Adding and subtracting fractions with different denominators. Addition and subtraction of algebraic fractions with different denominators (basic rules, simplest cases)

Actions with fractions.

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

So, what are fractions, types of fractions, transformations - we remembered. Let's get to the main issue.

What can you do with fractions? Yes, everything is the same as with ordinary numbers. Add, subtract, multiply, divide.

All these actions with decimal working with fractions is no different from working with whole numbers. Actually, that’s what’s good about them, decimal ones. The only thing is that you need to put the comma correctly.

Mixed numbers, as I already said, are of little use for most actions. They still need to be converted to ordinary fractions.

But the actions with ordinary fractions they will be more cunning. And much more important! Let me remind you: all actions with fractional expressions with letters, sines, unknowns, etc., etc. are no different from actions with ordinary fractions! Operations with ordinary fractions are the basis for all algebra. It is for this reason that we will analyze all this arithmetic in great detail here.

Adding and subtracting fractions.

Everyone can add (subtract) fractions with the same denominators (I really hope!). Well, let me remind those who are completely forgetful: when adding (subtracting), the denominator does not change. The numerators are added (subtracted) to give the numerator of the result. Type:

In short, in general terms:

What if the denominators are different? Then, using the basic property of a fraction (here it comes in handy again!), we make the denominators the same! For example:

Here we had to make the fraction 4/10 from the fraction 2/5. For the sole purpose of making the denominators the same. Let me note, just in case, that 2/5 and 4/10 are the same fraction! Only 2/5 are uncomfortable for us, and 4/10 are really okay.

By the way, this is the essence of solving any math problems. When we from uncomfortable we do expressions the same thing, but more convenient for solving.

Another example:

The situation is similar. Here we make 48 from 16. By simple multiplication by 3. This is all clear. But we came across something like:

How to be?! It's hard to make a nine out of a seven! But we are smart, we know the rules! Let's transform every fraction so that the denominators are the same. This is called “reduce to a common denominator”:

Wow! How did I know about 63? Very simple! 63 is a number that is divisible by 7 and 9 at the same time. Such a number can always be obtained by multiplying the denominators. If we multiply a number by 7, for example, then the result will certainly be divisible by 7!

If you need to add (subtract) several fractions, there is no need to do it in pairs, step by step. You just need to find the denominator common to all fractions and reduce each fraction to this same denominator. For example:

And what will be the common denominator? You can, of course, multiply 2, 4, 8, and 16. We get 1024. Nightmare. It’s easier to estimate that the number 16 is perfectly divisible by 2, 4, and 8. Therefore, from these numbers it’s easy to get 16. This number will be the common denominator. Let's turn 1/2 into 8/16, 3/4 into 12/16, and so on.

By the way, if you take 1024 as the common denominator, everything will work out, in the end everything will be reduced. But not everyone will get to this end, because of the calculations...

Complete the example yourself. Not some kind of logarithm... It should turn out to be 29/16.

So, the addition (subtraction) of fractions is clear, I hope? Of course, it’s easier to work in a shortened version, with additional multipliers. But this pleasure is available to those who worked honestly in the lower grades... And did not forget anything.

And now we will do the same actions, but not with fractions, but with fractional expressions. New rake will be revealed here, yes...

So, we need to add two fractional expressions:

We need to make the denominators the same. And only with the help multiplication! This is what the main property of a fraction dictates. Therefore, I cannot add one to X in the first fraction in the denominator. (that would be nice!). But if you multiply the denominators, you see, everything grows together! So we write down the line of the fraction, leave an empty space at the top, then add it, and write the product of the denominators below, so as not to forget:

And, of course, we don’t multiply anything on the right side, we don’t open the parentheses! And now, looking at the common denominator on the right side, we realize: in order to get the denominator x(x+1) in the first fraction, we need to multiply the numerator and denominator of this fraction by (x+1). And in the second fraction - to x. This is what you get:

Note! Here are the parentheses! This is the rake that many people step on. Not parentheses, of course, but their absence. The parentheses appear because we are multiplying all numerator and all denominator! And not their individual pieces...

In the numerator of the right side we write the sum of the numerators, everything is as in numerical fractions, then we open the brackets in the numerator of the right side, i.e. We multiply everything and give similar ones. There is no need to open the parentheses in the denominators or multiply anything! In general, in denominators (any) the product is always more pleasant! We get:

So we got the answer. The process seems long and difficult, but it depends on practice. Once you solve the examples, get used to it, everything will become simple. Those who have mastered fractions in due time do all these operations with one left hand, automatically!

And one more note. Many smartly deal with fractions, but get stuck on examples with whole numbers. Like: 2 + 1/2 + 3/4= ? Where to fasten the two-piece? You don’t need to fasten it anywhere, you need to make a fraction out of two. It's not easy, but very simple! 2=2/1. Like this. Any whole number can be written as a fraction. The numerator is the number itself, the denominator is one. 7 is 7/1, 3 is 3/1 and so on. It's the same with letters. (a+b) = (a+b)/1, x=x/1, etc. And then we work with these fractions according to all the rules.

Well, the knowledge of addition and subtraction of fractions was refreshed. Converting fractions from one type to another was repeated. You can also get checked. Shall we settle it a little?)

Calculate:

Answers (in disarray):

71/20; 3/5; 17/12; -5/4; 11/6

Multiplication/division of fractions - in the next lesson. There are also tasks for all operations with fractions.

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.

Lesson content

Adding fractions with like denominators

There are two types of addition of fractions:

Adding fractions with like denominators
Adding fractions with different denominators

First, let's learn the addition of fractions with like 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 . Add the numerators and leave the denominator unchanged:

This example can be easily understood if we remember the pizza, which is divided into four parts. If you add pizza to pizza, you get pizza:

Example 2. Add fractions and .

The answer turned out to be an improper fraction. When the end of the task comes, it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part of it. In our case, the whole part is easily isolated - two divided by two equals one:

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

Example 3. Add fractions and .

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

This example can be easily understood if we remember the pizza, which is divided into three parts. If you add more pizza to the pizza, you get pizza:

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 drawing. If you add pizza to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.

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

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

Adding fractions with different denominators

Now let's learn how to add fractions with different denominators. When adding fractions, the denominators of the 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 right away, since 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 look at only one of them, since the other methods may seem complicated for a beginner.

The essence of this method is that first the LCM of the denominators of both fractions is searched. The LCM is then divided by the denominator of the first fraction to obtain the first additional factor. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and a second additional factor is obtained.

The numerators and denominators of the fractions are then 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. Let's add the 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 let's return to fractions and . First, 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 multiplier. We write it down to the first fraction. To do this, make a small oblique line over the fraction and write down the additional factor found 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 multiplier. We write it down to the second fraction. Again, we make a small oblique line over the second fraction and write down the additional factor found above it:

Now we have everything ready for addition. It remains to multiply the numerators and denominators of the fractions by their additional factors:

Look carefully 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 take this example to the end:

This completes the example. It turns out to add .

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

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

The first drawing represents a fraction (four pieces out of six), and the second drawing represents a fraction (three pieces out of six). Adding these pieces we get (seven pieces out of six). This fraction is improper, so we highlighted the whole part of it. As a result, we got (one whole pizza and another sixth pizza).

Please note that we have described this example in too much detail. In educational institutions it is not customary to write in such detail. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the found additional factors by your numerators and denominators. If we were at school, we would have to write this example as follows:

But there is also another side to the coin. If you do not take detailed notes in the first stages of studying mathematics, then questions of the sort begin to appear. “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:

Find the LCM of the denominators of fractions;
Divide the LCM by the denominator of each fraction and obtain an additional factor for each fraction;
Multiply the numerators and denominators of fractions by their additional factors;
Add fractions that have the same denominators;
If the answer turns out to be an improper fraction, then select its whole part;

Example 2. Find the value of an expression .

Let's use the instructions given above.

Step 1. Find the LCM of the denominators of the fractions

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

Step 2. Divide the LCM by the denominator of each fraction and get an additional factor 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 above 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 get the second additional factor 4. We write it above 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 get the third additional factor 3. We write it above the third fraction:

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

We multiply the numerators and denominators by their additional factors:

Step 4. Add fractions with the same denominators

We came to the conclusion that fractions that had different denominators turned into fractions that had the same (common) denominators. All that remains is to add these fractions. Add it 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 moved 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 the 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 turns out to be an improper fraction, then highlight the whole part in it

Our answer turned out to be an improper fraction. We have to highlight a whole part of it. We highlight:

We received an answer

Subtracting fractions with like denominators

There are two types of subtraction of fractions:

Subtracting fractions with like denominators
Subtracting fractions with different denominators

First, let's learn how to subtract fractions with like 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, but leave the denominator the same.

For example, let's find the value of the expression . To solve this example, you need 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 remember the pizza, which 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 remember the pizza, which 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 about subtracting fractions with the same denominators. It is enough to understand the following rules:

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;
If the answer turns out to be an improper fraction, then you need to highlight the whole part of it.

Subtracting fractions with different denominators

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

The common denominator is found using 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 above 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 above the second fraction.

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

Example 1. Find the meaning of the expression:

These fractions have different denominators, so you need to reduce 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 let's return to fractions and

Let's find an additional factor for the first fraction. To do this, 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. Write a four above the first fraction:

We do the same with the second fraction. 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 three over the second fraction:

Now we are 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 had the same denominators. And we already know how to subtract such fractions. Let's take this example to the end:

We received an answer

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

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

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

The first picture shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting 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 first you need to reduce them to the same (common) denominator.

Let's 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, 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 above 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 above 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 above 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 had 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 regular fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it simpler. What can be done? You can shorten this fraction.

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

So, we find the gcd of 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

We received 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 that number and leave the denominator the same.

Example 1. Multiply a fraction by the number 1.

Multiply the numerator of the fraction by the number 1

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

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

This notation can be understood as taking half of one. 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 was an improper fraction. Let's highlight the whole part of it:

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

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

Multiplying fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer turns out to be an improper fraction, you need to highlight the whole part of it.

Example 1. Find the value of the expression.

We received an answer. It is advisable 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 make pizza. Remember what pizza looks like when divided into three parts:

One piece of this pizza and the two pieces we took will have the same dimensions:

In other words, we are talking about the same size pizza. 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 was an improper fraction. Let's highlight the 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 regular fraction, but it would be good if it was shortened. 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 numbers 105 and 450:

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

Representing a whole number as a fraction

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

Reciprocal numbers

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

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

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

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

Is it possible to find a number that, when multiplied by 5, gives one? It turns out it is possible. Let's imagine 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 upside down:

What will happen as a 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 you multiply 5 by you get one.

The reciprocal of a number can also be found for any other integer.

You can also find the reciprocal of any other fraction. To do this, just turn it over.

Dividing a fraction by a number

Let's say we have half a pizza:

Let's divide it equally between two. How much pizza will each person get?

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

Division of fractions is done using reciprocals. Reciprocal numbers allow you to replace division with multiplication.

To divide a fraction by a number, you need to multiply the fraction by the inverse of the divisor.

Using this rule, we will write down the division of our half of the pizza into two parts.

So, you need to divide the fraction by the number 2. Here the dividend is the fraction and the divisor is the number 2.

To divide a fraction by the number 2, you need to multiply this fraction by the reciprocal of the divisor 2. The reciprocal of the divisor 2 is the fraction. So you need to multiply by

Note! Before writing your final answer, see if you can shorten the fraction you received.

Subtracting fractions with like denominators, examples:

Subtracting a proper fraction from one.

If it is necessary to subtract a fraction from a unit that is proper, the unit is converted to the form of an improper fraction, its denominator is equal to the denominator of the subtracted fraction.

An example of subtracting a proper fraction from one:

Denominator of the fraction to be subtracted = 7 , i.e., we represent one as an improper fraction 7/7 and subtract it according to the rule for subtracting fractions with like denominators.

Subtracting a proper fraction from a whole number.

Rules for subtracting fractions - correct from a whole number (natural number):

We convert given fractions that contain an integer part into improper ones. We obtain normal terms (it doesn’t matter if they have different denominators), which we calculate according to the rules given above;
Next, we calculate the difference between the fractions that we received. As a result, we will almost find the answer;
We perform the inverse transformation, that is, we get rid of the improper fraction - we select the whole part in the fraction.

Subtract a proper fraction from a whole number: represent the natural number as a mixed number. Those. We take a unit in a natural number and convert it to the form of an improper fraction, the denominator being the same as that of the subtracted fraction.

Example of subtracting fractions:

In the example, we replaced one with the improper fraction 7/7 and instead of 3 we wrote down a mixed number and subtracted a fraction from the fractional part.

Subtracting fractions with different denominators.

Or, to put it another way, subtracting different fractions.

Rule for subtracting fractions with different denominators. In order to subtract fractions with different denominators, it is necessary, first, to reduce these fractions to the lowest common denominator (LCD), and only after this, perform the subtraction as with fractions with the same denominators.

The common denominator of several fractions is LCM (least common multiple) natural numbers that are the denominators of these fractions.

Attention! If in the final fraction the numerator and denominator have common factors, then the fraction must be reduced. An improper fraction is best represented as a mixed fraction. Leaving the subtraction result without reducing the fraction where possible is an incomplete solution to the example!

Procedure for subtracting fractions with different denominators.

find the LCM for all denominators;
put additional factors for all fractions;
multiply all numerators by an additional factor;
We write the resulting products into the numerator, signing the common denominator under all fractions;
subtract the numerators of fractions, signing the common denominator under the difference.

In the same way, addition and subtraction of fractions is carried out if there are letters in the numerator.

Subtracting fractions, examples:

Subtracting mixed fractions.

At subtracting mixed fractions (numbers) separately, the integer part is subtracted from the integer part, and the fractional part is subtracted from the fractional part.

The first option for subtracting mixed fractions.

If the fractional parts the same denominators and numerator of the fractional part of the minuend (we subtract it from it) ≥ numerator of the fractional part of the subtrahend (we subtract it).

For example:

The second option for subtracting mixed fractions.

When fractional parts different denominators. To begin with, we bring the fractional parts to a common denominator, and after that we subtract the whole part from the whole part, and the fractional part from the fractional part.

For example:

The third option for subtracting mixed fractions.

The fractional part of the minuend is less than the fractional part of the subtrahend.

Example:

Because Fractional parts have different denominators, which means, as in the second option, we first bring ordinary fractions to a common denominator.

The numerator of the fractional part of the minuend is less than the numerator of the fractional part of the subtrahend.3 < 14. This means we take a unit from the whole part and reduce this unit to the form of an improper fraction with the same denominator and numerator = 18.

In the numerator from the right side we write the sum of the numerators, then we open the brackets in the numerator from the right side, that is, we multiply everything and give similar ones. We do not open the parentheses in the denominator. It is customary to leave the product in the denominators. We get:

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 an object not as a whole, but in separate pieces. Start studying this topic - shares. Shares are equal parts, into which this or that object is divided. After all, it is not always possible to express, for example, the length or price of a product as a whole number; parts or shares of some measure should be taken into account. Formed from the verb “to split” - to divide into parts, and having Arabic roots, the word “fraction” itself arose in the Russian language in the 8th century.

Fractional expressions have long been considered the most difficult branch of mathematics. In the 17th century, when first textbooks on mathematics appeared, they were called “broken numbers,” which was very difficult for people to understand.

The modern form of simple fractional remainders, the parts of which are separated by a horizontal line, was first promoted by Fibonacci - Leonardo of Pisa. His works are dated to 1202. But the purpose of this article is to simply and clearly explain to the reader how mixed fractions with different denominators are multiplied.

Multiplying fractions with different denominators

Initially it is worth determining types of fractions:

correct;
incorrect;
mixed.

Next, you need to remember how fractional numbers with the same denominators are multiplied. The very rule of this process is not difficult to formulate independently: the result of multiplying simple fractions with identical 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 initially existing ones.

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 line will be a product of different numbers and, naturally, 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 methods for reducing fractional expressions. You can only reduce numerator numbers with denominator numbers; adjacent factors above or below the fraction line cannot be reduced.

Along with simple fractions, 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 multiplication of a number by ordinary fractional part, the rule for this action can be written as:

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 solution to multiplying a number by a fractional remainder. You just need to divide the denominator by this number:

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

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

Convert mixed numbers to improper fractions and obtain 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 of representing a mixed fraction as an improper fraction, and 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 whole part with the denominator and adding it with the numerator of the original fractional remainder, and the denominator remains the same.

This process also works in the opposite direction. To separate the whole part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator using a “corner”.

Multiplying improper fractions produced in a generally accepted way. When writing under a single fraction line, you need to reduce fractions as necessary in order to reduce numbers using this method and make it easier to calculate the result.

There are many helpers on the Internet to solve even complex mathematical problems in various variations of programs. A sufficient number of such services offer their assistance in calculating the multiplication of fractions with different numbers in the denominators - 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’s easy to work with; you fill in the appropriate fields on the website page, select the sign of the mathematical operation, and click “calculate.” The program calculates automatically.

The topic of arithmetic operations with fractions is relevant throughout the education of middle and high school students. In high school, they no longer consider 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-mastered basic knowledge gives complete confidence in successfully solving the most complex problems.

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

One of the most important sciences, the application of which can be seen in disciplines such as chemistry, physics and even biology, is mathematics. Studying this science allows you to develop some mental qualities and improve your ability to concentrate. One of the topics that deserve special attention in the Mathematics course is adding and subtracting fractions. Many students find it difficult to study. Perhaps our article will help to better understand this topic.

How to subtract fractions whose denominators are the same

Fractions are the same numbers with which you can perform various operations. Their difference from whole numbers lies in the presence of a denominator. That is why, when performing operations with fractions, you need to study some of their features and rules. The simplest case is the subtraction of ordinary fractions whose denominators are represented as the same number. Performing this action will not be difficult if you know a simple rule:

In order to subtract a second from one fraction, it is necessary to subtract the numerator of the subtracted fraction from the numerator of the fraction being reduced. We write this number into the numerator of the difference, and leave the denominator the same: k/m - b/m = (k-b)/m.

Examples of subtracting fractions whose denominators are the same

7/19 - 3/19 = (7 - 3)/19 = 4/19.

From the numerator of the fraction “7” we subtract the numerator of the fraction “3” to be subtracted, we get “4”. We write this number in the numerator of the answer, and in the denominator we put the same number that was in the denominators of the first and second fractions - “19”.

The picture below shows several more similar examples.

Let's consider a more complex example where fractions with like denominators are subtracted:

29/47 - 3/47 - 8/47 - 2/47 - 7/47 = (29 - 3 - 8 - 2 - 7)/47 = 9/47.

From the numerator of the fraction “29” being reduced by subtracting in turn the numerators of all subsequent fractions - “3”, “8”, “2”, “7”. As a result, we get the result “9”, which we write down in the numerator of the answer, and in the denominator we write down the number that is in the denominators of all these fractions - “47”.

Adding fractions that have the same denominator

Adding and subtracting ordinary fractions follows the same principle.

In order to add fractions whose denominators are the same, you need to add the numerators. The resulting number is the numerator of the sum, and the denominator will remain the same: k/m + b/m = (k + b)/m.

Let's see what this looks like using an example:

1/4 + 2/4 = 3/4.

To the numerator of the first term of the fraction - “1” - add the numerator of the second term of the fraction - “2”. The result - “3” - is written into the numerator of the sum, and the denominator is left the same as that present in the fractions - “4”.

Fractions with different denominators and their subtraction

We have already considered the operation with fractions that have the same denominator. As you can see, knowing simple rules, solving such examples is quite easy. But what if you need to perform an operation with fractions that have different denominators? Many secondary school students are confused by such examples. But even here, if you know the principle of the solution, the examples will no longer be difficult for you. There is also a rule here, without which solving such fractions is simply impossible.

To subtract fractions with different denominators, they must be reduced to the same smallest denominator.

We will talk in more detail about how to do this.

Property of a fraction

In order to bring several fractions to the same denominator, you need to use the main property of a fraction in the solution: after dividing or multiplying the numerator and denominator by the same number, you get a fraction equal to the given one.

So, for example, the fraction 2/3 can have denominators such as “6”, “9”, “12”, etc., that is, it can have the form of any number that is a multiple of “3”. After we multiply the numerator and denominator by “2”, we get the fraction 4/6. After we multiply the numerator and denominator of the original fraction by “3”, we get 6/9, and if we perform a similar operation with the number “4”, we get 8/12. One equality can be written as follows:

2/3 = 4/6 = 6/9 = 8/12…

How to convert multiple fractions to the same denominator

Let's look at how to reduce multiple fractions to the same denominator. For example, let's take the fractions shown in the picture below. First you need to determine which number can become the denominator for all of them. To make things easier, let's factorize the existing denominators.

The denominator of the fraction 1/2 and the fraction 2/3 cannot be factorized. The denominator 7/9 has two factors 7/9 = 7/(3 x 3), the denominator of the fraction 5/6 = 5/(2 x 3). Now we need to determine which factors will be the smallest for all these four fractions. Since the first fraction has the number “2” in the denominator, it means that it must be present in all denominators; in the fraction 7/9 there are two triplets, which means that both of them must also be present in the denominator. Taking into account the above, we determine that the denominator consists of three factors: 3, 2, 3 and is equal to 3 x 2 x 3 = 18.

Let's consider the first fraction - 1/2. There is a “2” in its denominator, but there is not a single “3”, but there should be two. To do this, we multiply the denominator by two triples, but, according to the property of a fraction, we must multiply the numerator by two triples:
1/2 = (1 x 3 x 3)/(2 x 3 x 3) = 9/18.

We do the same with the remaining fractions.

2/3 - one three and one two are missing in the denominator:
2/3 = (2 x 3 x 2)/(3 x 3 x 2) = 12/18.
7/9 or 7/(3 x 3) - the denominator is missing a two:
7/9 = (7 x 2)/(9 x 2) = 14/18.
5/6 or 5/(2 x 3) - the denominator is missing a three:
5/6 = (5 x 3)/(6 x 3) = 15/18.

All together it looks like this:

How to subtract and add fractions that have different denominators

As mentioned above, in order to add or subtract fractions that have different denominators, they must be reduced to the same denominator, and then use the rules for subtracting fractions that have the same denominator, which have already been discussed.

Let's look at this using an example: 4/18 - 3/15.

Finding the multiple of numbers 18 and 15:

The number 18 is made up of 3 x 2 x 3.
The number 15 is made up of 5 x 3.
The common multiple will be the following factors: 5 x 3 x 3 x 2 = 90.

After the denominator has been found, it is necessary to calculate the factor that will be different for each fraction, that is, the number by which not only the denominator, but also the numerator will need to be multiplied. To do this, we divide the number that we found (the common multiple) by the denominator of the fraction for which we need to determine additional factors.

90 divided by 15. The resulting number “6” will be a multiplier for 3/15.
90 divided by 18. The resulting number “5” will be a multiplier for 4/18.

The next stage of our solution is to reduce each fraction to the denominator “90”.

We have already talked about how this is done. Let's see how this is written in an example:

(4 x 5)/(18 x 5) - (3 x 6)/(15 x 6) = 20/90 - 18/90 = 2/90 = 1/45.

If fractions have small numbers, then you can determine the common denominator, as in the example shown in the picture below.

The same is true for those with different denominators.

Subtraction and having integer parts

We have already discussed in detail the subtraction of fractions and their addition. But how to subtract if a fraction has an integer part? Again, let's use a few rules:

Convert all fractions that have an integer part to improper ones. In simple words, remove an entire part. To do this, multiply the number of the integer part by the denominator of the fraction, and add the resulting product to the numerator. The number that comes out after these actions is the numerator of the improper fraction. The denominator remains unchanged.
If fractions have different denominators, they should be reduced to the same denominator.
Perform addition or subtraction with the same denominators.
When receiving an improper fraction, select the whole part.

There is another way in which you can add and subtract fractions with whole parts. To do this, actions are performed separately with whole parts, and actions with fractions separately, and the results are recorded together.

The example given consists of fractions that have the same denominator. In the case when the denominators are different, they must be brought to the same value, and then perform the actions as shown in the example.

Subtracting fractions from whole numbers

Another type of operation with fractions is the case when a fraction must be subtracted from. At first glance, such an example seems difficult to solve. However, everything is quite simple here. To solve it, you need to convert the integer into a fraction, and with the same denominator that is in the subtracted fraction. Next, we perform a subtraction similar to subtraction with identical denominators. In an example it looks like this:

7 - 4/9 = (7 x 9)/9 - 4/9 = 53/9 - 4/9 = 49/9.

The subtraction of fractions (grade 6) presented in this article is the basis for solving more complex examples that are covered in subsequent grades. Knowledge of this topic is subsequently used to solve functions, derivatives, and so on. Therefore, it is very important to understand and understand the operations with fractions discussed above.