Fractions. Multiplying and dividing fractions

Fractions are ordinary numbers and can also be added and subtracted. But because they have a denominator, they require more complex rules than for integers.

Let's consider the simplest case, when there are two fractions with the same denominators. Then:

To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged.

To subtract fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.

Within each expression, the denominators of the fractions are equal. By definition of adding and subtracting fractions we get:

As you can see, it’s nothing complicated: we just add or subtract the numerators and that’s it.

But even in such simple actions people manage to make mistakes. What is most often forgotten is that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.

Get rid of bad habit Adding the denominators is quite simple. Try the same thing when subtracting. As a result, the denominator will be zero, and the fraction will (suddenly!) lose its meaning.

Therefore, remember once and for all: when adding and subtracting, the denominator does not change!

Many people also make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus and where to put a plus.

This problem is also very easy to solve. It is enough to remember that the minus before the sign of a fraction can always be transferred to the numerator - and vice versa. And of course, don’t forget two simple rules:

1. Plus by minus gives minus;
2. Two negatives make an affirmative.

Let's look at all this with specific examples:

Task. Find the meaning of the expression:

In the first case, everything is simple, but in the second, let’s add minuses to the numerators of the fractions:

What to do if the denominators are different

You cannot add fractions with different denominators directly. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.

There are many ways to convert fractions. Three of them are discussed in the lesson “Reducing fractions to a common denominator”, so we will not dwell on them here. Let's look at some examples:

Task. Find the meaning of the expression:

In the first case, we reduce the fractions to a common denominator using the “criss-cross” method. In the second we will look for the NOC. Note that 6 = 2 · 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are relatively prime. Therefore, LCM(6, 9) = 2 3 3 = 18.

What to do if a fraction has an integer part

I can please you: different denominators in fractions are not the biggest evil. Much more errors occur when fractions-terms are highlighted whole part.

Of course, there are own addition and subtraction algorithms for such fractions, but they are quite complex and require a long study. Better use simple diagram, given below:

1. Convert all fractions containing an integer part to improper ones. We obtain normal terms (even with different denominators), which are calculated according to the rules discussed above;
2. Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
3. If this is all that was required in the problem, we perform the inverse transformation, i.e. We get rid of an improper fraction by highlighting the whole part.

The rules for moving to improper fractions and highlighting the whole part are described in detail in the lesson “What is a numerical fraction”. If you don’t remember, be sure to repeat it. Examples:

Task. Find the meaning of the expression:

Everything is simple here. The denominators inside each expression are equal, so all that remains is to convert all fractions to improper ones and count. We have:

To simplify the calculations, I have skipped some obvious steps in the last examples.

A small note about the last two examples, where fractions with the integer part highlighted are subtracted. The minus before the second fraction means that the entire fraction is subtracted, and not just its whole part.

Re-read this sentence again, look at the examples - and think about it. This is where beginners admit great amount errors. They love to give such problems on tests. You will also encounter them several times in the tests for this lesson, which will be published shortly.

Summary: general calculation scheme

In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:

1. If one or more fractions have an integer part, convert these fractions to improper ones;
2. Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the writers of the problems did this);
3. Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with like denominators;
4. If possible, shorten the result. If the fraction is incorrect, select the whole part.

Remember that it is better to highlight the whole part at the very end of the task, immediately before writing down the answer.

Online calculator.
Evaluating an expression with numerical fractions.
Multiplying, subtracting, dividing, adding and reducing fractions with different denominators.

With this online calculator you can multiply, subtract, divide, add and reduce fractions with different denominators.

The program works with regular, improper and mixed number fractions.

This program (online calculator) can:
- perform addition of mixed fractions with different denominators
- perform subtraction of mixed fractions with different denominators
- divide mixed fractions with different denominators
- multiply mixed fractions with different denominators
- reduce fractions to a common denominator
- transform mixed fractions in the wrong
- reduce fractions

You can also enter not an expression with fractions, but one single fraction.
In this case, the fraction will be reduced and the whole part will be separated from the result.

The online calculator for calculating expressions with numerical fractions does not just give the answer to the problem, it provides a detailed solution with explanations, i.e. displays the process of finding a solution.

This program may be useful for high school students secondary schools in preparation for tests and exams, when testing knowledge before the Unified State Exam, for parents to control the solution of many problems in mathematics and algebra. Or maybe it’s too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with detailed solutions.

This way you can conduct your own training and/or training of yours. younger brothers or sisters, while the level of education in the field of problems being solved increases.

If you are not familiar with the rules for entering expressions with numerical fractions, we recommend that you familiarize yourself with them.

Rules for entering expressions with numerical fractions

Only a whole number can act as the numerator, denominator and integer part of a fraction.

The denominator cannot be negative.

When entering a numerical fraction, the numerator is separated from the denominator by a division sign: /
Input: -2/3 + 7/5
Result: \(-\frac(2)(3) + \frac(7)(5)\)

The whole part is separated from the fraction by the ampersand sign: &
Input: -1&2/3 * 5&8/3
Result: \(-1\frac(2)(3) \cdot 5\frac(8)(3)\)

The division of fractions is introduced by the colon sign: :
Input: -9&37/12: -3&5/14
Result: \(-9\frac(37)(12) : \left(-3\frac(5)(14) \right) \)
Remember that you cannot divide by zero!

You can use parentheses when entering expressions with numeric fractions.
Input: -2/3 * (6&1/2-5/9) : 2&1/4 + 1/3
Result: \(-\frac(2)(3) \cdot \left(6 \frac(1)(2) - \frac(5)(9) \right) : 2\frac(1)(4) + \frac(1)(3)\)

Enter an expression using numerical fractions.

Calculate

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A little theory.

Ordinary fractions. Division with remainder

If we need to divide 497 by 4, then when dividing we will see that 497 is not evenly divisible by 4, i.e. the remainder of the division remains. In such cases it is said that it is completed division with remainder, and the solution is written as follows:
497: 4 = 124 (1 remainder).

The division components on the left side of the equality are called the same as in division without a remainder: 497 - dividend, 4 - divider. The result of division when divided with a remainder is called incomplete private. In our case, this is the number 124. And finally, the last component, which is not in ordinary division, is remainder. In cases where there is no remainder, one number is said to be divided by another without a trace, or entirely. It is believed that with such a division the remainder is zero. In our case, the remainder is 1.

The remainder is always less than the divisor.

Division can be checked by multiplication. If, for example, there is an equality 64: 32 = 2, then the check can be done like this: 64 = 32 * 2.

Often in cases where division with a remainder is performed, it is convenient to use the equality
a = b * n + r,
where a is the dividend, b is the divisor, n is the partial quotient, r is the remainder.

The quotient of natural numbers can be written as a fraction.

The numerator of a fraction is the dividend, and the denominator is the divisor.

Since the numerator of a fraction is the dividend and the denominator is the divisor, believe that the line of a fraction means the action of division. Sometimes it is convenient to write division as a fraction without using the ":" sign.

The quotient of the division of natural numbers m and n can be written as a fraction \(\frac(m)(n)\), where the numerator m is the dividend, and the denominator n is the divisor:
\(m:n = \frac(m)(n)\)

The following rules are true:

To get the fraction \(\frac(m)(n)\), you need to divide one by n equal parts(shares) and take m such parts.

To get the fraction \(\frac(m)(n)\), you need to divide the number m by the number n.

To find a part of a whole, you need to divide the number corresponding to the whole by the denominator and multiply the result by the numerator of the fraction that expresses this part.

To find a whole from its part, you need to divide the number corresponding to this part by the numerator and multiply the result by the denominator of the fraction that expresses this part.

If both the numerator and denominator of a fraction are multiplied by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a \cdot n)(b \cdot n) \)

If both the numerator and denominator of a fraction are divided by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a: m)(b: m) \)
This property is called main property of a fraction.

The last two transformations are called reducing a fraction.

If fractions need to be represented as fractions with the same denominator, then this action is called bringing fractions to a common denominator.

Proper and improper fractions. Mixed numbers

You already know that a fraction can be obtained by dividing a whole into equal parts and taking several such parts. For example, the fraction \(\frac(3)(4)\) means three-quarters of one. In many of the problems in the previous paragraph, fractions were used to represent parts of a whole. Common sense dictates that the part should always be less than the whole, but what about fractions such as \(\frac(5)(5)\) or \(\frac(8)(5)\)? It is clear that this is no longer part of the unit. This is probably why fractions whose numerator is greater than or equal to the denominator are called improper fractions. The remaining fractions, i.e. fractions whose numerator is less than the denominator, are called correct fractions.

As you know, any common fraction, both proper and improper, can be thought of as the result of dividing the numerator by the denominator. Therefore, in mathematics, unlike ordinary language, the term “improper fraction” does not mean that we did something wrong, but only that the numerator of this fraction is greater than or equal to the denominator.

If a number consists of an integer part and a fraction, then such fractions are called mixed.

For example:
\(5:3 = 1\frac(2)(3) \) : 1 is the integer part, and \(\frac(2)(3) \) is the fractional part.

If the numerator of the fraction \(\frac(a)(b) \) is divisible by a natural number n, then in order to divide this fraction by n, its numerator must be divided by this number:
\(\large \frac(a)(b) : n = \frac(a:n)(b) \)

If the numerator of the fraction \(\frac(a)(b)\) is not divisible by a natural number n, then to divide this fraction by n, you need to multiply its denominator by this number:
\(\large \frac(a)(b) : n = \frac(a)(bn) \)

Note that the second rule is also true when the numerator is divisible by n. Therefore, we can use it when it is difficult to determine at first glance whether the numerator of a fraction is divisible by n or not.

Actions with fractions. Adding fractions.

You can perform arithmetic operations with fractional numbers, just like with natural numbers. Let's look at adding fractions first. It's easy to add fractions with like denominators. Let us find, for example, the sum of \(\frac(2)(7)\) and \(\frac(3)(7)\). It is easy to understand that \(\frac(2)(7) + \frac(2)(7) = \frac(5)(7) \)

To add fractions with the same denominators, you need to add their numerators and leave the denominator the same.

Using letters, the rule for adding fractions with like denominators can be written as follows:
\(\large \frac(a)(c) + \frac(b)(c) = \frac(a+b)(c) \)

If you need to add fractions with different denominators, they must first be reduced to a common denominator. For example:
\(\large \frac(2)(3)+\frac(4)(5) = \frac(2\cdot 5)(3\cdot 5)+\frac(4\cdot 3)(5\cdot 3 ) = \frac(10)(15)+\frac(12)(15) = \frac(10+12)(15) = \frac(22)(15) \)

For fractions, as for natural numbers, the commutative and associative properties of addition are valid.

Adding mixed fractions

Notations such as \(2\frac(2)(3)\) are called mixed fractions. In this case, the number 2 is called whole part mixed fraction, and the number \(\frac(2)(3)\) is its fractional part. The entry \(2\frac(2)(3)\) is read as follows: “two and two thirds.”

When dividing the number 8 by the number 3, you can get two answers: \(\frac(8)(3)\) and \(2\frac(2)(3)\). They express the same fractional number, i.e. \(\frac(8)(3) = 2 \frac(2)(3)\)

Thus, the improper fraction \(\frac(8)(3)\) is represented as a mixed fraction \(2\frac(2)(3)\). In such cases they say that from an improper fraction highlighted the whole part.

Subtracting fractions (fractional numbers)

Subtraction fractional numbers, like natural numbers, is determined on the basis of the action of addition: subtracting another from one number means finding a number that, when added to the second, gives the first. For example:
\(\frac(8)(9)-\frac(1)(9) = \frac(7)(9) \) since \(\frac(7)(9)+\frac(1)(9 ) = \frac(8)(9)\)

The rule for subtracting fractions with like denominators is similar to the rule for adding such fractions:
To find the difference between fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and leave the denominator the same.

Using letters, this rule is written like this:
\(\large \frac(a)(c)-\frac(b)(c) = \frac(a-b)(c) \)

Multiplying fractions

To multiply a fraction by a fraction, you need to multiply their numerators and denominators and write the first product as the numerator, and the second as the denominator.

Using letters, the rule for multiplying fractions can be written as follows:
\(\large \frac(a)(b) \cdot \frac(c)(d) = \frac(a \cdot c)(b \cdot d) \)

Using the formulated rule, you can multiply a fraction by a natural number, by a mixed fraction, and also multiply mixed fractions. To do this, you need to write a natural number as a fraction with a denominator of 1, a mixed fraction - as an improper fraction.

The result of multiplication should be simplified (if possible) by reducing the fraction and isolating the whole part of the improper fraction.

For fractions, as for natural numbers, the commutative and combinative properties of multiplication, as well as the distributive property of multiplication relative to addition, are valid.

Division of fractions

Let's take the fraction \(\frac(2)(3)\) and “flip” it, swapping the numerator and denominator. We get the fraction \(\frac(3)(2)\). This fraction is called reverse fractions \(\frac(2)(3)\).

If we now “reverse” the fraction \(\frac(3)(2)\), we will get the original fraction \(\frac(2)(3)\). Therefore, fractions such as \(\frac(2)(3)\) and \(\frac(3)(2)\) are called mutually inverse.

For example, the fractions \(\frac(6)(5) \) and \(\frac(5)(6) \), \(\frac(7)(18) \) and \(\frac (18)(7)\).

Using letters, reciprocal fractions can be written as follows: \(\frac(a)(b) \) and \(\frac(b)(a) \)

It is clear that the product of reciprocal fractions is equal to 1. For example: \(\frac(2)(3) \cdot \frac(3)(2) =1 \)

Using reciprocal fractions, you can reduce division of fractions to multiplication.

The rule for dividing a fraction by a fraction is:
To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor.

This article is a general look at operating with fractions. Here we will formulate and justify the rules for addition, subtraction, multiplication, division and exponentiation of fractions of the general form A/B, where A and B are some numbers, numerical expressions or expressions with variables. As usual, we will provide the material with explanatory examples with detailed descriptions of solutions.

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Rules for performing operations with general numerical fractions

Let's agree on numerical fractions general view understand fractions in which the numerator and/or denominator can be represented not only by natural numbers, but also by other numbers or numerical expressions. For clarity, here are a few examples of such fractions: , .

We know the rules by which they are carried out. Using the same rules, you can perform operations with general fractions:

Rationale for the rules

To justify the validity of the rules for performing operations with numerical fractions of a general form, you can start from the following points:

• The slash is essentially a division sign,
• division by some nonzero number can be considered as multiplication by the inverse of the divisor (this immediately explains the rule dividing fractions),
• properties of operations with real numbers,
• and its general understanding,

They allow you to carry out the following transformations that justify the rules of addition, subtraction of fractions with like and unlike denominators, as well as the rule of multiplication of fractions:

Examples

Let us give examples of performing operations with general fractions according to the rules learned in the previous paragraph. Let's say right away that usually after performing operations with fractions, the resulting fraction requires simplification, and the process of simplifying a fraction is often more complicated than performing previous actions. We will not dwell in detail on simplifying fractions (the corresponding transformations are discussed in the article transforming fractions), so as not to be distracted from the topic that interests us.

Let's start with examples of adding and subtracting fractions with like denominators. First, let's add the fractions and . Obviously the denominators are equal. According to the corresponding rule, we write down a fraction whose numerator is equal to the sum of the numerators of the original fractions, and leave the denominator the same, we have. The addition is done, all that remains is to simplify the resulting fraction: . So, .

The solution could have been handled differently: first make the transition to ordinary fractions, and then carry out the addition. With this approach we have .

Now let's subtract from the fraction fraction . The denominators of the fractions are equal, therefore, we follow the rule for subtracting fractions with the same denominators:

Let's move on to examples of adding and subtracting fractions with different denominators. The main difficulty here is bringing fractions to a common denominator. For general fractions, this is a rather extensive topic; we will examine it in detail in a separate article. reducing fractions to a common denominator. Now let's limit ourselves to a couple general recommendations, since in this moment we are more interested in the technique of performing operations with fractions.

In general, the process is similar to reducing ordinary fractions to a common denominator. That is, the denominators are presented in the form of products, then all the factors from the denominator of the first fraction are taken and the missing factors from the denominator of the second fraction are added to them.

When the denominators of fractions being added or subtracted do not have common factors, then it is logical to take their product as the common denominator. Let's give an example.

Let's say we need to perform addition of fractions and 1/2. Here, as a common denominator, it is logical to take the product of the denominators of the original fractions, that is, . In this case, the additional factor for the first fraction will be 2. After multiplying the numerator and denominator by it, the fraction will take the form . And for the second fraction, the additional factor is the expression. With its help, the fraction 1/2 is reduced to the form . All that remains is to add the resulting fractions with the same denominators. Here's a summary of the entire solution:

In the case of general fractions, we are no longer talking about the lowest common denominator, to which ordinary fractions are usually reduced. Although in this matter it is still advisable to strive for some minimalism. By this we want to say that you should not immediately take the product of the denominators of the original fractions as a common denominator. For example, it is not at all necessary to take the common denominator of fractions and the product . Here we can take .

Let's move on to examples of multiplying general fractions. Let's multiply fractions and . The rule for performing this action instructs us to write down a fraction, the numerator of which is the product of the numerators of the original fractions, and the denominator is the product of the denominators. We have . Here, as in many other cases when multiplying fractions, you can reduce the fraction: .

The rule for dividing fractions allows you to move from division to multiplication by the reciprocal fraction. Here you need to remember that in order to get the inverse of a given fraction, you need to swap the numerator and denominator of the given fraction. Here is an example of the transition from division of general numerical fractions to multiplication: . All that remains is to perform the multiplication and simplify the resulting fraction (if necessary, see the transformation of irrational expressions):

Concluding the information in this paragraph, let us recall that any number or numeric expression can be represented as a fraction with a denominator of 1, therefore, adding, subtracting, multiplying and dividing numbers and fractions can be thought of as performing the corresponding operation with fractions, one of which has one in the denominator. For example, replacing in the expression root of three by a fraction, we move from multiplying a fraction by a number to multiplying two fractions: .

Doing things with fractions that contain variables

The rules from the first part of this article also apply to performing operations with fractions that contain variables. Let's justify the first of them - the rule for adding and subtracting fractions with identical denominators, the rest are proven in absolutely the same way.

Let us prove that for any expressions A, C and D (D is not identically equal to zero) the equality holds on its range of permissible values ​​of variables.

Let's take a certain set of variables from the ODZ. Let the expressions A, C and D take the values ​​a 0, c 0 and d 0 for these values ​​of the variables. Then substituting the values ​​of variables from the selected set into the expression turns it into a sum (difference) of numerical fractions with like denominators of the form , which, according to the rule of addition (subtraction) of numerical fractions with like denominators, is equal to . But substituting the values ​​of variables from the selected set into the expression turns it into the same fraction. This means that for the selected set of variable values ​​from the ODZ, the values ​​of the expressions and are equal. It is clear that the values ​​of the indicated expressions will be equal for any other set of values ​​of variables from the ODZ, which means that the expressions and are identically equal, that is, the equality being proved is true .

Examples of adding and subtracting fractions with variables

When the denominators of the fractions being added or subtracted are the same, then everything is quite simple - the numerators are added or subtracted, but the denominator remains the same. It is clear that the fraction obtained after this is simplified if necessary and possible.

Note that sometimes the denominators of fractions differ only at first glance, but in fact they are identically equal expressions, for example, and , or and . And sometimes it is enough to simplify the original fractions so that their identical denominators “appear.”

Example.

, b) , V) .

Solution.

a) We need to subtract fractions with like denominators. According to the corresponding rule, we leave the denominator the same and subtract the numerators, we have . The action has been completed. But you can also open the parentheses in the numerator and present similar terms: .

b) Obviously, the denominators of the fractions being added are the same. Therefore, we add up the numerators and leave the denominator the same: . Addition completed. But it is easy to see that the resulting fraction can be reduced. Indeed, the numerator of the resulting fraction can be collapsed using the formula square of the sum as (lgx+2) 2 (see formulas for abbreviated multiplication), thus the following transformations take place: .

c) Fractions in sum have different denominators. But, having transformed one of the fractions, you can move on to adding fractions with the same denominators. We will show two solutions.

First way. The denominator of the first fraction can be factorized using the difference of squares formula, and then reduce this fraction: . Thus, . It still doesn’t hurt to free yourself from irrationality in the denominator of the fraction: .

Second way. Multiplying the numerator and denominator of the second fraction by (this expression does not go to zero for any value of the variable x from the ODZ for the original expression) allows you to achieve two goals at once: free yourself from irrationality and move on to adding fractions with the same denominators. We have

Answer:

A) , b) , V) .

The last example brought us to the question of reducing fractions to a common denominator. There we almost accidentally arrived at the same denominators by simplifying one of the added fractions. But in most cases, when adding and subtracting fractions with different denominators, you have to purposefully bring the fractions to a common denominator. To do this, usually the denominators of fractions are presented in the form of products, all the factors from the denominator of the first fraction are taken and the missing factors from the denominator of the second fraction are added to them.

Example.

Perform operations with fractions: a) , b) , c) .

Solution.

a) There is no need to do anything with the denominators of the fractions. As a common denominator we take the product . In this case, the additional factor for the first fraction is the expression, and for the second fraction - the number 3. These additional factors bring the fractions to a common denominator, which later allows us to perform the action we need, we have

b) In this example, the denominators are already represented as products and do not require any additional transformations. Obviously, the factors in the denominators differ only in exponents, therefore, as a common denominator we take the product of the factors with the highest exponents, that is, . Then the additional factor for the first fraction will be x 4, and for the second – ln(x+1) . Now we're ready to subtract fractions:

c) And in this case, first we will work with the denominators of fractions. The formulas for the difference of squares and the square of the sum allow you to move from the original sum to the expression . Now it is clear that these fractions can be reduced to a common denominator . With this approach, the solution will have next view:

Answer:

A)

b)

V)

Examples of multiplying fractions with variables

Multiplying fractions produces a fraction whose numerator is the product of the numerators of the original fractions, and the denominator is the product of the denominators. Here, as you can see, everything is familiar and simple, and we can only add that the fraction obtained as a result of this action often turns out to be reducible. In these cases, it is reduced, unless, of course, it is necessary and justified.

Almost every fifth grader after the first acquaintance with ordinary fractions and is in a little shock. Not only do you need to understand the essence of fractions, but you also have to perform arithmetic operations with them. After this, the little students will systematically interrogate their teacher to find out when these fractions will end.

To avoid such situations, it is enough just to explain this difficult topic to children as simply as possible, and better yet, game form.

The essence of a fraction

Before learning what a fraction is, a child must become familiar with the concept share . The associative method is best suited here.

Imagine a whole cake that is divided into several equal parts, say four. Then each piece of the cake can be called a share. If you take one of the four pieces of cake, it will be one-fourth.

The shares are different, because the whole can be divided into completely different quantities parts. The more shares in general, the smaller they are, and vice versa.

So that the shares could be designated, we came up with this mathematical concept, How common fraction. The fraction will allow us to write down as many shares as needed.

The components of a fraction are the numerator and denominator, which are separated by a fraction line or a slash. Many children do not understand their meaning, and therefore the essence of the fraction is not clear to them. The fractional line indicates division, there is nothing complicated here.

It is customary to write the denominator below, under the fractional line or to the right of the forward line. It shows the number of parts of a whole. The numerator, it is written above the fraction line or to the left of the forward line, determines how many shares were taken. For example, the fraction 4/7. In this case, 7 is the denominator, showing that there are only 7 shares, and the numerator 4 indicates that four of the seven shares were taken.

Main shares and their writing in fractions:

In addition to the ordinary fraction, there is also a decimal fraction.

Operations with fractions 5th grade

In the fifth grade they learn to perform all arithmetic operations with fractions.

All operations with fractions are performed according to the rules, and you should not hope that without learning the rule everything will work out on its own. Therefore, you should not neglect oral part math homework.

We have already understood that the notation of a decimal and an ordinary fraction is different, therefore arithmetic operations will be performed differently. Actions with ordinary fractions depend on the numbers that are in the denominator, and in the decimal - after the decimal point to the right.

For fractions that have the same denominators, the algorithm for addition and subtraction is very simple. We perform actions only with numerators.

For fractions with different denominators you need to find Least Common Denominator (LCD). This is the number that will be divisible by all denominators without a remainder, and will be the smallest of such numbers if there are several of them.

To add or subtract decimal fractions, you need to write them in a column, with a comma under the comma, and equalize the number of decimal places if required.

To multiply ordinary fractions, simply find the product of the numerators and denominators. A very simple rule.

The division is performed according to the following algorithm:

1. Write the dividend unchanged
2. Turn division into multiplication
3. Reverse the divisor (write the reciprocal fraction to the divisor)
4. Perform multiplication

Addition of fractions, explanation

Let's take a closer look at how to add fractions and decimals.

As you can see in the image above, the fraction one third and two thirds has a common denominator of three. This means that you only need to add the numerators one and two, and leave the denominator unchanged. The result is a sum of three thirds. This answer, when the numerator and denominator of the fraction are equal, can be written as 1, since 3:3 = 1.

You need to find the sum of the fractions two thirds and two ninths. In this case, the denominators are different, 3 and 9. To perform addition, you need to find a common one. There is a very simple way. We choose the largest denominator, it is 9. We check whether it is divisible by 3. Since 9:3 = 3 without a remainder, therefore 9 is suitable as a common denominator.

The next step is to find additional factors for each numerator. To do this, we divide the common denominator 9 by the denominator of each fraction in turn, the resulting numbers will be additional. plural For the first fraction: 9:3 = 3, add 3 to the numerator of the first fraction. For the second fraction: 9:9 = 1, you don’t have to add one, since when multiplied by it you get the same number.

Now we multiply the numerators by their additional factors and add the results. The resulting amount is a fraction of eight-ninths.

Adding decimals follows the same rule as adding natural numbers. In a column, the digit is written under the digit. The only difference is that in decimal fractions you need to place the correct comma in the result. To do this, fractions are written with a comma under the comma, and in the total you only need to move the comma down.

Let's find the sum of the fractions 38, 251 and 1, 56. To make it more convenient to perform the actions, we equalized the number of decimal places on the right by adding 0.

Add fractions without paying attention to the comma. And in the resulting amount we simply lower the comma down. Answer: 39, 811.

Subtracting fractions, explanation

To find the difference between the fractions two-thirds and one-third, you need to calculate the difference of the numerators 2-1 = 1, and leave the denominator unchanged. The answer gives a difference of one third.

Let's find the difference between the fractions five-sixths and seven-tenths. Finding a common denominator. We use the selection method, from 6 and 10 the largest is 10. We check: 10: 6 is not divisible without a remainder. We add another 10, it turns out 20:6, which is also not divisible without a remainder. Again we increase by 10, we get 30:6 = 5. The common denominator is 30. Also, the NOZ can be found using the multiplication table.

Finding additional factors. 30:6 = 5 - for the first fraction. 30:10 = 3 - for the second. We multiply the numerators and their additional multiplicities. We get the minuend 25/30 and the subtract 21/30. Next, we subtract the numerators and leave the denominator unchanged.

The result was a difference of 4/30. The fraction is reducible. Divide it by 2. The answer is 2/15.

Dividing decimals grade 5

This topic discusses two options:

Multiplying decimals grade 5

Remember how you multiply integers, in exactly the same way the product of decimal fractions is found. First, let's figure out how to multiply a decimal fraction by a natural number. For this:

When multiplying a decimal fraction by a decimal, we act in exactly the same way.

Mixed Fractions Grade 5

Fifth graders like to call such fractions not mixed, but<<смешные>>It's probably easier to remember this way. Mixed fractions are so called because they are made by combining a whole natural number and an ordinary fraction.

A mixed fraction consists of an integer and a fractional part.

When reading such fractions, first they name the whole part, then the fractional part: one whole two thirds, two whole one fifth, three whole two fifths, four point three quarters.

How are they obtained, these mixed fractions? It's quite simple. When we receive an improper fraction in an answer (a fraction whose numerator is greater than the denominator), we must always convert it to a mixed fraction. It is enough to divide the numerator by the denominator. This action is called selecting an entire part:

Converting a mixed fraction back to an improper fraction is also easy:

Examples with decimal fractions grade 5 with explanation

Examples of several actions raise many questions in children. Let's look at a couple of such examples.

(0.4 8.25 - 2.025) : 0.5 =

The first step is to find the product of the numbers 8.25 and 0.4. We perform multiplication according to the rule. In the answer, count three digits from right to left and put a comma.

The second action is there in brackets, this is the difference. From 3,300 we subtract 2,025. We record the action in a column with a comma under the comma.

The third action is division. The resulting difference in the second step is divided by 0.5. The comma is moved one place. Result 2.55.

Answer: 2.55.

(0, 93 + 0, 07) : (0, 93 — 0, 805) =

The first step is the amount in brackets. Add it in a column, remember that the comma is under the comma. We get the answer 1.00.

The second action is the difference from the second bracket. Since the minuend has fewer decimal places than the subtrahend, we add the missing one. The result of the subtraction is 0.125.

The third step is to divide the sum by the difference. The comma is moved three places. The result is a division of 1000 by 125.

Answer: 8.

Examples with ordinary fractions with different denominators grade 5 with explanation

In the first In this example, we find the sum of the fractions 5/8 and 3/7. The common denominator will be the number 56. Find additional factors, divide 56:8 = 7 and 56:7 = 8. Add them to the first and second fractions, respectively. We multiply the numerators and their factors, we get the sum of the fractions 35/56 and 24/56. The result was 59/56. The fraction is improper, we convert it to a mixed number. The remaining examples are solved similarly.

Examples with fractions grade 5 for training

For convenience, convert mixed fractions to improper fractions and perform the operations.

How to teach your child to solve fractions easily using Legos

With the help of such a constructor, you can not only develop a child’s imagination, but also explain clearly in a playful way what a share and a fraction are.

The picture below shows that one part with eight circles is a whole. This means that if you take a puzzle with four circles, you get half, or 1/2. The picture clearly shows how to solve examples with Lego, if you count the circles on the parts.

You can build towers from a certain number of parts and label each of them, as in the picture below. For example, let's take a seven-piece turret. Each piece of the green construction set will be 1/7. If you add two more to one such part, you get 3/7. A visual explanation of the example 1/7+2/7 = 3/7.

To get A's in math, don't forget to learn the rules and practice them.

This article examines operations on fractions. Rules for addition, subtraction, multiplication, division or exponentiation of fractions of the form A B will be formed and justified, where A and B can be numbers, numerical expressions or expressions with variables. In conclusion, examples of solutions with detailed descriptions will be considered.

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Rules for performing operations with general numerical fractions

General fractions have a numerator and a denominator that contain natural numbers or numerical expressions. If we consider fractions such as 3 5, 2, 8 4, 1 + 2 3 4 (5 - 2), 3 4 + 7 8 2, 3 - 0, 8, 1 2 2, π 1 - 2 3 + π, 2 0, 5 ln 3, then it is clear that the numerator and denominator can have not only numbers, but also expressions of various types.

Definition 1

There are rules by which operations with ordinary fractions are carried out. It is also suitable for general fractions:

• When subtracting fractions with like denominators, only the numerators are added, and the denominator remains the same, namely: a d ± c d = a ± c d, the values ​​a, c and d ≠ 0 are some numbers or numerical expressions.
• When adding or subtracting a fraction with different denominators, it is necessary to reduce it to a common denominator, and then add or subtract the resulting fractions with the same exponents. Literally it looks like this: a b ± c d = a · p ± c · r s, where the values ​​a, b ≠ 0, c, d ≠ 0, p ≠ 0, r ≠ 0, s ≠ 0 are real numbers, and b · p = d · r = s . When p = d and r = b, then a b ± c d = a · d ± c · d b · d.
• When multiplying fractions, the action is performed with numerators, after which with denominators, then we get a b · c d = a · c b · d, where a, b ≠ 0, c, d ≠ 0 act as real numbers.
• When dividing a fraction by a fraction, we multiply the first by the second inverse, that is, we swap the numerator and denominator: a b: c d = a b · d c.

Rationale for the rules

Definition 2

There are the following mathematical points that you should rely on when calculating:

• the slash means the division sign;
• division by a number is treated as multiplication by its reciprocal value;
• application of the property of operations with real numbers;
• application of the basic property of fractions and numerical inequalities.

With their help, you can perform transformations of the form:

a d ± c d = a · d - 1 ± c · d - 1 = a ± c · d - 1 = a ± c d ; a b ± c d = a · p b · p ± c · r d · r = a · p s ± c · e s = a · p ± c · r s ; a b · c d = a · d b · d · b · c b · d = a · d · a · d - 1 · b · c · b · d - 1 = = a · d · b · c · b · d - 1 · b · d - 1 = a · d · b · c b · d · b · d - 1 = = (a · c) · (b · d) - 1 = a · c b · d

Examples

In the previous paragraph it was said about operations with fractions. It is after this that the fraction needs to be simplified. This topic was discussed in detail in the paragraph on converting fractions.

First, let's look at an example of adding and subtracting fractions with the same denominator.

Example 1

Given the fractions 8 2, 7 and 1 2, 7, then according to the rule it is necessary to add the numerator and rewrite the denominator.

Solution

Then we get a fraction of the form 8 + 1 2, 7. After performing the addition, we obtain a fraction of the form 8 + 1 2, 7 = 9 2, 7 = 90 27 = 3 1 3. So, 8 2, 7 + 1 2, 7 = 8 + 1 2, 7 = 9 2, 7 = 90 27 = 3 1 3.

Answer: 8 2 , 7 + 1 2 , 7 = 3 1 3

There is another solution. To begin with, we switch to the form of an ordinary fraction, after which we perform a simplification. It looks like this:

8 2 , 7 + 1 2 , 7 = 80 27 + 10 27 = 90 27 = 3 1 3

Example 2

Let's subtract from 1 - 2 3 · log 2 3 · log 2 5 + 1 a fraction of the form 2 3 3 · log 2 3 · log 2 5 + 1 .

Since equal denominators are given, it means that we are calculating a fraction with the same denominator. We get that

1 - 2 3 log 2 3 log 2 5 + 1 - 2 3 3 log 2 3 log 2 5 + 1 = 1 - 2 - 2 3 3 log 2 3 log 2 5 + 1

There are examples of calculating fractions with different denominators. An important point is reduction to a common denominator. Without this we will not be able to carry out further actions with fractions.

The process is vaguely reminiscent of reduction to a common denominator. That is, the least common divisor in the denominator is searched for, after which the missing factors are added to the fractions.

If the fractions being added do not have common factors, then their product can become one.

Example 3

Let's look at the example of adding fractions 2 3 5 + 1 and 1 2.

Solution

In this case, the common denominator is the product of the denominators. Then we get that 2 · 3 5 + 1. Then, when setting additional factors, we have that for the first fraction it is equal to 2, and for the second it is 3 5 + 1. After multiplication, the fractions are reduced to the form 4 2 · 3 5 + 1. The general reduction of 1 2 will be 3 5 + 1 2 · 3 5 + 1. We add the resulting fractional expressions and get that

2 3 5 + 1 + 1 2 = 2 2 2 3 5 + 1 + 1 3 5 + 1 2 3 5 + 1 = = 4 2 3 5 + 1 + 3 5 + 1 2 3 5 + 1 = 4 + 3 5 + 1 2 3 5 + 1 = 5 + 3 5 2 3 5 + 1

Answer: 2 3 5 + 1 + 1 2 = 5 + 3 5 2 3 5 + 1

When we are dealing with general fractions, then we usually do not talk about the lowest common denominator. It is unprofitable to take the product of the numerators as the denominator. First you need to check if there is a number that is less in value than their product.

Example 4

Let's consider the example of 1 6 · 2 1 5 and 1 4 · 2 3 5, when their product is equal to 6 · 2 1 5 · 4 · 2 3 5 = 24 · 2 4 5. Then we take 12 · 2 3 5 as the common denominator.

Let's look at examples of multiplying general fractions.

Example 5

To do this, you need to multiply 2 + 1 6 and 2 · 5 3 · 2 + 1.

Solution

Following the rule, it is necessary to rewrite and write the product of the numerators as a denominator. We get that 2 + 1 6 2 5 3 2 + 1 2 + 1 2 5 6 3 2 + 1. Once a fraction has been multiplied, you can make reductions to simplify it. Then 5 · 3 3 2 + 1: 10 9 3 = 5 · 3 3 2 + 1 · 9 3 10.

Using the rule for transition from division to multiplication by a reciprocal fraction, we obtain a fraction that is the reciprocal of the given one. To do this, the numerator and denominator are swapped. Let's look at an example:

5 3 3 2 + 1: 10 9 3 = 5 3 3 2 + 1 9 3 10

Then they must multiply and simplify the resulting fraction. If necessary, get rid of irrationality in the denominator. We get that

5 3 3 2 + 1: 10 9 3 = 5 3 3 9 3 10 2 + 1 = 5 2 10 2 + 1 = 3 2 2 + 1 = 3 2 - 1 2 2 + 1 2 - 1 = 3 2 - 1 2 2 2 - 1 2 = 3 2 - 1 2

Answer: 5 3 3 2 + 1: 10 9 3 = 3 2 - 1 2

This paragraph is applicable when a number or numerical expression can be represented as a fraction with a denominator equal to 1, then the operation with such a fraction is considered a separate paragraph. For example, the expression 1 6 · 7 4 - 1 · 3 shows that the root of 3 can be replaced by another 3 1 expression. Then this entry will look like multiplying two fractions of the form 1 6 · 7 4 - 1 · 3 = 1 6 · 7 4 - 1 · 3 1.

Performing Operations on Fractions Containing Variables

The rules discussed in the first article are applicable to operations with fractions containing variables. Consider the subtraction rule when the denominators are the same.

It is necessary to prove that A, C and D (D not equal to zero) can be any expressions, and the equality A D ± C D = A ± C D is equivalent to its domain acceptable values.

It is necessary to take a set of ODZ variables. Then A, C, D must take the corresponding values ​​a 0 , c 0 and d 0. Substitution of the form A D ± C D results in a difference of the form a 0 d 0 ± c 0 d 0 , where, using the addition rule, we obtain a formula of the form a 0 ± c 0 d 0 . If we substitute the expression A ± C D, then we get the same fraction of the form a 0 ± c 0 d 0. From here we conclude that the selected value that satisfies the ODZ, A ± C D and A D ± C D are considered equal.

For any value of the variables, these expressions will be equal, that is, they are called identically equal. This means that this expression is considered a provable equality of the form A D ± C D = A ± C D .

Examples of adding and subtracting fractions with variables

When you have the same denominators, you only need to add or subtract the numerators. This fraction can be simplified. Sometimes you have to work with fractions that are identically equal, but at first glance this is not noticeable, since some transformations must be performed. For example, x 2 3 x 1 3 + 1 and x 1 3 + 1 2 or 1 2 sin 2 α and sin a cos a. Most often, a simplification of the original expression is required in order to see the same denominators.

Example 6

Calculate: 1) x 2 + 1 x + x - 2 - 5 - x x + x - 2, 2) l g 2 x + 4 x · (l g x + 2) + 4 · l g x x · (l g x + 2) , x - 1 x - 1 + x x + 1 .

Solution

1. To make the calculation, you need to subtract fractions that have the same denominator. Then we get that x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + 1 - 5 - x x + x - 2 . After which you can expand the brackets and add similar terms. We get that x 2 + 1 - 5 - x x + x - 2 = x 2 + 1 - 5 + x x + x - 2 = x 2 + x - 4 x + x - 2
2. Since the denominators are the same, all that remains is to add the numerators, leaving the denominator: l g 2 x + 4 x (l g x + 2) + 4 l g x x (l g x + 2) = l g 2 x + 4 + 4 x (l g x + 2)
   The addition has been completed. It can be seen that it is possible to reduce the fraction. Its numerator can be folded using the formula for the square of the sum, then we get (l g x + 2) 2 from abbreviated multiplication formulas. Then we get that
   l g 2 x + 4 + 2 l g x x (l g x + 2) = (l g x + 2) 2 x (l g x + 2) = l g x + 2 x
3. Given fractions of the form x - 1 x - 1 + x x + 1 with different denominators. After the transformation, you can move on to addition.

Let's consider a twofold solution.

The first method is that the denominator of the first fraction is factorized using squares, with its subsequent reduction. We get a fraction of the form

x - 1 x - 1 = x - 1 (x - 1) x + 1 = 1 x + 1

So x - 1 x - 1 + x x + 1 = 1 x + 1 + x x + 1 = 1 + x x + 1 .

In this case, it is necessary to get rid of irrationality in the denominator.

1 + x x + 1 = 1 + x x - 1 x + 1 x - 1 = x - 1 + x x - x x - 1

The second method is to multiply the numerator and denominator of the second fraction by the expression x - 1. Thus, we get rid of irrationality and move on to adding fractions with the same denominator. Then

x - 1 x - 1 + x x + 1 = x - 1 x - 1 + x x - 1 x + 1 x - 1 = = x - 1 x - 1 + x x - x x - 1 = x - 1 + x · x - x x - 1

Answer: 1) x 2 + 1 x + x - 2 - 5 - x x + x - 2 = x 2 + x - 4 x + x - 2, 2) l g 2 x + 4 x · (l g x + 2) + 4 · l g x x · (l g x + 2) = l g x + 2 x, 3) x - 1 x - 1 + x x + 1 = x - 1 + x · x - x x - 1 .

In the last example we found that reduction to a common denominator is inevitable. To do this, you need to simplify the fractions. When adding or subtracting, you always need to look for a common denominator, which looks like the product of the denominators with additional factors added to the numerators.

Example 7

Calculate the values ​​of the fractions: 1) x 3 + 1 x 7 + 2 2, 2) x + 1 x ln 2 (x + 1) (2 x - 4) - sin x x 5 ln (x + 1) (2 x - 4) , 3) ​​1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x

Solution

1. The denominator does not require any complex calculations, so you need to choose their product of the form 3 x 7 + 2 · 2, then choose x 7 + 2 · 2 for the first fraction as an additional factor, and 3 for the second. When multiplying, we get a fraction of the form x 3 + 1 x 7 + 2 2 = x x 7 + 2 2 3 x 7 + 2 2 + 3 1 3 x 7 + 2 2 = = x x 7 + 2 2 + 3 3 x 7 + 2 2 = x x 7 + 2 2 x + 3 3 x 7 + 2 2
2. It can be seen that the denominators are presented in the form of a product, which means that additional transformations are unnecessary. The common denominator will be considered to be a product of the form x 5 · ln 2 x + 1 · 2 x - 4 . Hence x 4 is an additional factor to the first fraction, and ln(x + 1) to the second. Then we subtract and get:
   x + 1 x · ln 2 (x + 1) · 2 x - 4 - sin x x 5 · ln (x + 1) · 2 x - 4 = = x + 1 · x 4 x 5 · ln 2 (x + 1 ) · 2 x - 4 - sin x · ln x + 1 x 5 · ln 2 (x + 1) · (2 ​​x - 4) = = x + 1 · x 4 - sin x · ln (x + 1) x 5 · ln 2 (x + 1) · (2 ​​x - 4) = x · x 4 + x 4 - sin x · ln (x + 1) x 5 · ln 2 (x + 1) · (2 ​​x - 4 )
3. This example makes sense when working with fraction denominators. It is necessary to apply the formulas for the difference of squares and the square of the sum, since they will make it possible to move on to an expression of the form 1 cos x - x · cos x + x + 1 (cos x + x) 2. It can be seen that the fractions are reduced to a common denominator. We get that cos x - x · cos x + x 2 .

Then we get that

1 cos 2 x - x + 1 cos 2 x + 2 cos x x + x = = 1 cos x - x cos x + x + 1 cos x + x 2 = = cos x + x cos x - x cos x + x 2 + cos x - x cos x - x cos x + x 2 = = cos x + x + cos x - x cos x - x cos x + x 2 = 2 cos x cos x - x cos x + x 2

Answer:

1) x 3 + 1 x 7 + 2 2 = x x 7 + 2 2 x + 3 3 x 7 + 2 2, 2) x + 1 x ln 2 (x + 1) 2 x - 4 - sin x x 5 · ln (x + 1) · 2 x - 4 = = x · x 4 + x 4 - sin x · ln (x + 1) x 5 · ln 2 (x + 1) · ( 2 x - 4) , 3) ​​1 cos 2 x - x + 1 cos 2 x + 2 · cos x · x + x = 2 · cos x cos x - x · cos x + x 2 .

Examples of multiplying fractions with variables

When multiplying fractions, the numerator is multiplied by the numerator and the denominator by the denominator. Then you can apply the reduction property.

Example 8

Multiply the fractions x + 2 · x x 2 · ln x 2 · ln x + 1 and 3 · x 2 1 3 · x + 1 - 2 sin 2 · x - x.

Solution

Multiplication needs to be done. We get that

x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x) = = x - 2 x 3 x 2 1 3 x + 1 - 2 x 2 ln x 2 ln x + 1 sin (2 x - x)

The number 3 is moved to the first place for the convenience of calculations, and you can reduce the fraction by x 2, then we get an expression of the form

3 x - 2 x x 1 3 x + 1 - 2 ln x 2 ln x + 1 sin (2 x - x)

Answer: x + 2 x x 2 ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x) = 3 x - 2 x x 1 3 x + 1 - 2 ln x 2 · ln x + 1 · sin (2 · x - x) .

Division

Division of fractions is similar to multiplication, since the first fraction is multiplied by the second reciprocal. If we take for example the fraction x + 2 x x 2 ln x 2 ln x + 1 and divide by 3 x 2 1 3 x + 1 - 2 sin 2 x - x, then it can be written as

x + 2 · x x 2 · ln x 2 · ln x + 1: 3 · x 2 1 3 · x + 1 - 2 sin (2 · x - x) , then replace with a product of the form x + 2 · x x 2 · ln x 2 ln x + 1 3 x 2 1 3 x + 1 - 2 sin (2 x - x)

Exponentiation

Let's move on to considering operations with general fractions with exponentiation. If you have a degree with natural indicator, then the action is considered as multiplication of equal fractions. But it is recommended to use general approach, based on the properties of degrees. Any expressions A and C, where C is not identically equal to zero, and any real r on the ODZ for an expression of the form A C r the equality A C r = A r C r is valid. The result is a fraction raised to a power. For example, consider:

x 0, 7 - π · ln 3 x - 2 - 5 x + 1 2, 5 = = x 0, 7 - π · ln 3 x - 2 - 5 2, 5 x + 1 2, 5

Procedure for performing operations with fractions

Operations on fractions are performed according to certain rules. In practice, we notice that an expression may contain several fractions or fractional expressions. Then it is necessary to perform all actions in strict order: raise to a power, multiply, divide, then add and subtract. If there are parentheses, the first action is performed in them.

Example 9

Calculate 1 - x cos x - 1 c o s x · 1 + 1 x .

Solution

Since we have the same denominator, then 1 - x cos x and 1 c o s x, but subtractions cannot be performed according to the rule; first, the actions in parentheses are performed, then multiplication, and then addition. Then when calculating we get that

1 + 1 x = 1 1 + 1 x = x x + 1 x = x + 1 x

When substituting the expression into the original one, we get that 1 - x cos x - 1 cos x · x + 1 x. When multiplying fractions we have: 1 cos x · x + 1 x = x + 1 cos x · x. Having made all the substitutions, we get 1 - x cos x - x + 1 cos x · x. Now you need to work with fractions that have different denominators. We get:

x · 1 - x cos x · x - x + 1 cos x · x = x · 1 - x - 1 + x cos x · x = = x - x - x - 1 cos x · x = - x + 1 cos x x

Answer: 1 - x cos x - 1 c o s x · 1 + 1 x = - x + 1 cos x · x .

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