How to solve an example with fractions. Fractions, operations with fractions. A fraction is the number of parts of a whole

Now that we have learned how to add and multiply individual fractions, we can look at more complex structures. For example, what if the same problem involves adding, subtracting, and multiplying fractions?

First of all, you need to convert all fractions to improper ones. Then we perform the required actions sequentially - in the same order as for ordinary numbers. Namely:

Exponentiation is done first - get rid of all expressions containing exponents;
Then - division and multiplication;
The last step is addition and subtraction.

Of course, if there are parentheses in the expression, the order of operations changes - everything that is inside the parentheses must be counted first. And remember about improper fractions: you need to highlight the whole part only when all other actions have already been completed.

Let's convert all the fractions from the first expression to improper ones, and then perform the following steps:

Now let's find the value of the second expression. There are no fractions with an integer part, but there are parentheses, so first we perform addition, and only then division. Note that 14 = 7 · 2. Then:

Finally, consider the third example. There are brackets and a degree here - it is better to count them separately. Considering that 9 = 3 3, we have:

Pay attention to the last example. To raise a fraction to a power, you must separately raise the numerator to this power, and separately, the denominator.

You can decide differently. If we recall the definition of a degree, the problem will be reduced to the usual multiplication of fractions:

Multistory fractions

Until now, we have considered only “pure” fractions, when the numerator and denominator are ordinary numbers. This is quite consistent with the definition of a number fraction given in the very first lesson.

But what if you put a more complex object in the numerator or denominator? For example, another numerical fraction? Such constructions arise quite often, especially when working with long expressions. Here are a couple of examples:

There is only one rule for working with multi-level fractions: you must get rid of them immediately. Removing “extra” floors is quite simple, if you remember that the slash means the standard division operation. Therefore, any fraction can be rewritten as follows:

Using this fact and following the procedure, we can easily reduce any multi-story fraction to an ordinary one. Take a look at the examples:

Task. Convert multistory fractions to ordinary ones:

In each case, we rewrite the main fraction, replacing the dividing line with a division sign. Also remember that any integer can be represented as a fraction with a denominator of 1. That is 12 = 12/1; 3 = 3/1. We get:

In the last example, the fractions were canceled before the final multiplication.

Specifics of working with multi-level fractions

There is one subtlety in multi-level fractions that must always be remembered, otherwise you can get the wrong answer, even if all the calculations were correct. Take a look:

The numerator contains the single number 7, and the denominator contains the fraction 12/5;
The numerator contains the fraction 7/12, and the denominator contains the separate number 5.

So, for one recording we got two completely different interpretations. If you count, the answers will also be different:

To ensure that the record is always read unambiguously, use a simple rule: the dividing line of the main fraction must be longer than the line of the nested fraction. Preferably several times.

If you follow this rule, then the above fractions should be written as follows:

Yes, it's probably unsightly and takes up too much space. But you will count correctly. Finally, a couple of examples where multi-story fractions actually arise:

Task. Find the meanings of the expressions:

So, let's work with the first example. Let's convert all fractions to improper ones, and then perform addition and division operations:

Let's do the same with the second example. Let's convert all fractions to improper ones and perform the required operations. In order not to bore the reader, I will omit some obvious calculations. We have:

Due to the fact that the numerator and denominator of the basic fractions contain sums, the rule for writing multi-story fractions is observed automatically. Also, in the last example, we intentionally left 46/1 in fraction form to perform division.

I will also note that in both examples the fraction bar actually replaces the parentheses: first of all, we found the sum, and only then the quotient.

Some will say that the transition to improper fractions in the second example was clearly redundant. Perhaps this is true. But by doing this we insure ourselves against mistakes, because next time the example may turn out to be much more complicated. Choose for yourself what is more important: speed or reliability.

How to learn to solve fractions?

I myself was faced with the fact that fractions turned out to be a rather difficult topic for my children.

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

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

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

What are the parts of the circles called? or

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

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

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

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

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

And here's how to explain fractions using Lego.

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

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

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

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

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

Same thing with addition.

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

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

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

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

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

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

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

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

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

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

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

Also: I advise you to use visual examples only at the initial stage of explanation, then quickly abstract and move on to the solution algorithm. Otherwise, clarity may be detrimental when solving more complex problems. For example, if you need to add fractions with denominators 29 and 121, what kind of visual aid will help? It will only confuse.

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

Problem formulation: Find the meaning of the expression (operations with fractions).

The problem is part of the Unified State Examination in basic level mathematics for grade 11 under number 1 (Actions with fractions).

Let's look at how such problems are solved using examples.

Example task 1:

Find the value of the expression 5/4 + 7/6: 2/3.

Let's calculate the value of the expression. To do this, we determine the order of operations: first multiplication and division, then addition and subtraction. And perform the necessary actions in the right order:

Answer: 3

Example task 2:

Find the value of the expression (3.9 – 2.4) ∙ 8.2

Answer: 12.3

Example task 3:

Find the value of the expression 27 ∙ (1/3 – 4/9 – 5/27).

Let's calculate the value of the expression. To do this, we determine the order of operations: first multiplication and division, then addition and subtraction. In this case, actions in brackets are executed before actions outside the brackets. And perform the necessary actions in the right order:

Answer: –8

Example task 4:

Find the value of the expression 2.7 / (1.4 + 0.1)

Answer: 1.8

Example problem 5:

Find the value of the expression 1 / (1/9 – 1/12).

Answer: 36

Example problem 6:

Find the value of the expression (0.24 ∙ 10^6) / (0.6 ∙ 10^4).

Answer: 40

Example problem 7:

Find the value of the expression (1.23 ∙ 45.7) / (12.3 ∙ 0.457).

Answer: 10

Example problem 8:

Find the value of the expression (728^2 – 26^2) : 754.

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 you 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” digit, 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 perform the same operations 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 as 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 the 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.

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 study 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 will be 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 denominators, 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 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 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: