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

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 may be hidden under the letter entry, numeric expression, numerical-letter expression.

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 a 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:

  1. Finding gcd for both sides of the fraction.
  2. 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.

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 definitely need to solve. 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.

This section covers actions with ordinary fractions. If it is necessary to carry out a mathematical operation with mixed numbers, then it is enough to translate mixed fraction into the extraordinary, spend necessary operations and, if necessary, end result represent again as mixed number. This operation will be described below.

Reducing a fraction

Mathematical operation. Reducing a fraction

To reduce the fraction \frac(m)(n) you need to find the greatest common divisor of its numerator and denominator: gcd(m,n), and then divide the numerator and denominator of the fraction by this number. If GCD(m,n)=1, then the fraction cannot be reduced. Example: \frac(20)(80)=\frac(20:20)(80:20)=\frac(1)(4)

Usually, immediately finding the greatest common divisor seems to be a difficult task, and in practice the fraction is reduced in several stages, step by step isolating the obvious ones from the numerator and denominator common factors. \frac(140)(315)=\frac(28\cdot5)(63\cdot5)=\frac(4\cdot7\cdot5)(9\cdot7\cdot5)=\frac(4)(9)

Reducing fractions to a common denominator

Mathematical operation. Reducing fractions to a common denominator

To bring two fractions \frac(a)(b) and \frac(c)(d) to a common denominator you need:

  • find the least common multiple of the denominators: M=LMK(b,d);
  • multiply the numerator and denominator of the first fraction by M/b (after which the denominator of the fraction becomes equal to the number M);
  • multiply the numerator and denominator of the second fraction by M/d (after which the denominator of the fraction becomes equal to the number M).

Thus, we transform the original fractions to fractions with the same denominators (which will be equal to the number M).

For example, the fractions \frac(5)(6) and \frac(4)(9) have LCM(6,9) = 18. Then: \frac(5)(6)=\frac(5\cdot3)(6 \cdot3)=\frac(15)(18);\quad\frac(4)(9)=\frac(4\cdot2)(9\cdot2)=\frac(8)(18) . Thus, the resulting fractions have a common denominator.

In practice, finding the least common multiple (LCM) of denominators is not always a simple task. Therefore, a number equal to the product of the denominators of the original fractions is chosen as the common denominator. For example, the fractions \frac(5)(6) and \frac(4)(9) are reduced to a common denominator N=6\cdot9:

\frac(5)(6)=\frac(5\cdot9)(6\cdot9)=\frac(45)(54);\quad\frac(4)(9)=\frac(4\cdot6)( 9\cdot6)=\frac(24)(54)

Comparison of fractions

Mathematical operation. Comparison of fractions

To compare two ordinary fractions you need:

  • compare the numerators of the resulting fractions; a fraction with a larger numerator will be larger.
For example, \frac(9)(14)

When comparing fractions, there are several special cases:

  1. From two fractions with the same denominators The fraction whose numerator is greater is greater. For example, \frac(3)(15)
  2. From two fractions with the same numerators The larger is the fraction whose denominator is smaller. For example, \frac(4)(11)>\frac(4)(13)
  3. That fraction which simultaneously larger numerator and smaller denominator, more. For example, \frac(11)(3)>\frac(10)(8)

Attention! Rule 1 applies to any fractions if their common denominator is positive number. Rules 2 and 3 apply to positive fractions (those with both the numerator and denominator greater than zero).

Adding and subtracting fractions

Mathematical operation. Adding and subtracting fractions

To add two fractions you need:

  • bring them to a common denominator;
  • add their numerators and leave the denominator unchanged.

Example: \frac(7)(9)+\frac(4)(7)=\frac(7\cdot7)(9\cdot7)+\frac(4\cdot9)(7\cdot9)=\frac(49 )(63)+\frac(36)(63)=\frac(49+36)(63)=\frac(85)(63)

To subtract another from one fraction, you need:

  • reduce fractions to a common denominator;
  • Subtract the numerator of the second fraction from the numerator of the first fraction and leave the denominator unchanged.

Example: \frac(4)(15)-\frac(3)(5)=\frac(4)(15)-\frac(3\cdot3)(5\cdot3)=\frac(4)(15) -\frac(9)(15)=\frac(4-9)(15)=\frac(-5)(15)=-\frac(5)(3\cdot5)=-\frac(1)( 3)

If the original fractions initially have a common denominator, then step 1 (reduction to a common denominator) is skipped.

Converting a mixed number to an improper fraction and vice versa

Mathematical operation. Converting a mixed number to an improper fraction and vice versa

To convert a mixed fraction to an improper fraction, simply sum the whole part of the mixed fraction with the fraction part. The result of such a sum will be an improper fraction, the numerator of which is equal to the sum of the product of the whole part by the denominator of the fraction with the numerator of the mixed fraction, and the denominator will remain the same. For example, 2\frac(6)(11)=2+\frac(6)(11)=\frac(2\cdot11)(11)+\frac(6)(11)=\frac(2\cdot11+ 6)(11)=\frac(28)(11)

To convert an improper fraction to a mixed number:

  • divide the numerator of a fraction by its denominator;
  • write the remainder of the division into the numerator and leave the denominator the same;
  • write the result of the division as an integer part.

For example, the fraction \frac(23)(4) . When dividing 23:4=5.75, that is, the whole part is 5, the remainder of the division is 23-5*4=3. Then the mixed number will be written: 5\frac(3)(4) . \frac(23)(4)=\frac(5\cdot4+3)(4)=5\frac(3)(4)

Converting a Decimal to a Fraction

Mathematical operation. Converting a Decimal to a Fraction

In order to convert a decimal fraction to a common fraction, you need to:

  1. take the nth power of ten as the denominator (here n is the number of decimal places);
  2. as the numerator take the number after the decimal point (if the integer part original number is not equal to zero, then take all leading zeros as well);
  3. the non-zero integer part is written in the numerator at the very beginning; the zero integer part is omitted.

Example 1: 0.0089=\frac(89)(10000) (there are 4 decimal places, so the denominator has 10 4 =10000, since the integer part is 0, the numerator contains the number after the decimal point without leading zeros)

Example 2: 31.0109=\frac(310109)(10000) (in the numerator we write the number after the decimal point with all zeros: “0109”, and then before it we add the whole part of the original number “31”)

If the whole part of a decimal fraction is non-zero, then it can be converted to a mixed fraction. To do this, we convert the number into an ordinary fraction as if the whole part were equal to zero (points 1 and 2), and simply rewrite the whole part in front of the fraction - this will be the whole part of the mixed number. Example:

3.014=3\frac(14)(100)

To convert a fraction to a decimal, simply divide the numerator by the denominator. Sometimes you end up with an infinite decimal. In this case, it is necessary to round to the desired decimal place. Examples:

\frac(401)(5)=80.2;\quad \frac(2)(3)\approx0.6667

Multiplying and dividing fractions

Mathematical operation. Multiplying and dividing fractions

To multiply two ordinary fractions, you need to multiply the numerators and denominators of the fractions.

\frac(5)(9)\cdot\frac(7)(2)=\frac(5\cdot7)(9\cdot2)=\frac(35)(18)

To divide one common fraction by another, you need to multiply the first fraction by the reciprocal of the second ( reciprocal fraction- a fraction in which the numerator and denominator are swapped.

\frac(5)(9):\frac(7)(2)=\frac(5)(9)\cdot\frac(2)(7)=\frac(5\cdot2)(9\cdot7)= \frac(10)(63)

If one of the fractions is natural number, then the above rules of multiplication and division remain in force. You just need to take into account that an integer is the same fraction, the denominator of which is equal to one. For example: 3:\frac(3)(7)=\frac(3)(1):\frac(3)(7)=\frac(3)(1)\cdot\frac(7)(3)= \frac(3\cdot7)(1\cdot3)=\frac(7)(1)=7

    I myself was faced with the fact that the fractions turned out to be quite complex topic for my children.

    There are very good game Nikitin's fractions, it is intended for preschoolers, but also at school it will perfectly help the child figure out what they are - fractions, their relationship to each other..., and all in an accessible, visual and exciting form.

    It consists of twelve multi-colored circles. One circle is whole, and all the rest are divided into equal parts - two, three.... (up to twelve).

    The child is asked to complete simple game tasks, for example:

    What are the parts of the circles called? or

    Which part is bigger? (put the smaller one on top of the larger one.)

    This technique helped me. In general, I really regret that all these Nikitin developments did not catch my eye when the children were still babies.

    You can make the game yourself or buy a ready-made one, and find out more about everything -.

    Solving fractions can also be explained using Lego bricks. It develops not only imagination, but also creative and logical thinking, which means it can also be used as a teaching aid.

    Alicia Zimmerman came up with the idea of ​​​​using the blocks of the famous designer to teach children the basics of mathematics.

    And here's how to explain fractions using Lego.

    Practice shows that the most difficulties arise when adding (subtracting) fractions with different denominators and when dividing fractions.

    Difficulties arise due to incorrect instructions in the textbook, such as dividing a fraction by a fraction.

    To divide a fraction by a fraction, you multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction.

    Can a child in 4th grade understand this and not get confused? NO!

    And the teacher explained it to us in an elementary way: we need to turn the second fraction over and then multiply it!

    Same thing with addition.

    To add two fractions, you need to multiply the numerator of the first fraction by the denominator of the second fraction, and multiply the numerator of the second fraction by the denominator of the first fraction, add the resulting numbers and write them in the numerator. And in the denominator you need to write the product of the denominators of the fractions. After this, the resulting fraction can (or should) be reduced.

    And it’s simpler: Reduce the fractions to a common denominator, which is equal to the LCM of the denominators, and then add the numerators.

    Show them with a clear example. For example, cut an apple into 4 parts, put it into 8 parts, add 12 parts into a whole, add several parts, subtract. At the same time, explain on paper using rules. Rules for addition and subtraction. dividing fractions, as well as from improper fraction highlight the whole - learn all this while manipulating the apple. Do not rush the children, let them carefully sort out the slices with your help.

    Teaching children to solve fractions, in particular, is quite common and will not create much trouble. The simplest thing you can do is take something whole, for example a tangerine, or any other fruit, divide it into parts, and use an example to show subtraction, addition and other operations with pieces of this fruit, which will be fractions from the whole. Everything needs to be explained and shown, and the final factor will be to explain and solve problems together using mathematical examples until the child learns to do these tasks himself.

    The figure clearly shows what corresponds to what and how the fraction looks on real subject, this is exactly how it needs to be explained.

    You need to approach this issue thoroughly, since solving fractions will come in handy in life. It is necessary in this matter, as they say, to be on equal terms with children, and explain the theory to them accessible language, for example in the language of cake or mandarin. You need to divide the cake into do and give it to friends, after which the child will begin to understand the essence of solving fractions. Don't start with heavy fractions, start with the concepts of 1/2, 1/3, 1/10. First, subtract and add, and then move on to more complex concepts like multiplication and division.

    There are different types of problems with fractions. One child cannot understand that one second and five tenths are the same thing, others are perplexed by bringing different fractions to the same denominator, and still others are confused by the division of fractions. Therefore, there is no one rule for all occasions.

    The main thing in problems involving fractions is not to miss the moment when what is understandable ceases to be so. Return to the stove and repeat everything all over again, even if it seems wretchedly primitive. For example, go back to what is one second.

    The child must understand that mathematical concepts- abstract, that the same phenomenon can be described in different words, expressed in different numbers.

    I like the answer given by Mefody66. I will add from many years of personal practice: teaching how to solve problems with fractions (and not solving fractions; solving fractions is impossible, just as it is impossible to solve numbers) is quite simple, you just need to be close to the child when he first starts solving such problems, and correct his solution in time , so that mistakes, which are inevitable in any learning, do not have time to take hold in the child’s mind. Relearning is more difficult than learning something new. And solve such problems as much as possible. Bringing the solution of such tasks to automaticity would be a good thing to do. The ability to solve problems with ordinary fractions is as important in a school mathematics course as knowledge of the multiplication table. So you need to take the time to watch how your child solves such problems.

    And don’t rely too much on the textbook: teachers in schools explain exactly as Mefody66 wrote in his answer. It is better to talk with the teacher, find out in what words the teacher explained this topic. And use the same words and phrases if possible (so as not to confuse the child too much)

    More: illustrative examples I recommend using it only for initial stage explanations, then quickly abstract and move on to the solution algorithm. Otherwise, clarity may be detrimental when solving more complex tasks. For example, if you need to add fractions with denominators 29 and 121, what kind of visual aid will help? It will only confuse.

    Fractions are one of those blessed mathematical topics where there are no abstractions that are not applicable. Products should be used (on cakes, like Juanita Solis in Desperate Housewives - a really cool method of explanation). All these numerator-denominators come later. Then it is necessary for the child to understand that dividing by a fraction is no longer a decrease at all, and multiplication is not an increase. Here it is better to show how to divide by a fraction in the form of multiplication by inversion. IN game form submit a reduction, if they are divisible by one number, then divide, it’s almost a Sudoku, if you’re interested. The main thing is to notice misunderstandings in time, because further on there will be more interesting topics that are not easy to understand. Therefore, have more practice solving fractions and everything will get better quickly. To me, the purest humanist, far from the slightest degree of abstraction, fractions have always been clearer than other topics.

Let’s agree that “actions with fractions” in our lesson will mean actions with ordinary fractions. A common fraction is a fraction that has attributes such as a numerator, a fraction line, and a denominator. This distinguishes a common fraction from a decimal, which is obtained from a common fraction by reducing the denominator to a multiple of 10. Decimal written with a comma separating the whole part from the fractional part. We will talk about operations with ordinary fractions, since they are the ones that cause the greatest difficulties for students who have forgotten the basics of this topic, covered in the first half of the school mathematics course. At the same time, when transforming expressions in higher mathematics, it is mainly operations with ordinary fractions that are used. The fraction abbreviations alone are worth it! Decimal fractions do not cause any particular difficulties. So, go ahead!

Two fractions are said to be equal if .

For example, since

Fractions and (since), and (since) are also equal.

Obviously, both fractions and are equal. This means that if the numerator and denominator of a given fraction are multiplied or divided by the same natural number, you will get a fraction equal to the given one: .

This property is called the basic property of a fraction.

The basic property of a fraction can be used to change the signs of the numerator and denominator of a fraction. If the numerator and denominator of a fraction are multiplied by -1, we get . This means that the value of a fraction will not change if the signs of the numerator and denominator are changed at the same time. If you change the sign of only the numerator or only the denominator, then the fraction will change its sign:

Reducing Fractions

Using the basic property of a fraction, you can replace a given fraction with another fraction that is equal to the given one, but with a smaller numerator and denominator. This substitution is called fraction reduction.

Let, for example, be given a fraction. The numbers 36 and 48 have a greatest common divisor of 12. Then

.

In general, reducing a fraction is always possible if the numerator and denominator are not mutually prime numbers. If the numerator and denominator are mutual prime numbers, then the fraction is called irreducible.

So, to reduce a fraction means to divide the numerator and denominator of the fraction by a common factor. All of the above also applies to fractional expressions containing variables.

Example 1. Reduce a fraction

Solution. To factorize the numerator, first presenting the monomial - 5 xy as a sum - 2 xy - 3xy, we get

To factorize the denominator, we use the difference of squares formula:

As a result

.

Reducing fractions to a common denominator

Let two fractions and . They have different denominators: 5 and 7. Using the basic property of fractions, you can replace these fractions with others that are equal to them, and such that the resulting fractions will have the same denominators. Multiplying the numerator and denominator of the fraction by 7, we get

Multiplying the numerator and denominator of the fraction by 5, we get

So, the fractions are reduced to a common denominator:

.

But it's not the only solution given task: for example, these fractions can also be reduced to a common denominator of 70:

,

and in general to any denominator divisible by both 5 and 7.

Let's consider another example: let's bring the fractions and to a common denominator. Arguing as in the previous example, we get

,

.

But in in this case You can reduce fractions to a common denominator that is less than the product of the denominators of these fractions. Let's find the least common multiple of the numbers 24 and 30: LCM(24, 30) = 120.

Since 120:4 = 5, to write a fraction with a denominator of 120, you need to multiply both the numerator and the denominator by 5, this number is called an additional factor. Means .

Next, we get 120:30=4. Multiplying the numerator and denominator of the fraction by an additional factor of 4, we get .

So, these fractions are reduced to a common denominator.

The least common multiple of the denominators of these fractions is the smallest possible common denominator.

For fractional expressions that involve variables, the common denominator is a polynomial that is divided by the denominator of each fraction.

Example 2. Find the common denominator of the fractions and.

Solution. The common denominator of these fractions is a polynomial, since it is divisible by both and. However, this polynomial is not the only one that can be a common denominator of these fractions. It can also be a polynomial , and polynomial , and polynomial etc. Usually they take such a common denominator that any other common denominator is divided by the chosen one without a remainder. This denominator is called the lowest common denominator.

In our example, the lowest common denominator is . Received:

;

.

We were able to reduce fractions to their lowest common denominator. This happened by multiplying the numerator and denominator of the first fraction by , and the numerator and denominator of the second fraction by . Polynomials are called additional factors, respectively for the first and second fractions.

Adding and subtracting fractions

Addition of fractions is defined as follows:

.

For example,

.

If b = d, That

.

This means that to add fractions with same denominator It is enough to add the numerators and leave the denominator the same. For example,

.

If you add fractions with different denominators, you usually reduce the fractions to the lowest common denominator, and then add the numerators. For example,

.

Now let's look at an example of adding fractional expressions with variables.

Example 3. Convert expression to one fraction

.

Solution. Let's find the lowest common denominator. To do this, we first factorize the denominators.