Rules for dividing fractional numbers. Multiplication and division of fractions

With fractions, you can perform all actions, including division. This article shows the division ordinary fractions. Definitions will be given, examples will be considered. Let us dwell on the division of fractions by natural numbers and vice versa. The division of an ordinary fraction by a mixed number will be considered.

Division of ordinary fractions

Division is the inverse of multiplication. When dividing, the unknown factor is at the known product and another factor, where its given meaning is preserved with ordinary fractions.

If it is necessary to divide the ordinary fraction a b by c d, then to determine such a number, you need to multiply by the divisor c d, this will eventually give the dividend a b. Let's get a number and write it a b · d c , where d c is the reciprocal of c d number. Equalities can be written using the properties of multiplication, namely: a b d c c d = a b d c c d = a b 1 = a b , where the expression a b d c is the quotient of dividing a b by c d .

From here we obtain and formulate the rule for dividing ordinary fractions:

Definition 1

To divide an ordinary fraction a b by c d, it is necessary to multiply the dividend by the reciprocal of the divisor.

Let's write the rule as an expression: a b: c d = a b d c

The rules of division are reduced to multiplication. To stick to it, you need to be well versed in performing multiplication of ordinary fractions.

Let's move on to the division of ordinary fractions.

Example 1

Perform division 9 7 by 5 3 . Write the result as a fraction.

Solution

The number 5 3 is the reciprocal of 3 5 . You must use the rule for dividing ordinary fractions. We write this expression as follows: 9 7: 5 3 \u003d 9 7 3 5 \u003d 9 3 7 5 \u003d 27 35.

Answer: 9 7: 5 3 = 27 35 .

When reducing fractions, you should highlight the whole part if the numerator is greater than the denominator.

Example 2

Divide 8 15: 24 65 . Write the answer as a fraction.

Solution

The solution is to switch from division to multiplication. We write it in this form: 8 15: 24 65 = 2 2 2 5 13 3 5 2 2 2 3 = 13 3 3 = 13 9

It is necessary to make a reduction, and this is done as follows: 8 65 15 24 \u003d 2 2 2 5 13 3 5 2 2 2 3 \u003d 13 3 3 \u003d 13 9

We select the integer part and get 13 9 = 1 4 9 .

Answer: 8 15: 24 65 = 1 4 9 .

Division of an extraordinary fraction by a natural number

We use the rule for dividing a fraction by natural number: to divide a b by a natural number n , you need to multiply only the denominator by n . From here we get the expression: a b: n = a b · n .

The division rule is a consequence of the multiplication rule. Therefore, representing a natural number as a fraction will give an equality of this type: a b: n \u003d a b: n 1 \u003d a b 1 n \u003d a b n.

Consider this division of a fraction by a number.

Example 3

Divide the fraction 1645 by the number 12.

Solution

Apply the rule for dividing a fraction by a number. We get an expression like 16 45: 12 = 16 45 12 .

Let's reduce the fraction. We get 16 45 12 = 2 2 2 2 (3 3 5) (2 2 3) = 2 2 3 3 3 5 = 4 135 .

Answer: 16 45: 12 = 4 135 .

Division of a natural number by a common fraction

The division rule is similar O the rule of dividing a natural number by an ordinary fraction: to divide a natural number n by an ordinary a b , it is necessary to multiply the number n by the reciprocal of the fraction a b .

Based on the rule, we have n: a b \u003d n b a, and thanks to the rule of multiplying a natural number by an ordinary fraction, we get our expression in the form n: a b \u003d n b a. It is necessary to consider this division with an example.

Example 4

Divide 25 by 15 28 .

Solution

We need to move from division to multiplication. We write in the form of an expression 25: 15 28 = 25 28 15 = 25 28 15 . Let's reduce the fraction and get the result in the form of a fraction 46 2 3 .

Answer: 25: 15 28 = 46 2 3 .

Division of a common fraction by a mixed number

When dividing an ordinary fraction by a mixed number, you can easily shine to dividing ordinary fractions. You need to convert a mixed number to an improper fraction.

Example 5

Divide the fraction 35 16 by 3 1 8 .

Solution

Since 3 1 8 is a mixed number, let's represent it as an improper fraction. Then we get 3 1 8 = 3 8 + 1 8 = 25 8 . Now let's divide the fractions. We get 35 16: 3 1 8 = 35 16: 25 8 = 35 16 8 25 = 35 8 16 25 = 5 7 2 2 2 2 2 2 2 (5 5) = 7 10

Answer: 35 16: 3 1 8 = 7 10 .

Dividing a mixed number is done in the same way as ordinary numbers.

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Adding fractions with the same denominators

Adding fractions is of two types:

1. Adding fractions with the same denominators
2. Adding fractions with different denominators

Let's start with adding fractions with the same denominators. Everything is simple here. To add fractions with the same denominators, you need to add their numerators, and leave the denominator unchanged. For example, let's add the fractions and . We add the numerators, and leave the denominator unchanged:

This example can be easily understood if we think of a pizza that is divided into four parts. If you add pizza to pizza, you get pizza:

Example 2 Add fractions and .

The answer is an improper fraction. If the task ends, then improper fractions accepted to get rid of. To get rid of an improper fraction, you need to select the whole part in it. In our case whole part stands out easily - two divided by two is equal to one:

This example can be easily understood if we think of a pizza that is divided into two parts. If you add more pizzas to the pizza, you get one whole pizza:

Example 3. Add fractions and .

Again, add the numerators, and leave the denominator unchanged:

This example can be easily understood if we think of a pizza that is divided into three parts. If you add more pizzas to pizza, you get pizzas:

Example 4 Find the value of an expression

This example is solved in exactly the same way as the previous ones. The numerators must be added and the denominator left unchanged:

Let's try to depict our solution using a picture. If you add pizzas to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.

As you can see, adding fractions with the same denominators is not difficult. It is enough to understand the following rules:

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

Adding fractions with different denominators

Now we will learn how to add fractions with different denominators. When adding fractions, the denominators of those fractions must be the same. But they are not always the same.

For example, fractions can be added because they have the same denominators.

But fractions cannot be added at once, because these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

There are several ways to reduce fractions to the same denominator. Today we will consider only one of them, since the rest of the methods may seem complicated for a beginner.

The essence of this method lies in the fact that first (LCM) of the denominators of both fractions is sought. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and the second additional factor is obtained.

Then the numerators and denominators of the fractions are multiplied by their additional factors. As a result of these actions, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to add such fractions.

Example 1. Add fractions and

First of all, we find the least common multiple of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 2. The least common multiple of these numbers is 6

LCM (2 and 3) = 6

Now back to fractions and . First, we divide the LCM by the denominator of the first fraction and get the first additional factor. LCM is the number 6, and the denominator of the first fraction is the number 3. Divide 6 by 3, we get 2.

The resulting number 2 is the first additional factor. We write it down to the first fraction. To do this, we make a small oblique line above the fraction and write down the found additional factor above it:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction and get the second additional factor. LCM is the number 6, and the denominator of the second fraction is the number 2. Divide 6 by 2, we get 3.

The resulting number 3 is the second additional factor. We write it to the second fraction. Again, we make a small oblique line above the second fraction and write the found additional factor above it:

Now we are all set to add. It remains to multiply the numerators and denominators of fractions by their additional factors:

Look closely at what we have come to. We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to add such fractions. Let's complete this example to the end:

Thus the example ends. To add it turns out.

Let's try to depict our solution using a picture. If you add pizzas to a pizza, you get one whole pizza and another sixth of a pizza:

Reduction of fractions to the same (common) denominator can also be depicted using a picture. Bringing the fractions and to a common denominator, we get the fractions and . These two fractions will be represented by the same slices of pizzas. The only difference will be that this time they will be divided into equal shares (reduced to the same denominator).

The first drawing shows a fraction (four pieces out of six) and the second picture shows a fraction (three pieces out of six). Putting these pieces together we get (seven pieces out of six). This fraction is incorrect, so we have highlighted the integer part in it. The result was (one whole pizza and another sixth pizza).

Note that we have painted this example in too much detail. IN educational institutions it is not customary to write in such a detailed manner. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the additional factors found by your numerators and denominators. While at school, we would have to write this example as follows:

But there is also the other side of the coin. If detailed notes are not made at the first stages of studying mathematics, then questions of the kind “Where does that number come from?”, “Why do fractions suddenly turn into completely different fractions? «.

To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:

1. Find the LCM of the denominators of fractions;
2. Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction;
3. Multiply the numerators and denominators of fractions by their additional factors;
4. Add fractions that have the same denominators;
5. If the answer turned out to be an improper fraction, then select its whole part;

Example 2 Find the value of an expression .

Let's use the instructions above.

Step 1. Find the LCM of the denominators of fractions

Find the LCM of the denominators of both fractions. The denominators of the fractions are the numbers 2, 3 and 4

Step 2. Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction

Divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it over the first fraction:

Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. Divide 12 by 3, we get 4. We got the second additional factor 4. We write it over the second fraction:

Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We got the third additional factor 3. We write it over the third fraction:

Step 3. Multiply the numerators and denominators of fractions by your additional factors

We multiply the numerators and denominators by our additional factors:

Step 4. Add fractions that have the same denominators

We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. It remains to add these fractions. Add up:

The addition didn't fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is carried over to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning of a new line. The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.

Step 5. If the answer turned out to be an improper fraction, then select the whole part in it

Our answer is an improper fraction. We must single out the whole part of it. We highlight:

Got an answer

Subtraction of fractions with the same denominators

There are two types of fraction subtraction:

1. Subtraction of fractions with the same denominators
2. Subtraction of fractions with different denominators

First, let's learn how to subtract fractions with the same denominators. Everything is simple here. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same.

For example, let's find the value of the expression . To solve this example, it is necessary to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged. Let's do this:

This example can be easily understood if we think of a pizza that is divided into four parts. If you cut pizzas from a pizza, you get pizzas:

Example 2 Find the value of the expression .

Again, from the numerator of the first fraction, subtract the numerator of the second fraction, and leave the denominator unchanged:

This example can be easily understood if we think of a pizza that is divided into three parts. If you cut pizzas from a pizza, you get pizzas:

Example 3 Find the value of an expression

This example is solved in exactly the same way as the previous ones. From the numerator of the first fraction, you need to subtract the numerators of the remaining fractions:

As you can see, there is nothing complicated in subtracting fractions with the same denominators. It is enough to understand the following rules:

1. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged;
2. If the answer turned out to be an improper fraction, then you need to select the whole part in it.

Subtraction of fractions with different denominators

For example, a fraction can be subtracted from a fraction, since these fractions have the same denominators. But a fraction cannot be subtracted from a fraction, because these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

The common denominator is found according to the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written over the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written over the second fraction.

The fractions are then multiplied by their additional factors. As a result of these operations, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to subtract such fractions.

Example 1 Find the value of an expression:

These fractions have different denominators, so you need to bring them to the same (common) denominator.

First, we find the LCM of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 4. The least common multiple of these numbers is 12

LCM (3 and 4) = 12

Now back to fractions and

Let's find an additional factor for the first fraction. To do this, we divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. We write the four over the first fraction:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. Write a triple over the second fraction:

Now we are all set for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to subtract such fractions. Let's complete this example to the end:

Got an answer

Let's try to depict our solution using a picture. If you cut pizzas from a pizza, you get pizzas.

This is the detailed version of the solution. Being at school, we would have to solve this example in a shorter way. Such a solution would look like this:

Reduction of fractions and to a common denominator can also be depicted using a picture. Bringing these fractions to a common denominator, we get the fractions and . These fractions will be represented by the same pizza slices, but this time they will be divided into the same fractions (reduced to the same denominator):

The first drawing shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting off three pieces from eight pieces, we get five pieces out of twelve. The fraction describes these five pieces.

Example 2 Find the value of an expression

These fractions have different denominators, so you first need to bring them to the same (common) denominator.

Find the LCM of the denominators of these fractions.

The denominators of the fractions are the numbers 10, 3 and 5. The least common multiple of these numbers is 30

LCM(10, 3, 5) = 30

Now we find additional factors for each fraction. To do this, we divide the LCM by the denominator of each fraction.

Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is the number 10. Divide 30 by 10, we get the first additional factor 3. We write it over the first fraction:

Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it over the second fraction:

Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is the number 5. Divide 30 by 5, we get the third additional factor 6. We write it over the third fraction:

Now everything is ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. And we already know how to subtract such fractions. Let's finish this example.

The continuation of the example will not fit on one line, so we move the continuation to the next line. Don't forget about the equal sign (=) on the new line:

The answer turned out to be a correct fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it easier. What can be done? You can reduce this fraction.

To reduce a fraction, you need to divide its numerator and denominator by (gcd) the numbers 20 and 30.

So, we find the GCD of the numbers 20 and 30:

Now we return to our example and divide the numerator and denominator of the fraction by the found GCD, that is, by 10

Got an answer

Multiplying a fraction by a number

To multiply a fraction by a number, you need to multiply the numerator of the given fraction by this number, and leave the denominator unchanged.

Example 1. Multiply the fraction by the number 1.

Multiply the numerator of the fraction by the number 1

The entry can be understood as taking half 1 time. For example, if you take pizza 1 time, you get pizza

From the laws of multiplication, we know that if the multiplicand and the multiplier are interchanged, then the product will not change. If the expression is written as , then the product will still be equal to . Again, the rule for multiplying an integer and a fraction works:

This entry can be understood as taking half of the unit. For example, if there is 1 whole pizza and we take half of it, then we will have pizza:

Example 2. Find the value of an expression

Multiply the numerator of the fraction by 4

The answer is an improper fraction. Let's take a whole part of it:

The expression can be understood as taking two quarters 4 times. For example, if you take pizzas 4 times, you get two whole pizzas.

And if we swap the multiplicand and the multiplier in places, we get the expression. It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:

The number that is being multiplied by a fraction, and the denominator of the fraction is resolved if they have common divisor, greater than unity.

For example, an expression can be evaluated in two ways.

First way. Multiply the number 4 by the numerator of the fraction, and leave the denominator of the fraction unchanged:

Second way. The quadruple being multiplied and the quadruple in the denominator of the fraction can be reduced. You can reduce these fours by 4, since the greatest common divisor for two fours is the four itself:

We got the same result 3. After reducing the fours, new numbers are formed in their place: two ones. But multiplying one with a triple, and then dividing by one does not change anything. Therefore, the solution can be written shorter:

The reduction can be performed even when we decided to use the first method, but at the stage of multiplying the number 4 and the numerator 3, we decided to use the reduction:

But for example, the expression can only be calculated in the first way - multiply 7 by the denominator of the fraction, and leave the denominator unchanged:

This is due to the fact that the number 7 and the denominator of the fraction do not have a common divisor greater than one, and, accordingly, are not reduced.

Some students mistakenly abbreviate the number being multiplied and the numerator of the fraction. You can't do this. For example, the following entry is not correct:

The reduction of the fraction implies that and numerator and denominator will be divided by the same number. In the situation with the expression, the division is performed only in the numerator, since writing this is the same as writing . We see that the division is performed only in the numerator, and no division occurs in the denominator.

Multiplication of fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer is an improper fraction, you need to select the whole part in it.

Example 1 Find the value of the expression .

Got an answer. It is desirable to reduce this fraction. The fraction can be reduced by 2. Then the final solution will take the following form:

The expression can be understood as taking a pizza from half a pizza. Let's say we have half a pizza:

How to take two-thirds from this half? First you need to divide this half into three equal parts:

And take two from these three pieces:

We'll get pizza. Remember what a pizza looks like divided into three parts:

One slice from this pizza and the two slices we took will have the same dimensions:

In other words, we are talking about the same pizza size. Therefore, the value of the expression is

Example 2. Find the value of an expression

Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer is an improper fraction. Let's take a whole part of it:

Example 3 Find the value of an expression

Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer turned out to be a correct fraction, but it will be good if it is reduced. To reduce this fraction, you need to divide the numerator and denominator of this fraction by the greatest common divisor (GCD) of the numbers 105 and 450.

So, let's find the GCD of the numbers 105 and 450:

Now we divide the numerator and denominator of our answer to the GCD that we have now found, that is, by 15

Representing an integer as a fraction

Any whole number can be represented as a fraction. For example, the number 5 can be represented as . From this, the five will not change its meaning, since the expression means “the number five divided by one”, and this, as you know, is equal to five:

Reverse numbers

Now we will get acquainted with interesting topic in mathematics. It's called "reverse numbers".

Definition. Reverse to numbera is the number that, when multiplied bya gives a unit.

Let's substitute in this definition instead of a variable a number 5 and try to read the definition:

Reverse to number 5 is the number that, when multiplied by 5 gives a unit.

Is it possible to find a number that, when multiplied by 5, gives one? It turns out you can. Let's represent five as a fraction:

Then multiply this fraction by itself, just swap the numerator and denominator. In other words, let's multiply the fraction by itself, only inverted:

What will be the result of this? If we continue to solve this example, we get one:

This means that the inverse of the number 5 is the number, since when 5 is multiplied by one, one is obtained.

The reciprocal can also be found for any other integer.

You can also find the reciprocal for any other fraction. To do this, it is enough to turn it over.

Division of a fraction by a number

Let's say we have half a pizza:

Let's divide it equally between two. How many pizzas will each get?

It can be seen that after splitting half of the pizza, two equal pieces were obtained, each of which makes up a pizza. So everyone gets a pizza.

Multiplication and division of fractions.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

This operation is much nicer than addition-subtraction! Because it's easier. I remind you: 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! Don't need it here...

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

For example:

If multiplication or division with integers and fractions is caught, it's okay. As with addition, we make a fraction from a whole number with a unit in the denominator - and go! For example:

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

How to bring this fraction to a decent form? Yes, very easy! Use division through two points:

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

In the first case (expression on the left):

In the second (expression on the right):

Feel the difference? 4 and 1/9!

What is the order of division? Or brackets, or (as here) the length of horizontal dashes. Develop an eye. And if there are no brackets or dashes, like:

then divide-multiply in order, left to right!

And another very simple and important trick. In actions with degrees, it will come in handy for you! Let's divide the unit by any fraction, for example, by 13/15:

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

That's all the actions with fractions. The thing is quite simple, but gives more than enough errors. Note practical advice, and they (errors) will be less!

Practical Tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a severe need! Do all the calculations on the exam as a full-fledged task, with concentration and clarity. It is better to write two extra lines in a draft than to mess up when calculating in your head.

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

3. We reduce all fractions to the stop.

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

5. We divide the unit into a fraction in our mind, simply by turning the fraction over.

Here are the tasks you need to complete. Answers are given after all tasks. Use the materials of this topic and practical advice. Estimate how many examples you could solve correctly. The first time! Without a calculator! And draw the right conclusions...

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

So, solve in exam mode ! This is preparation for the exam, by the way. We solve an example, we check, we solve the following. We decided everything - we checked again from the first to the last. But only Then look at the answers.

Calculate:

Did you decide?

Looking for answers that match yours. I specifically wrote them down in a mess, away from the temptation, so to speak ... Here they are, the answers, written down with a semicolon.

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

And now we draw conclusions. If everything worked out - happy for you! Elementary 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. Learning - with interest!)

you can get acquainted with functions and derivatives.

Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use some object not entirely, but in separate pieces. The beginning of the study of this topic - share. Shares are equal parts into which an object is divided. After all, it is not always possible to express, for example, the length or price of a product as an integer; one should take into account parts or shares of any measure. Formed from the verb "to crush" - to divide into parts, and having Arabic roots, in the VIII century the word "fraction" itself appeared in Russian.

Fractional expressions have long been considered the most difficult section of mathematics. In the 17th century, when first textbooks in mathematics appeared, they were called "broken numbers", which was very difficult to display in people's understanding.

modern look simple fractional residues, parts of which are separated precisely by a horizontal line, were first contributed to Fibonacci - Leonardo of Pisa. His writings are dated 1202. But the purpose of this article is to simply and clearly explain to the reader how the multiplication of mixed fractions with different denominators occurs.

Multiplying fractions with different denominators

Initially, it is necessary to determine varieties of fractions:

• correct;
• wrong;
• mixed.

Next, you need to remember how fractional numbers with the same denominators are multiplied. The very rule of this process is easy to formulate independently: the result of multiplying simple fractions with the same denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in fact, the new denominator is the square of one of the existing ones initially.

When multiplying simple fractions with different denominators for two or more factors, the rule does not change:

a/b * c/d = a*c / b*d.

The only difference is that the formed number under the fractional bar will be the product of different numbers and, of course, it cannot be called the square of one numerical expression.

It is worth considering the multiplication of fractions with different denominators using examples:

• 8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
• 4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .

The examples use ways to reduce fractional expressions. You can reduce only the numbers of the numerator with the numbers of the denominator; adjacent factors above or below the fractional bar cannot be reduced.

Along with simple fractional numbers, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:

1 4/ 11 =1 + 4/ 11.

How does multiplication work?

Several examples are provided for consideration.

2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.

The example uses the multiplication of a number by ordinary fractional part, you can write down the rule for this action by the formula:

a * b/c = a*b /c.

In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. special case:

4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.

There is another option for solving the multiplication of a number by a fractional remainder. Simply divide the denominator by this number:

d* e/f = e/f: d.

It is useful to use this technique when the denominator is divided by a natural number without a remainder or, as they say, completely.

Convert mixed numbers to improper fractions and get the product in the previously described way:

1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.

This example involves a way to represent a mixed fraction as an improper fraction, it can also be represented as a general formula:

a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the integer part with the denominator and adding it to the numerator of the original fractional remainder, and the denominator remains the same.

This process also works in reverse side. To select the integer part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator with a “corner”.

Multiplication of improper fractions produced in the usual way. When the entry goes under a single fractional line, as necessary, you need to reduce the fractions in order to reduce the numbers using this method and it is easier to calculate the result.

There are many helpers on the Internet to solve even complex problems. math problems in various programs. A sufficient number of such services offer their help in counting the multiplication of fractions with different numbers in denominators - the so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions and mixed numbers. It is not difficult to work with it, the corresponding fields are filled in on the site page, the sign of the mathematical action is selected and “calculate” is pressed. The program counts automatically.

The topic of arithmetic operations with fractional numbers is relevant throughout the education of middle and senior schoolchildren. In high school, they are no longer considering the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations, obtained earlier, is applied in its original form. Well-learned basic knowledge gives full confidence in the successful solution of the most complex tasks.

In conclusion, it makes sense to cite the words of Leo Tolstoy, who wrote: “Man is a fraction. It is not in the power of man to increase his numerator - his own merits, but anyone can decrease his denominator - his opinion of himself, and by this decrease come closer to his perfection.

A fraction is one or more parts of a whole, which is usually taken as a unit (1). As with natural numbers, you can perform all basic arithmetic operations with fractions (addition, subtraction, division, multiplication), for this you need to know the features of working with fractions and distinguish between their types. There are several types of fractions: decimal and ordinary, or simple. Each type of fractions has its own specifics, but once you have thoroughly figured out how to deal with them once, you will be able to solve any examples with fractions, since you will know the basic principles for performing arithmetic calculations with fractions. Let's look at examples of how to divide a fraction by an integer using different types fractions.

How to divide a decimal by an integer?
A decimal fraction is a fraction that is obtained by dividing a unit into ten, a thousand, and so on parts. Arithmetic operations with decimal fractions are quite simple.

Consider an example of how to divide a fraction by an integer. Let's say we need to divide the decimal fraction 0.925 by the natural number 5.

• to separate decimal fraction division into a column is applied to a natural number;
  
• a comma is placed in the private when the division of the integer part of the dividend is completed.