Adding simple fractions. Operations with fractions

Adding and subtracting fractions with like denominators
Adding and subtracting fractions with different denominators
Concept of NOC
Reducing fractions to the same denominator
How to add a whole number and a fraction

1 Adding and subtracting fractions with like denominators

To add fractions with the same denominators, you need to add their numerators, but leave the denominator the same, for example:

To subtract fractions with the same denominators, 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:

To add mixed fractions, you need to separately add their whole parts, and then add their fractional parts, and write the result as a mixed fraction,

If, when adding the fractional parts, the result is Not proper fraction, select a whole part from it and add it to the whole part, for example:

2 Adding and subtracting fractions with different denominators

In order to add or subtract fractions with different denominators, you must first reduce them to the same denominator, and then proceed as indicated at the beginning of this article. The common denominator of several fractions is the LCM (least common multiple). For the numerator of each fraction, additional factors are found by dividing the LCM by the denominator of this fraction. We will look at an example later, after we understand what an NOC is.

3 Least common multiple (LCM)

The least common multiple of two numbers (LCM) is the smallest natural number that is divisible by both numbers without leaving a remainder. Sometimes the NOC can be selected orally, but more often, especially when working with large numbers, you have to find the LOC in writing using the following algorithm:

In order to find the LCM of several numbers, you need:

Divide these numbers into prime factors
Take the largest expansion and write these numbers as a product
Identify numbers in other expansions that do not appear in the largest expansion (or occur in it smaller number times), and add them to the work.
Multiply all the numbers in the product, this will be the LCM.

For example, let's find the LCM of the numbers 28 and 21:

4Reducing fractions to the same denominator

Let's return to adding fractions with different denominators.

When we reduce fractions to the same denominator, equal to the LCM of both denominators, we must multiply the numerators of these fractions by additional multipliers. You can find them by dividing the LCM by the denominator of the corresponding fraction, for example:

Thus, to reduce fractions to the same exponent, you must first find the LCM (that is, smallest number, which is divisible by both denominators) of the denominators of these fractions, then add additional factors to the numerators of the fractions. You can find them by dividing the common denominator (CLD) by the denominator of the corresponding fraction. Then you need to multiply the numerator of each fraction by an additional factor, and put the LCM as the denominator.

5How to add a whole number and a fraction

In order to add a whole number and a fraction, you just need to add this number before the fraction, which will result in a mixed fraction, for example.

To add a whole number to a fraction, it is enough to perform a series of actions, or rather calculations.

For example, you have 7 - an integer; you need to add it to the fraction 1/2.

We proceed as follows:

We multiply 7 by the denominator (2), we get 14,
add to 14 top part(1), comes out 15,
and substitute the denominator.
the result is 15/2.

In this simple way you can add whole numbers to fractions.

And to isolate a whole number from a fraction, you need to divide the numerator by the denominator, and the remainder - and there will be a fraction.

The operation of adding an integer to a proper ordinary fraction is not complicated and sometimes simply involves the formation of a mixed fraction in which whole part placed to the left of the fractional part, for example, such a fraction will be mixed:

However, more often than not, adding a whole number to a fraction results in an improper fraction in which the numerator is greater than the denominator. This operation is performed as follows: the whole number is represented as an improper fraction with the same denominator as the fraction being added, and then the numerators of both fractions are simply added. In an example it will look like this:

5+1/8 = 5*8/8+1/8 = 40/8+1/8 = 41/8

I think it's very simple.

For example, we have the fraction 1/4 (this is the same as 0.25, that is, a quarter of the whole number).

And to this quarter you can add any integer, for example 3. You get three and a quarter:

3.25. Or in fraction it is expressed like this: 3 1/4

Using this example, you can add any fractions with any integers.

You need to raise a whole number to a fraction with a denominator of 10 (6/10). Next, bring the existing fraction to a common denominator of 10 (35=610). Well, perform the operation as with ordinary fractions 610+610=1210 total 12.

There are two ways to do this.

1). A fraction can be converted to a whole number and addition can be performed. For example, 1/2 is 0.5; 1/4 equals 0.25; 2/5 is 0.4, etc.

Take the integer 5, to which you need to add the fraction 4/5. Let's transform the fraction: 4/5 is 4 divided by 5 and we get 0.8. Adds 0.8 to 5 and we get 5.8 or 5 4/5.

2). Second method: 5 + 4/5 = 29/5 = 5 4/5.

Adding fractions is simple mathematical operation, example, you need to add the integer 3 and the fraction 1/7. To add these two numbers you must have the same denominator, so you must multiply three by seven and divide by that figure, then you get 21/7+1/7, denominator one, add 21 and 1, you get the answer 22/7 .

Just take and add an integer to this fraction. Let's say you need 6 + 1/2 = 6 1/2. Well, if this decimal then you can do it like this: 6+1.2=7.2.

To add a fraction and a whole number, you need to add the fraction to the whole number and write them in the form complex number, for example, when adding an ordinary fraction with an integer, we get: 1/2 +3 =3 1/2; when adding a decimal fraction: 0.5 +3 =3.5.

A fraction in itself is not a whole number, because its quantity does not reach it, and therefore there is no need to convert the whole number into this fraction. Therefore, the integer remains an integer and fully demonstrates the full value, and the fraction is added to it, and demonstrates how much this integer is missing before adding the next full point.

Academic example.

10 + 7/3 = 10 whole and 7/3.

If, of course, there are integers, then they are summed with integers.

12 + 5 7/9 = 17 and 7/9.

It depends on which integer and which fraction.

If both terms are positive, this fraction should be added to the whole number. The result will be a mixed number. Moreover, there may be 2 cases.

Case 1.

The fraction is correct, i.e. numerator less than the denominator. Then the mixed number obtained after the assignment will be the answer.

4/9 + 10 = 10 4/9 (ten point four ninths).

Case 2.

The fraction is improper, i.e. the numerator is greater than the denominator. Then a little conversion is required. An improper fraction should be turned into a mixed number, in other words, the whole part should be separated. This is done like this:

After this, you need to add the whole part of the improper fraction to the whole number and add its fractional part to the resulting amount. In the same way, a whole is added to a mixed number.

1) 11/4 + 5 = 2 3/4 + 5 = 7 3/4 (7 point three quarters).

2) 5 1/2 + 6 = 11 1/2 (11 point one).

If one of the terms or both negative, then we perform the addition according to the rules for adding numbers with different or identical signs. A whole number is represented as the ratio of that number and 1, and then both the numerator and the denominator are multiplied by a number equal to the denominator of the fraction to which the whole number is added.

3) 1/5 + (-2)= 1/5 + -2/1 = 1/5 + -10/5 = -9/5 = -1 4/5 (minus 1 point four fifths).

4) -13/3 + (-4) = -13/3 + -4/1 = -13/3 + -12/3 = -25/3 = -8 1/3 (minus 8 point one third).

Comment.

After meeting negative numbers, when studying actions with them, 6th grade students should understand that negative fraction adding a positive integer is the same as subtracting from natural number fraction. This action is known to be performed like this:

In fact, in order to add a fraction and an integer, you simply need to convert the existing integer to a fraction, and doing this is as easy as shelling pears. You just need to take the denominator of the fraction (in the example) and make it the denominator of a whole number by multiplying it by that denominator and dividing, here's an example:

2+2/3 = 2*3/3+2/3 = 6/3+2/3 = 8/3

Find the numerator and denominator. A fraction includes two numbers: the number that is located above the line is called the numerator, and the number that is located below the line is called the denominator. The denominator stands for total parts into which a whole is divided, and the numerator is the number of such parts being considered.

For example, in the fraction ½ the numerator is 1 and the denominator is 2.

Determine the denominator. If two or more fractions have a common denominator, such fractions have the same number under the line, that is, in this case, a certain whole is divided into the same number of parts. Adding fractions with a common denominator is very simple, since the denominator of the total fraction will be the same as the fractions being added. For example:

The fractions 3/5 and 2/5 have a common denominator of 5.
The fractions 3/8, 5/8, 17/8 have a common denominator of 8.

Determine the numerators. To add fractions with a common denominator, add their numerators and write the result above the denominator of the fractions being added.

The fractions 3/5 and 2/5 have numerators 3 and 2.
Fractions 3/8, 5/8, 17/8 have numerators 3, 5, 17.

Add up the numerators. In problem 3/5 + 2/5, add the numerators 3 + 2 = 5. In problem 3/8 + 5/8 + 17/8, add the numerators 3 + 5 + 17 = 25.

Write the total fraction. Remember that when adding fractions with a common denominator, it remains unchanged - only the numerators are added.

3/5 + 2/5 = 5/5
3/8 + 5/8 + 17/8 = 25/8

Convert the fraction if necessary. Sometimes a fraction can be written as a whole number rather than as a fraction or decimal. For example, the fraction 5/5 is easily converted to 1, since any fraction whose numerator is equal to its denominator is 1. Imagine a pie cut into three parts. If you eat all three parts, you will have eaten the whole (one) pie.

Any fraction can be converted to a decimal; To do this, divide the numerator by the denominator. For example, the fraction 5/8 can be written as follows: 5 ÷ 8 = 0.625.

If possible, simplify the fraction. A simplified fraction is a fraction whose numerator and denominator do not have common factors.

For example, consider the fraction 3/6. Here both the numerator and the denominator have common divisor, equal to 3, that is, the numerator and denominator are completely divisible by 3. Therefore, the fraction 3/6 can be written as follows: 3 ÷ 3/6 ÷ 3 = ½.

If necessary, convert improper fraction to mixed fraction(mixed number). An improper fraction has a numerator greater than its denominator, for example, 25/8 (a proper fraction has a numerator less than its denominator). An improper fraction can be converted to a mixed fraction, which consists of an integer part (that is, a whole number) and a fraction part (that is, a proper fraction). To convert an improper fraction, such as 25/8, to a mixed number, follow these steps:

Divide the numerator of an improper fraction by its denominator; write down the partial quotient (whole answer). In our example: 25 ÷ 8 = 3 plus some remainder. IN in this case the whole answer is the whole part of the mixed number.
Find the remainder. In our example: 8 x 3 = 24; subtract the resulting result from the original numerator: 25 - 24 = 1, that is, the remainder is 1. In this case, the remainder is the numerator of the fractional part of the mixed number.
Write a mixed fraction. The denominator does not change (that is, it is equal to the denominator of the improper fraction), so 25/8 = 3 1/8.

Some of the most difficult for a student to understand are different actions with simple fractions. This is due to the fact that it is still difficult for children to think abstractly, and fractions, in fact, look exactly like that to them. Therefore, when presenting the material, teachers often resort to analogies and explain subtraction and addition of fractions literally on their fingers. Although not a single school mathematics lesson is complete without rules and definitions.

Basic Concepts

Before you begin, it is advisable to learn a few basic definitions and rules. Initially, it is important to understand what a fraction is. It refers to a number that represents one or more fractions of a unit. For example, if you cut a loaf into 8 pieces and put 3 slices of them on a plate, then 3/8 will be a fraction. Moreover, in this writing it will be a simple fraction, where the number above the line is the numerator, and below it is the denominator. But if you write it down as 0.375, it will already be a decimal fraction.

In addition, simple fractions are divided into proper, improper and mixed. The first include all those whose numerator is less than the denominator. If, on the contrary, the denominator is less than the numerator, it will already be an improper fraction. If the correct number is preceded by an integer, they are called mixed numbers. Thus, the fraction 1/2 is proper, but 7/2 is not. And if you write it in this form: 3 1/2, then it will become mixed.

To make it easier to understand what adding fractions is and to perform it with ease, it is also important to remember its essence in the following. If the numerator and denominator are multiplied by the same number, the fraction will not change. It is this property that allows you to perform simple operations with ordinary and other fractions. In fact, this means that 1/15 and 3/45 are essentially the same number.

Adding fractions with like denominators

Performing this action usually does not cause much difficulty. Adding fractions in this case is very similar to a similar operation with integers. The denominator remains unchanged, and the numerators are simply added together. For example, if you need to add the fractions 2/7 and 3/7, then the solution to the school problem in your notebook will be like this:

2/7 + 3/7 = (2+3)/7 = 5/7.

In addition, this addition of fractions can be explained using simple example. Take a regular apple and cut it, for example, into 8 pieces. First lay out 3 parts separately, and then add 2 more to them. As a result, the cup will contain 5/8 of a whole apple. Samu arithmetic problem written as below:

3/8 + 2/8 = (3+2)/8 = 5/8.

But often there are more complex problems where you need to add together, for example, 5/9 and 3/5. This is where the first difficulties arise in working with fractions. After all, adding such numbers will require additional knowledge. Now you will fully need to remember their main property. To add fractions from the example, you first need to bring them to one common denominator. To do this, you simply need to multiply 9 and 5 together, multiply the numerator “5” by 5, and “3”, respectively, by 9. Thus, the following fractions are already added: 25/45 and 27/45. Now all that remains is to add the numerators and get the answer 52/45. On a piece of paper, an example would look like this:

5/9 + 3/5 = (5 x 5)/(9 x 5) + (3 x 9)/(5 x 9) = 25/45 + 27/45 = (25+27)/45 = 52/ 45 = 1 7 / 45.

But adding fractions with such denominators does not always require simply multiplying the numbers under the line. First they look for the lowest common denominator. For example, as for the fractions 2/3 and 5/6. For them it will be the number 6. But the answer is not always obvious. In this case, it is worth remembering the rule for finding the least common multiple (abbreviated LCM) of two numbers.

It is understood as the least common factor of two integers. To find it, they decompose each into prime factors. Now write down those of them that appear at least once in each number. They multiply them together and get the same denominator. In reality, everything looks a little simpler.

For example, you need to add the fractions 4/15 and 1/6. So, 15 is obtained by multiplying the simple numbers 3 and 5, and six is obtained by multiplying the simple numbers two and three. This means that the LCM for them will be 5 x 3 x 2 = 30. Now, dividing 30 by the denominator of the first fraction, we get the multiplier for its numerator - 2. And for the second fraction it will be the number 5. Thus, it remains to add the ordinary fractions 8/30 and 5/30 and receive an answer of 13/30. Everything is extremely simple. In your notebook you should write this task down like this:

4/15 + 1/6 = (4 x 2)/(15 x 2) + (1 x 5)/(6 x 5) = 8/30 + 5/30 = 13/30.

LCM(15, 6) = 30.

Addition of mixed numbers

Now, knowing all the basic addition techniques simple fractions, you can try your hand at more complex examples. And these will be mixed numbers, by which we mean a fraction of this form: 2 2 / 3. Here the whole part is written before the proper fraction. And many people get confused when performing actions with such numbers. In reality, the same rules apply here.

To add mixed numbers, add whole parts and proper fractions separately. And then these 2 results are summed up. In practice, everything is much simpler, you just need to practice a little. For example, the problem requires adding the following mixed numbers: 1 1/3 and 4 2/5. To do this, first add 1 and 4 to get 5. Then add 1/3 and 2/5 using lowest common denominator techniques. The solution will be 11/15. And the final answer is 5 11/15. In a school notebook it will look much shorter:

1 1 / 3 + 4 2 / 5 = (1 + 4) + (1/3 + 2/5) = 5 + 5/15 + 6/15 = 5 + 11/15 = 5 11 / 15 .

Adding Decimals

Besides ordinary fractions, there are also decimal ones. By the way, they are much more common in life. For example, the price in a store often looks like this: 20.3 rubles. This is the same fraction. Of course, these are much easier to fold than ordinary ones. Basically, you just need to add 2 ordinary numbers, the main thing is to put a comma in the right place. This is where difficulties arise.

For example, you need to add 2.5 and 0.56. To do this correctly, you need to add a zero to the first one at the end, and everything will be fine.

2,50 + 0,56 = 3,06.

It's important to know that any decimal can be converted to a fraction, but not every fraction can be written as a decimal. So, from our example, 2.5 = 2 1/2 and 0.56 = 14/25. But a fraction like 1/6 will only be approximately equal to 0.16667. The same situation will happen with other similar numbers - 2/7, 1/9 and so on.

Conclusion

Many schoolchildren, not understanding the practical side of working with fractions, treat this topic carelessly. However, in more these basic knowledge will make you crack like nuts complex examples with logarithms and finding derivatives. Therefore, it’s worth once to thoroughly understand the operations with fractions, so that later you don’t bite your elbows in frustration. After all, it is unlikely that a teacher in high school will return to this already covered topic. Any high school student should be able to perform such exercises.

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 pizzas 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 same denominators.

But fractions cannot be added right away, since these fractions 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 too detailed. IN educational institutions It’s 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 back side medals. 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 select the whole part of 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 fraction by that number and leave the denominator unchanged.

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 pizza once, you get pizza

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:

The number being multiplied by the fraction and the denominator of the fraction are resolved if they have a common factor greater than one.

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 four being multiplied and the four in the denominator of the fraction can be reduced. These fours can be reduced 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 three, and then dividing by one does not change anything. Therefore, the solution can be written briefly:

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 do not cancel.

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

Reducing a fraction means that both numerator and denominator will be divided by the same number. In the situation with the expression, division is performed only in the numerator, since writing this is the same as writing . We see that division is performed only in the numerator, and no division occurs in the denominator.

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 given fraction. The fraction can be reduced by 2. Then final decision 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're talking about 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 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.