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

As you know from mathematics, a fractional number consists of a numerator and a denominator. The numerator is at the top and the denominator at the bottom.

It is quite simple to perform mathematical operations on the addition or subtraction of fractional quantities with the same denominator. You just need to be able to add or subtract the numbers in the numerator (top), and the same bottom number remains unchanged.

For example, let's take the fractional number 7/9, here:

• the number "seven" on top is the numerator;
• the number "nine" below is the denominator.

Example 1. Addition:

5/49 + 4/49 = (5+4) / 49 =9/49.

Example 2. Subtraction:

6/35−3/35 = (6−3) / 35 = 3/35.

Subtraction of simple fractional values ​​\u200b\u200bthat have a different denominator

To perform a mathematical operation to subtract values ​​that have a different denominator, you must first bring them to a common denominator. When performing this task, it is necessary to adhere to the rule that this common denominator must be the smallest of all possible options.

Example 3

Given two simple quantities with different denominators (lower numbers): 7/8 and 2/9.

Subtract the second from the first value.

The solution consists of several steps:

1. Find the common lower number, i.e. that which is divisible both by the lower value of the first fraction and the second. This will be the number 72, since it is a multiple of the numbers "eight" and "nine".

2. The bottom digit of each fraction has increased:

• the number "eight" in the fraction 7/8 increased nine times - 8*9=72;
• the number "nine" in the fraction 2/9 has increased eight times - 9*8=72.

3. If the denominator (lower number) has changed, then the numerator (upper number) must also change. According to the existing mathematical rule, the upper figure must be increased by exactly the same amount as the lower one. I.e:

• the numerator "seven" in the first fraction (7/8) is multiplied by the number "nine" - 7*9=63;
• the numerator "two" in the second fraction (2/9) is multiplied by the number "eight" - 2*8=16.

4. As a result of the actions, we got two new values, which, however, are identical to the original ones.

• first: 7/8 = 7*9 / 8*9 = 63/72;
• second: 2/9 = 2*8 / 9*8 = 16/72.

5. Now it is allowed to subtract one fractional number from another:

7/8−2/9 = 63/72−16/72 =?

6. Performing this action, we return to the topic of subtracting fractions with the same lower numbers (denominators). And this means that the subtraction action will be carried out from above, in the numerator, and the lower figure is transferred without changes.

63/72−16/72 = (63−16) / 72 = 47/72.

7/8−2/9 = 47/72.

Example 4

Let's complicate the problem by taking several fractions for solving with different, but multiple digits at the bottom.

Values ​​given: 5/6; 1/3; 1/12; 7/24.

They must be taken away from each other in this sequence.

1. We bring the fractions in the above way to a common denominator, which will be the number "24":

• 5/6 = 5*4 / 6*4 = 20/24;
• 1/3 = 1*8 / 3*8 = 8/24;
• 1/12 = 1*2 / 12*2 = 2/24.

7/24 - we leave this last value unchanged, since the denominator is the total number "24".

2. Subtract all values:

20/24−8/2−2/24−7/24 = (20−8−2−7)/24 = 3/24.

3. Since the numerator and denominator of the resulting fraction are divisible by one number, they can be reduced by dividing by the number "three":

3:3 / 24:3 = 1/8.

4. We write the answer like this:

5/6−1/3−1/12−7/24 = 1/8.

Example 5

Given three fractions with non-multiple denominators: 3/4; 2/7; 1/13.

You need to find the difference.

1. We bring the first two numbers to a common denominator, it will be the number "28":

• ¾ \u003d 3 * 7 / 4 * 7 \u003d 21/28;
• 2/7 = 2*4 / 7*4 = 8/28.

2. Subtract the first two fractions between each other:

¾−2/7 = 21/28−8/28 = (21−8) / 28 = 13/28.

3. Subtract the third given fraction from the resulting value:

4. We bring the numbers to a common denominator. If it is not possible to select the same denominator in an easier way, then you just need to perform the steps by multiplying all the denominators in series with each other, not forgetting to increase the value of the numerator by the same figure. In this example, we do this:

• 13/28 \u003d 13 * 13 / 28 * 13 \u003d 169/364, where 13 is the lower digit from 5/13;
• 5/13 \u003d 5 * 28 / 13 * 28 \u003d 140/364, where 28 is the lower digit from 13/28.

5. Subtract the resulting fractions:

13/28−5/13 = 169/364−140/364 = (169−140) / 364 = 29/364.

Answer: ¾-2/7-5/13 = 29/364.

Mixed fractional numbers

In the examples discussed above, only proper fractions were used.

As an example:

• 8/9 is a proper fraction;
• 9/8 is wrong.

It is impossible to turn an improper fraction into a proper one, but it is possible to turn it into mixed. Why is the top number (numerator) divided by the bottom number (denominator) to get a number with a remainder. The integer resulting from division is written down in this way, the remainder is written in the numerator at the top, and the denominator, which is at the bottom, remains the same. To make it clearer, consider a specific example:

Example 6

We convert the improper fraction 9/8 into the proper one.

To do this, we divide the number "nine" by "eight", as a result we get a mixed fraction with an integer and a remainder:

9: 8 = 1 and 1/8 (in another way it can be written as 1 + 1/8), where:

• the number 1 is the integer resulting from the division;
• another number 1 - the remainder;
• the number 8 is the denominator, which has remained unchanged.

An integer is also called a natural number.

The remainder and denominator are a new, but already correct fraction.

When writing the number 1, it is written before the correct fraction 1/8.

Subtracting mixed numbers with different denominators

From the above, we give the definition of a mixed fractional number: "Mixed number - this is a value that is equal to the sum of a whole number and a proper ordinary fraction. In this case, the whole part is called natural number, and the number that is in the remainder is its fractional part».

Example 7

Given: two mixed fractional quantities, consisting of a whole number and a proper fraction:

• the first value is 9 and 4/7, that is, (9 + 4/7);
• the second value is 3 and 5/21, i.e. (3+5/21).

It is required to find the difference between these values.

1. To subtract 3+5/21 from 9+4/7, you must first subtract integer values ​​from each other:

4/7−5/21 = 4*3 / 7*3−5/21 =12/21−5/21 = (12−5) / 21 = 7/21.

3. The result of the difference between two mixed numbers will consist of a natural (integer) number 6 and a proper fraction 7/21 = 1/3:

(9 + 4/7) - (3 + 5/21) = 6 + 1/3.

Mathematicians of all countries have agreed that the “+” sign when writing mixed quantities can be omitted and only the whole number in front of the fraction without any sign can be left.

Fractions are ordinary numbers, they can also be added and subtracted. But due to the fact that they have a denominator, more complex rules are required here than for integers.

Consider the simplest case, when there are two fractions with the same denominators. Then:

To add fractions with the same denominators, add their numerators and leave the denominator unchanged.

To subtract fractions with the same denominators, it is necessary to subtract the numerator of the second from the numerator of the first fraction, and again leave the denominator unchanged.

Within each expression, the denominators of the fractions are equal. By definition of addition and subtraction of fractions, we get:

As you can see, nothing complicated: just add or subtract the numerators - and that's it.

But even in such simple actions, people manage to make mistakes. Most often they forget that the denominator does not change. For example, when adding them, they also begin to add up, and this is fundamentally wrong.

Getting rid of the bad habit of adding denominators is quite simple. Try to do the same when subtracting. As a result, the denominator will be zero, and the fraction (suddenly!) will lose its meaning.

Therefore, remember once and for all: when adding and subtracting, the denominator does not change!

Also, many people make mistakes when adding several negative fractions. There is confusion with the signs: where to put a minus, and where - a plus.

This problem is also very easy to solve. It is enough to remember that the minus before the fraction sign can always be transferred to the numerator - and vice versa. And of course, do not forget two simple rules:

1. Plus times minus gives minus;
2. Two negatives make an affirmative.

Let's analyze all this with specific examples:

Task. Find the value of the expression:

In the first case, everything is simple, and in the second, we will add minuses to the numerators of fractions:

What if the denominators are different

You cannot directly add fractions with different denominators. At least, this method is unknown to me. However, the original fractions can always be rewritten so that the denominators become the same.

There are many ways to convert fractions. Three of them are discussed in the lesson " Bringing fractions to a common denominator", so we will not dwell on them here. Let's take a look at some examples:

Task. Find the value of the expression:

In the first case, we bring the fractions to a common denominator using the "cross-wise" method. In the second, we will look for the LCM. Note that 6 = 2 3; 9 = 3 · 3. The last factors in these expansions are equal, and the first ones are coprime. Therefore, LCM(6; 9) = 2 3 3 = 18.

What if the fraction has an integer part

I can please you: different denominators of fractions are not the greatest evil. Much more errors occur when the whole part is highlighted in the fractional terms.

Of course, for such fractions there are own addition and subtraction algorithms, but they are rather complicated and require a long study. Better use the simple diagram below:

1. Convert all fractions containing an integer part to improper. We get normal terms (even if with different denominators), which are calculated according to the rules discussed above;
2. Actually, calculate the sum or difference of the resulting fractions. As a result, we will practically find the answer;
3. If this is all that was required in the task, we perform the inverse transformation, i.e. we get rid of the improper fraction, highlighting the integer part in it.

The rules for switching to improper fractions and highlighting the integer part are described in detail in the lesson "What is a numerical fraction". If you don't remember, be sure to repeat. Examples:

Task. Find the value of the expression:

Everything is simple here. The denominators inside each expression are equal, so it remains to convert all fractions to improper ones and count. We have:

To simplify the calculations, I skipped some obvious steps in the last examples.

A small note to the last two examples, where fractions with a highlighted integer part are subtracted. The minus before the second fraction means that it is the whole fraction that is subtracted, and not just its whole part.

Reread this sentence again, look at the examples, and think about it. This is where beginners make a lot of mistakes. They like to give such tasks at control work. You will also meet them repeatedly in the tests for this lesson, which will be published shortly.

Summary: General Scheme of Computing

In conclusion, I will give a general algorithm that will help you find the sum or difference of two or more fractions:

1. If an integer part is highlighted in one or more fractions, convert these fractions to improper ones;
2. Bring all the fractions to a common denominator in any way convenient for you (unless, of course, the compilers of the problems did this);
3. Add or subtract the resulting numbers according to the rules for adding and subtracting fractions with the same denominators;
4. Reduce the result if possible. If the fraction turned out to be incorrect, select the whole part.

Remember that it is better to highlight the whole part at the very end of the task, just before writing the answer.

The next action that can be performed with ordinary fractions is subtraction. As part of this material, we will consider how to correctly calculate the difference between fractions with the same and different denominators, how to subtract a fraction from a natural number and vice versa. All examples will be illustrated with tasks. Let us clarify in advance that we will analyze only cases where the difference of fractions results in a positive number.

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How to find the difference between fractions with the same denominator

Let's start right away with an illustrative example: let's say we have an apple that has been divided into eight parts. Let's leave five parts on the plate and take two of them. This action can be written like this:

We end up with 3 eighths because 5 − 2 = 3 . It turns out that 5 8 - 2 8 = 3 8 .

With this simple example, we have seen exactly how the subtraction rule works for fractions with the same denominators. Let's formulate it.

Definition 1

To find the difference between fractions with the same denominators, you need to subtract the numerator of one from the numerator of the other, and leave the denominator the same. This rule can be written as a b - c b = a - c b .

We will use this formula in what follows.

Let's take specific examples.

Example 1

Subtract from the fraction 24 15 the common fraction 17 15 .

Decision

We see that these fractions have the same denominators. So all we have to do is subtract 17 from 24. We get 7 and add a denominator to it, we get 7 15 .

Our calculations can be written like this: 24 15 - 17 15 \u003d 24 - 17 15 \u003d 7 15

If necessary, you can reduce a complex fraction or separate the whole part from an improper one to make it more convenient to count.

Example 2

Find the difference 37 12 - 15 12 .

Decision

Let's use the formula described above and calculate: 37 12 - 15 12 = 37 - 15 12 = 22 12

It is easy to see that the numerator and denominator can be divided by 2 (we already talked about this earlier when we analyzed the signs of divisibility). Reducing the answer, we get 11 6 . This is an improper fraction, from which we will select the whole part: 11 6 \u003d 1 5 6.

How to find the difference between fractions with different denominators

Such a mathematical operation can be reduced to what we have already described above. To do this, simply bring the desired fractions to the same denominator. Let's formulate the definition:

Definition 2

To find the difference between fractions that have different denominators, you need to bring them to the same denominator and find the difference between the numerators.

Let's look at an example of how this is done.

Example 3

Subtract 1 15 from 2 9 .

Decision

The denominators are different, and you need to reduce them to the smallest common value. In this case, the LCM is 45. For the first fraction, an additional factor of 5 is required, and for the second - 3.

Let's calculate: 2 9 = 2 5 9 5 = 10 45 1 15 = 1 3 15 3 = 3 45

We got two fractions with the same denominator, and now we can easily find their difference using the algorithm described earlier: 10 45 - 3 45 = 10 - 3 45 = 7 45

A brief record of the solution looks like this: 2 9 - 1 15 \u003d 10 45 - 3 45 \u003d 10 - 3 45 \u003d 7 45.

Do not neglect the reduction of the result or the selection of a whole part from it, if necessary. In this example, we do not need to do this.

Example 4

Find the difference 19 9 - 7 36 .

Decision

We bring the fractions indicated in the condition to the lowest common denominator 36 and obtain 76 9 and 7 36 respectively.

We consider the answer: 76 36 - 7 36 \u003d 76 - 7 36 \u003d 69 36

The result can be reduced by 3 to get 23 12 . The numerator is greater than the denominator, which means we can extract the whole part. The final answer is 1 11 12 .

The summary of the whole solution is 19 9 - 7 36 = 1 11 12 .

How to subtract a natural number from a common fraction

Such an action can also be easily reduced to a simple subtraction of ordinary fractions. This can be done by representing a natural number as a fraction. Let's show an example.

Example 5

Find the difference 83 21 - 3 .

Decision

3 is the same as 3 1 . Then you can calculate like this: 83 21 - 3 \u003d 20 21.

If in the condition it is necessary to subtract an integer from an improper fraction, it is more convenient to first extract the integer from it, writing it as a mixed number. Then the previous example can be solved differently.

From the fraction 83 21, when you select the integer part, you get 83 21 \u003d 3 20 21.

Now just subtract 3 from it: 3 20 21 - 3 = 20 21 .

How to subtract a fraction from a natural number

This action is done similarly to the previous one: we rewrite a natural number as a fraction, bring both to a common denominator and find the difference. Let's illustrate this with an example.

Example 6

Find the difference: 7 - 5 3 .

Decision

Let's make 7 a fraction 7 1 . We do the subtraction and transform the final result, extracting the integer part from it: 7 - 5 3 = 5 1 3 .

There is another way to make calculations. It has some advantages that can be used in cases where the numerators and denominators of the fractions in the problem are large numbers.

Definition 3

If the fraction to be subtracted is correct, then the natural number from which we are subtracting must be represented as the sum of two numbers, one of which is equal to 1. After that, you need to subtract the desired fraction from unity and get the answer.

Example 7

Calculate the difference 1 065 - 13 62 .

Decision

The fraction to be subtracted is correct, because its numerator is less than the denominator. Therefore, we need to subtract one from 1065 and subtract the desired fraction from it: 1065 - 13 62 \u003d (1064 + 1) - 13 62

Now we need to find the answer. Using the properties of subtraction, the resulting expression can be written as 1064 + 1 - 13 62 . Let's calculate the difference in brackets. To do this, we represent the unit as a fraction 1 1 .

It turns out that 1 - 13 62 \u003d 1 1 - 13 62 \u003d 62 62 - 13 62 \u003d 49 62.

Now let's remember about 1064 and formulate the answer: 1064 49 62 .

We use the old way to prove that it is less convenient. Here are the calculations we would get:

1065 - 13 62 = 1065 1 - 13 62 = 1065 62 1 62 - 13 62 = 66030 62 - 13 62 = = 66030 - 13 62 = 66017 62 = 1064 4 6

The answer is the same, but the calculations are obviously more cumbersome.

We considered the case when you need to subtract the correct fraction. If it's wrong, we replace it with a mixed number and subtract according to the familiar rules.

Example 8

Calculate the difference 644 - 73 5 .

Decision

The second fraction is improper, and the whole part must be separated from it.

Now we calculate similarly to the previous example: 630 - 3 5 = (629 + 1) - 3 5 = 629 + 1 - 3 5 = 629 + 2 5 = 629 2 5

Subtraction properties when working with fractions

The properties that the subtraction of natural numbers possesses also apply to the cases of subtracting ordinary fractions. Let's see how to use them when solving examples.

Example 9

Find the difference 24 4 - 3 2 - 5 6 .

Decision

We have already solved similar examples when we analyzed the subtraction of a sum from a number, so we act according to the already known algorithm. First, we calculate the difference 25 4 - 3 2, and then subtract the last fraction from it:

25 4 - 3 2 = 24 4 - 6 4 = 19 4 19 4 - 5 6 = 57 12 - 10 12 = 47 12

Let's transform the answer by extracting the integer part from it. The result is 3 11 12.

Brief summary of the whole solution:

25 4 - 3 2 - 5 6 = 25 4 - 3 2 - 5 6 = 25 4 - 6 4 - 5 6 = = 19 4 - 5 6 = 57 12 - 10 12 = 47 12 = 3 11 12

If the expression contains both fractions and natural numbers, it is recommended to group them by types when calculating.

Example 10

Find the difference 98 + 17 20 - 5 + 3 5 .

Decision

Knowing the basic properties of subtraction and addition, we can group numbers as follows: 98 + 17 20 - 5 + 3 5 = 98 + 17 20 - 5 - 3 5 = 98 - 5 + 17 20 - 3 5

Let's complete the calculations: 98 - 5 + 17 20 - 3 5 = 93 + 17 20 - 12 20 = 93 + 5 20 = 93 + 1 4 = 93 1 4

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Lesson content

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 end of the task comes, then it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part in it. In our case, the integer part is allocated 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. We 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, since 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 the same.

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:

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, 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 a very 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.

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

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

Your child brought homework from school and you don't know how to solve it? Then this mini tutorial is for you!

How to add decimals

It is more convenient to add decimal fractions in a column. To add decimals, you need to follow one simple rule:

• The digit must be under the digit, comma under the comma.

As you can see in the example, whole units are under each other, tenths and hundredths are under each other. Now we add the numbers, ignoring the comma. What to do with a comma? The comma is transferred to the place where it stood in the discharge of integers.

Adding fractions with equal denominators

To perform addition with a common denominator, you need to keep the denominator unchanged, find the sum of the numerators and get a fraction, which will be the total amount.

Adding fractions with different denominators by finding a common multiple

The first thing to pay attention to is the denominators. The denominators are different, whether one is divisible by the other, whether they are prime numbers. First you need to bring to one common denominator, there are several ways to do this:

• 1/3 + 3/4 = 13/12, to solve this example, we need to find the least common multiple (LCM) that will be divisible by 2 denominators. To denote the smallest multiple of a and b - LCM (a; b). In this example LCM (3;4)=12. Check: 12:3=4; 12:4=3.
• We multiply the factors and perform the addition of the resulting numbers, we get 13/12 - an improper fraction.

• In order to convert an improper fraction to a proper one, we divide the numerator by the denominator, we get the integer 1, the remainder 1 is the numerator and 12 is the denominator.

Adding fractions using cross multiplication

For adding fractions with different denominators, there is another way according to the “cross by cross” formula. This is a guaranteed way to equalize the denominators, for this you need to multiply the numerators with the denominator of one fraction and vice versa. If you are just at the initial stage of learning fractions, then this method is the easiest and most accurate way to get the right result when adding fractions with different denominators.