficient Is the number for which it is necessary to multiply the members of the fraction in order to bring it to a common denominator?
Multiply the numerator and denominator of each fraction with an additional factor.

Example.

1) Find the NOC names of these factions:

NOC(8, 12) = 24

2) Additional factors found:

24: 8 = 3 (for ) and 24: 12 = 2 (for )

3) Multiply the members of each faction with an additional factor:

The reduction of the common denominator can be written in a shorter form, indicating an additional coefficient in addition to the counter of each fraction (top right or top left) and not writing intermediate calculations:

The common denominator can be reduced more easily by multiplying the members of the first fraction with the second immanent share and the members of the second fraction with the denominator of the first.

Example. Get the common denominator of the fractions and :

The product of their denominators can be taken as the common denominator of the fractions.

Reducing fractions to a common denominator is used to add, subtract, and compare fractions with different denominators.

Common denominator reduction calculator

This calculator will help you bring common fractions down to the lowest common denominator.

Just enter two factions and click.

5.4.5. Examples of Converting Ordinary Fractions to the Least Common Denominator

The least common denominator of continued fractions is the smallest common denominator for those fractions. ( see section "Least common multiple search": 5.3.5. Find least amount multiples (NOC) of given numbers).

To reduce the share on the least common denominator, you must: 1) find the least common multiple of the denominators of these fractions, and this will be the least common denominator.

2) finds an additional coefficient for each of the factions, for which the new denominator is distributed with the name of each faction. 3) multiply the numerator and denominator of each fraction with an additional factor.

Examples. To reduce the following fractions to the lowest common denominator.

We find the lowest common denominator: LCM (5; 4) = 20, since 20 is the smallest number divided by 5 and 4.

For the first share, an additional coefficient 4 (20 : 5 = 4). For the second fraction, there is an additional coefficient of 5 (20 : 4 = 5). Multiply the number and denominator of the first fraction by 4, and the counter and denominator of the second fraction by 5.

20 ).

The least common denominator for these fractions is the number 8, since it is divisible by 4 and inside.

For the first share there is no additional factor (or we can say that it is equal to one), the second factor is an additional factor 2 (8 : 4 = 2). Multiply the numerator and denominator of the second fraction by 2.

Online calculator. Giving fractions to a common denominator

We have reduced these fractions to the lowest common denominator ( 8th place).

These factions are not intolerable.

The first fraction has been reduced by 4 and the second fraction has been reduced by 2. (See Examples for reducing regular fractions: Sitemap → 5.4.2.

Examples of reduction of conventional fractions). NOC finds (16 ; 20) = 24· 5 = 16· 5 = 80. An additional factor for the 1st fraction is 5 (80 : 16 = 5). An additional factor for the second fraction is 4 (80 : 20 = 4).

We multiply the numerator and denominator of the first fraction with 5, and the counter and denominator of the second fraction with 4. Fractional information has been given to the lowest common denominator ( 80 ).

Find the lowest common denominator of NOx (5 ; 6 and 15) = NOK (5 ; 6 and 15) = 30. An additional factor for the first fraction is 6 (30 : 5 = 6) is an additional factor in the second part of 5 (30 : 6 = 5), is an additional factor for the third fraction 2 (30 : 15 = 2).

The number and denominator of the first fraction are multiplied by 6, the count and denominator of the second fraction by 5, the count and denominator of the third fraction by 2. Partial data were given the lowest common denominator 30 ).

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Lowest common denominator.

What is the lowest common denominator?

Definition:
Lowest common denominator- is the least positive number multiple of the denominators of these fractions.

How to get the lowest common denominator? To answer this question, consider an example:

Reduce fractions with different denominators to the lowest common denominator.

Solution:
To find the least common denominator, you need to find the least common multiple (LCM) of the denominators of these fractions.

The first fraction has a denominator equal to 20, let's decompose it into prime factors.
20=2⋅5⋅2

We also expand the second denominator of the fraction 14 into simple factors.
14=7⋅2

LCM(14,20)= 2⋅5⋅2⋅7=140

Answer: The lowest common denominator is 140.

How to bring a fraction to a common denominator?

You need to multiply the first fraction \(\frac(1)(20)\) by 7 to get the denominator 140.

\(\frac(1)(20)=\frac(1 \times 7)(20 \times 7)=\frac(7)(140)\)
And multiply the second fraction by 10.

\(\frac(3)(14)=\frac(3 \times 10)(14 \times 10)=\frac(30)(140)\)

Rules or algorithm for reducing fractions to a common denominator.

Algorithm for reducing fractions to the lowest common denominator:

It is necessary to decompose the denominators of fractions into prime factors.
You need to find the least common multiple (LCM) for the denominators of these fractions.
Reduce fractions to a common denominator, that is, multiply both the numerator and denominator of the fraction by a factor.

Common denominator for several fractions.

How to find a common denominator for multiple fractions?

Consider an example:
Find the least common denominator for fractions \(\frac(2)(11), \frac(1)(15), \frac(3)(22)\)

Solution:
Let's decompose the denominators 11, 15 and 22 into prime factors.

The number 11 is already a prime number in itself, so there is no need to write it down.
Let's expand the number 15=5⋅3
Let's expand the number 22=11⋅2

Find the least common multiple (LCM) of the denominators 11, 15, and 22.
LCM(11, 15, 22)=11⋅2⋅5⋅3=330

We found the smallest common denominator for these fractions. Now we bring the data of the fraction \(\frac(2)(11), \frac(1)(15), \frac(3)(22)\) to a common denominator equal to 330.

\(\begin(align)
\frac(2)(11)=\frac(2 \times 30)(11 \times 30)=\frac(60)(330) \\\\
\frac(1)(15)=\frac(1 \times 22)(15 \times 22)=\frac(22)(330) \\\\
\frac(3)(22)=\frac(3 \times 15)(22 \times 15)=\frac(60)(330) \\\\
\end(align)\)

This method makes sense if the degree of the polynomial is not lower than the second. In this case, the common factor can be not only a binomial of the first degree, but also of higher degrees.

To find a common factor terms of the polynomial, it is necessary to perform a number of transformations. The simplest binomial or monomial that can be bracketed will be one of the roots of the polynomial. Obviously, in the case when the polynomial does not have a free term, there will be an unknown in the first degree - a polynomial equal to 0.

More difficult to find a common factor is the case when the free term is not equal to zero. Then simple selection or grouping methods are applicable. For example, let all the roots of the polynomial be rational, while all the coefficients of the polynomial are integers: y^4 + 3 y³ - y² - 9 y - 18.

Write down all integer divisors of the free term. If a polynomial has rational roots, then they are among them. As a result of the selection, the roots 2 and -3 are obtained. Hence, the common factors of this polynomial will be the binomials (y - 2) and (y + 3).

The method of taking out a common factor is one of the components of the factorization. The method described above is applicable if the coefficient at the highest degree is 1. If this is not the case, then a number of transformations must first be performed. For example: 2y³ + 19 y² + 41 y + 15.

Make a change of the form t = 2³ y³. To do this, multiply all the coefficients of the polynomial by 4:2³ y³ + 19 2² y² + 82 2 y + 60. After the replacement: t³ + 19 t² + 82 t + 60. Now, to find the common factor, apply the above method .

Besides, effective method search for a common factor is the elements of the polynomial. It is especially useful when the first method is not, i.e. The polynomial has no rational roots. However, groupings are not always obvious. For example: The polynomial y^4 + 4 y³ - y² - 8 y - 2 has no integer roots.

Use grouping: y^4 + 4 y³ - y² - 8 y - 2 = y^4 + 4 y³ - 2 y² + y² - 8 y - 2 = (y^4 - 2 y²) + ( 4 y³ - 8 y) + y² - 2 \u003d (y² - 2) * (y² + 4 y + 1). The common factor of the elements of this polynomial is (y² - 2).

Multiplication and division, just like addition and subtraction, are basic arithmetic operations. Without learning how to solve examples for multiplication and division, a person will face many difficulties not only when studying more complex sections of mathematics, but even in the most ordinary everyday affairs. Multiplication and division are closely related, and the unknown components of examples and problems for one of these actions are calculated using another action. At the same time, it is necessary to clearly understand that when solving examples, it does not matter what kind of objects you divide or multiply.

You will need

- multiplication table;
- a calculator or a piece of paper and a pencil.

Instruction

Write down the example you want. Designate unknown factor as an X. An example might look like this: a*x=b. Instead of the multiplier a and the product b in the example, there can be any or numbers. Remember the basic multiplication: the product does not change from changing the places of factors. So unknown factor x can be placed anywhere.

To find the unknown factor in an example where there are only two factors, you just need to divide the product by the known factor. That is, it is done as follows: x=b/a. If you find it difficult to operate with abstract quantities, try to represent this problem in the form of concrete objects. You, you have only apples and how many will eat them, but you don’t know how many apples everyone will get. For example, you have 5 family members, and the apples turned out to be 15. The number of apples intended for each, denote as x. Then the equation will look like this: 5(apples)*x=15(apples). Unknown factor is found in the same way as in the equation with letters, that is, divide 15 apples into five family members, in the end it turns out that each of them ate 3 apples.

The unknown is found in the same way. factor with the number of factors. For example, the example looks like a*b*c*x*=d. In theory, find factor it is possible and in the same way as in a more post example: x=d/a*b*c. But it is possible to reduce the equation to more plain sight, denoting the product of known factors with some other letter - for example, m. Find what m equals by multiplying numbers a,b and c: m=a*b*c. Then the whole example can be represented as m*x=d, and the unknown value will be equal to x=d/m.

If known factor and the product are fractions, the example is solved in the same way as with . But in this case it is necessary to remember actions. When multiplying fractions, the numerators and denominators are multiplied. When dividing fractions, the numerator of the dividend is multiplied by the denominator of the divisor, and the denominator of the dividend is multiplied by the numerator of the divisor. That is, in this case, the example will look like this: a/b*x=c/d. In order to find the unknown value, you need to divide the product by the known factor. That is x=a/b:c/d =a*d/b*c.

Multiplication "criss-cross"

Common divisor method

A task. Find expression values:

To appreciate how much of a win the least common multiple method gives, try calculating the same examples using the criss-cross method.

Common denominator of fractions

Of course, without a calculator. I think after that comments will be redundant.

See also:

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. And the desired numbers, "leveling" the denominators, are called.

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:

A task. Find expression values:

As additional factors, consider the denominators of neighboring fractions. We get:

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 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!

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 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 of 8 12 = 96.

The smallest number that is divisible by each of the denominators is called their (LCM).

Notation: the least common multiple of numbers 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:

How to find the lowest common denominator

Find expression values:

Note that 234 = 117 2; 351 = 117 3. The factors 2 and 3 are coprime (they 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 coprime, and factor 5 is common. Therefore LCM(15; 20) = 5 3 4 = 60.

Now we 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.

Don't think that these complex fractions in the real examples will not. 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.

See also:

Bringing fractions to a common denominator

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. And the desired numbers, "leveling" the denominators, are called.

Why do you need to bring fractions to a common denominator?

Common denominator, concept and definition.

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"

A task. Find expression values:

As additional factors, consider the denominators of neighboring fractions. We get:

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 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!

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

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 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 of 8 12 = 96.

The smallest number that is divisible by each of the denominators is called their (LCM).

Notation: the least common multiple of numbers 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 (they 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 coprime, and factor 5 is common. Therefore LCM(15; 20) = 5 3 4 = 60.

Now we 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!

See also:

Bringing fractions to a common denominator

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. And the desired numbers, "leveling" the denominators, are called.

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"

Take a look:

A task. Find expression values:

As additional factors, consider the denominators of neighboring fractions. We get:

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 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!

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

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 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 of 8 12 = 96.

The smallest number that is divisible by each of the denominators is called their (LCM).

Notation: the least common multiple of numbers 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 (they 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 coprime, and factor 5 is common. Therefore LCM(15; 20) = 5 3 4 = 60.

Now we 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.

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!

See also:

Bringing fractions to a common denominator

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. And the desired numbers, "leveling" the denominators, are called.

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"

A task. Find expression values:

As additional factors, consider the denominators of neighboring fractions. We get:

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.

Bringing fractions to a common denominator

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 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!

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

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 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 of 8 12 = 96.

The smallest number that is divisible by each of the denominators is called their (LCM).

Notation: the least common multiple of numbers 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 (they 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 coprime, and factor 5 is common. Therefore LCM(15; 20) = 5 3 4 = 60.

Now we 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.

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!

In this lesson, we will look at reducing fractions to a common denominator and solve problems on this topic. Let us define the concept of a common denominator and an additional factor, recall the mutual prime 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.

Third, multiply the numerator and denominator of each fraction by its additional factor.

a) Reduce the fractions and to a common denominator.

The lowest common denominator is 12. The additional factor for the first fraction is 4, for the second - 3. We 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 assignments: #270, #290