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How to solve subtraction of fractions. Online calculator. Calculating expressions with numerical fractions

This article is general view for operations 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 decisions.

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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 general numerical fractions, you can start from the following points:

The slash is essentially a division sign,
division by some non-zero 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. bringing fractions to a common denominator. Now let's limit ourselves to a couple general recommendations, since in at the 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 with these values expression variables A, C and D take the values a 0, c 0 and d 0. 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, as 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) .

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 factors with the highest rates, 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) A c in this case First, let's 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 look like this:

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.

When a student enters high school, mathematics is divided into two subjects: algebra and geometry. There are more and more concepts, the tasks are more and more difficult. Some people have difficulty understanding fractions. Missed the first lesson on this topic, and voila. fractions? A question that will torment throughout my school life.

The concept of an algebraic fraction

Let's start with a definition. Under algebraic fraction refers to the expressions P/Q, where P is the numerator and Q is the denominator. A number, a numerical expression, or a numerical-alphabetic expression can be hidden under a letter entry.

Before you wonder how to decide algebraic fractions, first you need to understand that such an expression is part of the whole.

As a rule, an integer is 1. The number in the denominator shows how many parts the unit is divided into. The numerator is needed to find out how many elements are taken. The fraction bar corresponds to the division sign. It is allowed to write a fractional expression as a mathematical operation “Division”. In this case, the numerator is the dividend, the denominator is the divisor.

Basic rule of common fractions

When students pass this topic at school, they are given examples to reinforce. To solve them correctly and find different paths from difficult situations, you need to apply the basic property of fractions.

It goes like this: If you multiply both the numerator and the denominator by the same number or expression (other than zero), the value of the common fraction does not change. A special case from of this rule is the division of both sides of an expression by the same number or polynomial. Similar transformations are called identical equalities.

Below we will look at how to solve addition and subtraction of algebraic fractions, multiplying, dividing and reducing fractions.

Mathematical operations with fractions

Let's look at how to solve, the main property of an algebraic fraction, and how to apply it in practice. If you need to multiply two fractions, add them, divide one by another, or subtract, you must always follow the rules.

Thus, for the operation of addition and subtraction, an additional factor must be found in order to bring the expressions to a common denominator. If the fractions are initially given with the same expressions Q, then this paragraph should be omitted. Once the common denominator is found, how do you solve algebraic fractions? You need to add or subtract numerators. But! It must be remembered that if there is a “-” sign in front of the fraction, all signs in the numerator are reversed. Sometimes you should not perform any substitutions or mathematical operations. It is enough to change the sign in front of the fraction.

The concept is often used as reducing fractions. This means the following: if the numerator and denominator are divided by an expression different from one (the same for both parts), then a new fraction is obtained. The dividend and divisor are smaller than before, but due to the basic rule of fractions they remain equal to the original example.

The purpose of this operation is to obtain a new irreducible expression. This problem can be solved by reducing the numerator and denominator by the largest common divisor. The operation algorithm consists of two points:

Finding gcd for both sides of the fraction.
Dividing the numerator and denominator by the found expression and obtaining an irreducible fraction equal to the previous one.

Below is a table showing the formulas. For convenience, you can print it out and carry it with you in a notebook. However, so that in the future, when solving a test or exam, there will be no difficulties in the question of how to solve algebraic fractions, these formulas must be learned by heart.

Several examples with solutions

From a theoretical point of view, the question of how to solve algebraic fractions is considered. The examples given in the article will help you better understand the material.

1. Convert fractions and bring them to a common denominator.

2. Convert fractions and bring them to a common denominator.

After studying the theoretical part and considering practical issues there shouldn't be any more.

Fraction- a number that consists of an integer number of fractions of a unit and is represented in the form: a/b

Numerator of fraction (a)- the number located above the fraction line and showing the number of shares into which the unit was divided.

Fraction denominator (b)- a number located under the fraction line and showing how many parts the unit is divided into.

2. Reducing fractions to a common denominator

3. Arithmetic operations on ordinary fractions And

3.1. Addition of ordinary fractions

3.2. Subtracting fractions

3.3. Multiplying common fractions

3.4. Dividing fractions

4. Reciprocal numbers

5. Decimals

6. Arithmetic operations on decimals

6.1. Adding Decimals

6.2. Subtracting Decimals

6.3. Multiplying Decimals

6.4. Decimal division

#1. The main property of a fraction

If the numerator and denominator of a fraction are multiplied or divided by the same number that is not equal to zero, you get a fraction equal to the given one.

3/7=3*3/7*3=9/21, that is, 3/7=9/21

**a/b=am/bm - this is what the main property of a fraction looks like.**

In other words, we get a fraction equal to the given one by multiplying or dividing the numerator and denominator of the original fraction by the same natural number.

If ad=bc, then two fractions a/b =c /d are considered equal.

For example, the fractions 3/5 and 9/15 will be equal, since 3*15=5*9, that is, 45=45

Reducing a fraction is the process of replacing a fraction in which the new fraction is equal to the original one, but with a smaller numerator and denominator.

It is customary to reduce fractions based on the basic property of the fraction.

For example, 45/60=15/ 20 =9/12=3/4 (the numerator and denominator are divided by the number 3, by 5 and by 15).

Irreducible fraction is a fraction of the form 3/4 , where the numerator and denominator are mutual prime numbers. The main purpose of reducing a fraction is to make the fraction irreducible.

2. Reducing fractions to a common denominator

To bring two fractions to a common denominator, you need to:

1) expand the denominator of each fraction into prime factors;

2) multiply the numerator and denominator of the first fraction by the missing ones

factors from the expansion of the second denominator;

3) multiply the numerator and denominator of the second fraction by the missing factors from the first expansion.

Examples: Reduce fractions to a common denominator.

Let's factor the denominators into simple factors: 18=3∙3∙2, 15=3∙5

Multiply the numerator and denominator of the fraction by the missing factor 5 from the second expansion.

numerator and denominator of the fraction into the missing factors 3 and 2 from the first expansion.

= , 90 – common denominator of fractions.

3. Arithmetic operations on ordinary fractions

3.1. Addition of ordinary fractions

a) When same denominators The numerator of the first fraction is added to the numerator of the second fraction, leaving the denominator the same. As you can see in the example:

a/b+c/b=(a+c)/b ;

b) For different denominators, fractions are first reduced to a common denominator, and then the numerators are added according to rule a):

7/3+1/4=7*4/12+1*3/12=(28+3)/12=31/12

3.2. Subtracting fractions

a) If the denominators are the same, subtract the numerator of the second fraction from the numerator of the first fraction, leaving the denominator the same:

a/b-c/b=(a-c)/b ;

b) If the denominators of the fractions are different, then first the fractions are brought to a common denominator, and then the actions are repeated as in point a).

3.3. Multiplying common fractions

Multiplying fractions obeys the following rule:

a/b*c/d=a*c/b*d,

that is, they multiply the numerators and denominators separately.

For example:

3/5*4/8=3*4/5*8=12/40.

3.4. Dividing fractions

Fractions are divided in the following way:

a/b:c/d=a*d/b*c,

that is, the fraction a/b is multiplied by the inverse fraction of the given one, that is, multiplied by d/c.

Example: 7/2:1/8=7/2*8/1=56/2=28

4. Reciprocal numbers

If a*b=1, then the number b is reciprocal number for the number a.

Example: for the number 9 the reciprocal is 1/9 , since 9*1/9 = 1 , for the number 5 - the inverse number 1/5 , because 5* 1/5 = 1 .

5. Decimals

Decimal called proper fraction, whose denominator is equal to 10, 1000, 10 000, …, 10^n 1 0 , 1 0 0 0 , 1 0 0 0 0 , . . . , 1 0 n .

For example: 6/10 =0,6; 44/1000=0,044 .

Incorrect ones with a denominator are written in the same way 10^n or mixed numbers.

For example: 51/10= 5,1; 763/100=7,63

Any ordinary fraction with a denominator that is a divisor of a certain power of 10 is represented as a decimal fraction.

a changer, which is a divisor of a certain power of the number 10.

Example: 5 is a divisor of 100, so it is a fraction 1/5=1 *20/5*20=20/100=0,2 0 = 0 , 2 .

6. Arithmetic operations on decimals

6.1. Adding Decimals

To add two decimal fractions, you need to arrange them so that there are identical digits under each other and a comma under the comma, and then add the fractions like ordinary numbers.

6.2. Subtracting Decimals

It is performed in the same way as addition.

6.3. Multiplying Decimals

When multiplying decimal numbers It is enough to multiply the given numbers, not paying attention to commas (like natural numbers), and in the resulting answer, a comma on the right separates as many digits as there are after the decimal point in both factors in total.

Let's multiply 2.7 by 1.3. We have 27\cdot 13=351 2 7 ⋅ 1 3 = 3 5 1 . We separate two digits on the right with a comma (the first and second numbers have one digit after the decimal point; 1+1=2 1 + 1 = 2 ). As a result we get 2.7\cdot 1.3=3.51 2 , 7 ⋅ 1 , 3 = 3 , 5 1 .

If the resulting result contains fewer digits than need to be separated by a comma, then the missing zeros are written in front, for example:

To multiply by 10, 100, 1000, you need to move the decimal point 1, 2, 3 digits to the right (if necessary, a certain number of zeros are assigned to the right).

For example: 1.47\cdot 10,000 = 14,700 1 , 4 7 ⋅ 1 0 0 0 0 = 1 4 7 0 0 .

6.4. Decimal division

Dividing a decimal fraction by a natural number is done in the same way as dividing a natural number by a natural number. The comma in the quotient is placed after the division of the whole part is completed.

If the integer part of the dividend is less than the divisor, then the answer is zero integers, for example:

Let's look at dividing a decimal by a decimal. Let's say we need to divide 2.576 by 1.12. First of all, let's multiply the dividend and divisor of the fraction by 100, that is, move the decimal point to the right in the dividend and divisor by as many decimal places as there are in the divisor after the decimal point (in this example, two). Then you need to divide the fraction 257.6 by the natural number 112, that is, the problem is reduced to the case already considered:

It happens that the final result is not always obtained decimal when dividing one number by another. The result is an infinite decimal fraction. In such cases, we move on to ordinary fractions.

For example, 2.8: 0.09= 28/10: 9/100= 28*100/10*9=2800/90=280/9= 31 1/9 .

Multiplying and dividing fractions.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

This operation is much nicer than addition-subtraction! Because it's easier. As a reminder, to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:

For example:

Everything is extremely simple. And please don't look for a common denominator! There is no need for him here...

To divide a fraction by a fraction, you need to reverse second(this is important!) fraction and multiply them, i.e.:

For example:

If you come across multiplication or division with integers and fractions, it’s okay. As with addition, we make a fraction from a whole number with one in the denominator - and go ahead! For example:

In high school, you often have to deal with three-story (or even four-story!) fractions. For example:

How can I make this fraction look decent? Yes, very simple! Use two-point division:

But don't forget about the order of division! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But it’s easy to make a mistake in a three-story fraction. Please note for example:

In the first case (expression on the left):

In the second (expression on the right):

Do you feel the difference? 4 and 1/9!

What determines the order of division? Either with brackets, or (as here) with the length of horizontal lines. Develop your eye. And if there are no brackets or dashes, like:

then divide and multiply in order, from left to right!

And also very simple and important technique. In actions with degrees, it will be so useful to you! Let's divide one by any fraction, for example, by 13/15:

The shot has turned over! And this always happens. When dividing 1 by any fraction, the result is the same fraction, only upside down.

That's it for operations with fractions. The thing is quite simple, but it gives more than enough errors. Please note practical advice, and there will be fewer of them (errors)!

Practical tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not general words, not good wishes! This is a dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.

2. In examples with different types fractions - go to ordinary fractions.

3. We reduce all fractions until they stop.

4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).

5. Divide a unit by a fraction in your head, simply turning the fraction over.

Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. Right the first time! Without a calculator! And draw the right conclusions...

Remember - the correct answer is received from the second (especially the third) time does not count! Such is the harsh life.

So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. And only Then look at the answers.

Calculate:

Have you decided?

We are looking for answers that match yours. I deliberately wrote them down in disarray, away from temptation, so to speak... Here they are, the answers, written with semicolons.

0; 17/22; 3/4; 2/5; 1; 25.

Now we draw conclusions. If everything worked out, I’m happy for you! Basic calculations with fractions are not your problem! You can do more serious things. If not...

So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But... This solvable problems.

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

In the article we will show how to solve fractions using simple, understandable examples. Let's figure out what a fraction is and consider solving fractions!

Concept fractions is introduced into mathematics courses starting from the 6th grade of secondary school.

Fractions have the form: ±X/Y, where Y is the denominator, it tells how many parts the whole was divided into, and X is the numerator, it tells how many such parts were taken. For clarity, let's take an example with a cake:

In the first case, the cake was cut equally and one half was taken, i.e. 1/2. In the second case, the cake was cut into 7 parts, of which 4 parts were taken, i.e. 4/7.

If the part of dividing one number by another is not a whole number, it is written as a fraction.

For example, the expression 4:2 = 2 gives an integer, but 4:7 is not divisible by a whole, so this expression is written as a fraction 4/7.

In other words fraction is an expression that denotes the division of two numbers or expressions, and which is written using a fractional slash.

If the numerator less than the denominator- a fraction is correct, if vice versa it is incorrect. A fraction can contain a whole number.

For example, 5 whole 3/4.

This entry means that in order to get the whole 6, one part of four is missing.

If you want to remember, how to solve fractions for 6th grade, you need to understand that solving fractions, basically, comes down to understanding a few simple things.

A fraction is essentially an expression of a fraction. That is, a numerical expression of what part is given value from one whole. For example, the fraction 3/5 expresses that if we divided something whole into 5 parts and the number of shares or parts of this whole is three.
The fraction can be less than 1, for example 1/2 (or essentially half), then it is correct. If the fraction is greater than 1, for example 3/2 (three halves or one and a half), then it is incorrect and to simplify the solution, it is better for us to select the whole part 3/2 = 1 whole 1/2.
Fractions are the same numbers as 1, 3, 10, and even 100, only the numbers are not whole numbers but fractions. You can perform all the same operations with them as with numbers. Counting fractions is no more difficult, and further on specific examples we will show it.

How to solve fractions. Examples.

A wide variety of arithmetic operations are applicable to fractions.

Reducing a fraction to a common denominator

For example, you need to compare the fractions 3/4 and 4/5.

To solve the problem, we first find the lowest common denominator, i.e. smallest number, which is divisible without a remainder by each of the denominators of the fractions

Least common denominator(4.5) = 20

Then the denominator of both fractions is reduced to the lowest common denominator

Answer: 15/20

Adding and subtracting fractions

If it is necessary to calculate the sum of two fractions, they are first brought to a common denominator, then the numerators are added, while the denominator remains unchanged. The difference between fractions is calculated in the same way, the only difference is that the numerators are subtracted.

For example, you need to find the sum of the fractions 1/2 and 1/3

Now let's find the difference between the fractions 1/2 and 1/4

Multiplying and dividing fractions

Here solving fractions is not difficult, everything is quite simple here:

Multiplication - numerators and denominators of fractions are multiplied together;
Division - first we get the fraction inverse of the second fraction, i.e. We swap its numerator and denominator, after which we multiply the resulting fractions.

For example:

That's about it how to solve fractions, All. If you still have any questions about solving fractions, if something is unclear, write in the comments and we will definitely answer you.

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