Actions with fractions.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
So, what are fractions, types of fractions, transformations - we remembered. Let's tackle the main question.
What can you do with fractions? Yes, everything is the same as with ordinary numbers. Add, subtract, multiply, divide.
All these actions with decimal operations with fractions are no different from operations with integers. Actually, this is what they are good for, decimal. The only thing is that you need to put the comma correctly.
mixed numbers, as I said, are of little use for most actions. They still need to be converted to ordinary fractions.
And here are the actions with ordinary fractions will be smarter. And much more important! Let me remind you: all actions with fractional expressions with letters, sines, unknowns, and so on and so forth are no different from actions with ordinary fractions! Operations with ordinary fractions are the basis for all algebra. It is for this reason that we will analyze all this arithmetic in great detail here.
Addition and subtraction of fractions.
Everyone can add (subtract) fractions with the same denominators (I really hope!). Well, let me remind you that I’m completely forgetful: when adding (subtracting), the denominator does not change. The numerators are added (subtracted) to give the numerator of the result. Type:
In short, in general terms:
What if the denominators are different? Then, using the main property of the fraction (here it came in handy again!), We make the denominators the same! For example:
Here we had to make the fraction 4/10 from the fraction 2/5. Solely for the purpose of making the denominators the same. I note, just in case, that 2/5 and 4/10 are the same fraction! Only 2/5 is uncomfortable for us, and 4/10 is even nothing.
By the way, this is the essence of solving any tasks in mathematics. When we're out uncomfortable expressions do the same, but more convenient to solve.
Another example:
The situation is similar. Here we make 48 out of 16. By simple multiplication by 3. This is all clear. But here we come across something like:
How to be?! It's hard to make a nine out of a seven! But we are smart, we know the rules! Let's transform every fraction so that the denominators are the same. This is called "reduce to a common denominator":
How! How did I know about 63? Very simple! 63 is a number that is evenly divisible by 7 and 9 at the same time. Such a number can always be obtained by multiplying the denominators. If we multiply some number by 7, for example, then the result will certainly be divided by 7!
If you need to add (subtract) several fractions, there is no need to do it in pairs, step by step. You just need to find the denominator that is common to all fractions, and bring each fraction to this same denominator. For example:
And what will be the common denominator? You can, of course, multiply 2, 4, 8, and 16. We get 1024. Nightmare. It is easier to estimate that the number 16 is perfectly divisible by 2, 4, and 8. Therefore, it is easy to get 16 from these numbers. This number will be the common denominator. 1/2 will turn into 8/16, 3/4 into 12/16, and so on.
By the way, if we take 1024 as a common denominator, everything will work out too, in the end everything will be reduced. Only not everyone will get to this end, because of the calculations ...
Solve the example yourself. Not a logarithm... It should be 29/16.
So, with the addition (subtraction) of fractions is clear, I hope? Of course, it is easier to work in a shortened version, with additional multipliers. But this pleasure is available to those who honestly worked in the lower grades ... And did not forget anything.
And now we will do the same actions, but not with fractions, but with fractional expressions. New rakes will be found here, yes ...
So, we need to add two fractional expressions:
We need to make the denominators the same. And only with the help multiplication! So the main property of the fraction says. Therefore, I cannot add one to x in the first fraction in the denominator. (But that would be nice!). But if you multiply the denominators, you see, everything will grow together! So we write down, the line of the fraction, leave an empty space on top, then add it, and write the product of the denominators below, so as not to forget:
And, of course, we don’t multiply anything on the right side, we don’t open brackets! And now, looking at the common denominator of the right side, we think: in order to get the denominator x (x + 1) in the first fraction, we need to multiply the numerator and denominator of this fraction by (x + 1). And in the second fraction - x. You get this:
Note! Parentheses are here! This is the rake that many step on. Not brackets, of course, but their absence. Parentheses appear because we multiply the whole numerator and the whole denominator! And not their individual pieces ...
In the numerator of the right side, we write the sum of the numerators, everything is as in numerical fractions, then we open the brackets in the numerator of the right side, i.e. multiply everything and give like. You don't need to open the brackets in the denominators, you don't need to multiply something! In general, in denominators (any) the product is always more pleasant! We get:
Here we got the answer. The process seems long and difficult, but it depends on practice. Solve examples, get used to it, everything will become simple. Those who have mastered the fractions in the allotted time, do all these operations with one hand, on the machine!
And one more note. Many famously deal with fractions, but hang on examples with whole numbers. Type: 2 + 1/2 + 3/4= ? Where to fasten a deuce? No need to fasten anywhere, you need to make a fraction out of a deuce. It's not easy, it's very simple! 2=2/1. Like this. Any whole number can be written as a fraction. The numerator is the number itself, the denominator is one. 7 is 7/1, 3 is 3/1 and so on. It's the same with letters. (a + b) \u003d (a + b) / 1, x \u003d x / 1, etc. And then we work with these fractions according to all the rules.
Well, on addition - subtraction of fractions, knowledge was refreshed. Transformations of fractions from one type to another - repeated. You can also check. Shall we settle a little?)
Calculate:
Answers (in disarray):
71/20; 3/5; 17/12; -5/4; 11/6
Multiplication / division of fractions - in the next lesson. There are also tasks for all actions with fractions.
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By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)
you can get acquainted with functions and derivatives.
Lesson contentAdding fractions with the same denominators
Adding fractions is of two types:
- Adding fractions with the same denominators
- 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:
- 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:
- Find the LCM of the denominators of fractions;
- Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction;
- Multiply the numerators and denominators of fractions by their additional factors;
- Add fractions that have the same denominators;
- If the answer turned out to be an improper fraction, then select its whole part;
Example 2 Find the value of an expression 
.
Let's use the instructions above.
Step 1. Find the LCM of the denominators of fractions
Find the LCM of the denominators of both fractions. The denominators of the fractions are the numbers 2, 3 and 4
Step 2. Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction
Divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it over the first fraction:
Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. Divide 12 by 3, we get 4. We got the second additional factor 4. We write it over the second fraction:
Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We got the third additional factor 3. We write it over the third fraction:
Step 3. Multiply the numerators and denominators of fractions by your additional factors
We multiply the numerators and denominators by our additional factors:
Step 4. Add fractions that have the same denominators
We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. It remains to add these fractions. Add up:
The addition didn't fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is carried over to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning of a new line. The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.
Step 5. If the answer turned out to be an improper fraction, then select the whole part in it
Our answer is an improper fraction. We must single out the whole part of it. We highlight:
Got an answer
Subtraction of fractions with the same denominators
There are two types of fraction subtraction:
- Subtraction of fractions with the same denominators
- 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:
- 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 turned out to be an improper fraction, then you need to select the whole part in it.
Subtraction of fractions with different denominators
For example, a fraction can be subtracted from a fraction, since these fractions have the same denominators. But a fraction cannot be subtracted from a fraction, because these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.
The common denominator is found according to the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written over the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written over the second fraction.
The fractions are then multiplied by their additional factors. As a result of these operations, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to subtract such fractions.
Example 1 Find the value of an expression:
These fractions have different denominators, so you need to bring them to the same (common) denominator.
First, we find the LCM of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 4. The least common multiple of these numbers is 12
LCM (3 and 4) = 12
Now back to fractions and
Let's find an additional factor for the first fraction. To do this, we divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. We write the four over the first fraction:
We do the same with the second fraction. We divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. Write a triple over the second fraction:
Now we are all set for subtraction. It remains to multiply the fractions by their additional factors:
We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to subtract such fractions. Let's complete this example to the end:
Got an answer
Let's try to depict our solution using a picture. If you cut pizzas from a pizza, you get pizzas.
This is the detailed version of the solution. Being at school, we would have to solve this example in a shorter way. Such a solution would look like this:
Reduction of fractions and to a common denominator can also be depicted using a picture. Bringing these fractions to a common denominator, we get the fractions and . These fractions will be represented by the same pizza slices, but this time they will be divided into the same fractions (reduced to the same denominator):
The first drawing shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting off three pieces from eight pieces, we get five pieces out of twelve. The fraction describes these five pieces.
Example 2 Find the value of an expression 
These fractions have different denominators, so you first need to bring them to the same (common) denominator.
Find the LCM of the denominators of these fractions.
The denominators of the fractions are the numbers 10, 3 and 5. The least common multiple of these numbers is 30
LCM(10, 3, 5) = 30
Now we find additional factors for each fraction. To do this, we divide the LCM by the denominator of each fraction.
Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is the number 10. Divide 30 by 10, we get the first additional factor 3. We write it over the first fraction:
Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it over the second fraction:
Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is the number 5. Divide 30 by 5, we get the third additional factor 6. We write it over the third fraction:
Now everything is ready for subtraction. It remains to multiply the fractions by their additional factors:
We came to the conclusion that fractions that had different denominators turned into fractions that have the same (common) denominators. And we already know how to subtract such fractions. Let's finish this example.
The continuation of the example will not fit on one line, so we move the continuation to the next line. Don't forget about the equal sign (=) on the new line:
The answer turned out to be a correct fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it easier. What can be done? You can reduce this fraction.
To reduce a fraction, you need to divide its numerator and denominator by (gcd) the numbers 20 and 30.
So, we find the GCD of the numbers 20 and 30:
Now we return to our example and divide the numerator and denominator of the fraction by the found GCD, that is, by 10
Got an answer
Multiplying a fraction by a number
To multiply a fraction by a number, you need to multiply the numerator of the given fraction by this number, and leave the denominator 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, the five will not change its meaning, since the expression means “the number five divided by one”, and this, as you know, is equal to five:
Reverse numbers
Now we will get acquainted with 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
In this lesson, we will consider the addition and subtraction of algebraic fractions with the same denominators. We already know how to add and subtract common fractions with the same denominators. It turns out that algebraic fractions follow the same rules. The ability to work with fractions with the same denominators is one of the cornerstones in learning the rules for working with algebraic fractions. In particular, understanding this topic will make it easy to master a more complex topic - addition and subtraction of fractions with different denominators. As part of the lesson, we will study the rules for adding and subtracting algebraic fractions with the same denominators, as well as analyze a number of typical examples
Rule for adding and subtracting algebraic fractions with the same denominators
Sfor-mu-li-ru-em pr-vi-lo slo-zhe-niya (you-chi-ta-niya) al-geb-ra-and-che-dro-bey with one-on-to-you -mi-know-on-te-la-mi (it is co-pa-yes-et with the ana-logic right-of-thumb for ordinary-but-ven-nyh-dr-bay): That is for the addition or you-chi-ta-niya al-geb-ra-and-che-dro-bey with one-to-you-mi-know-me-on-te-la-mi is necessary -ho-di-mo co-stand with-from-vet-stu-u-sche al-geb-ra-and-che-sum of the number of-li-te-lei, and the sign-me-on-tel leave without iz-me-no-ny.
We will analyze this right-vi-lo both on the example of ordinary-but-vein-shot-beats, and on the example of al-geb-ra-and-che-dro- bey.
Examples of applying the rule for ordinary fractions
Example 1. Add fractions:.
Solution
Let's add the number-whether-they-whether draw-beat, and let's leave the sign-me-on-tel the same. After that, we divide the numer-li-tel and the sign-me-on-tel into simple multipliers and so-kra-tim. Let's get it: 
.
Note: standard error, I’ll start up something when resolving in a good kind of example, for -key-cha-et-sya in the following-du-u-sch-so-so-be-so-she-tion: 
. This is a gross mistake, since the sign-on-tel remains the same as it was in the original fractions.
Example 2. Add fractions:.
Solution
This za-da-cha is nothing from-whether-cha-et-sya from the previous one:.
Examples of applying the rule for algebraic fractions
From the usual-but-vein-nyh dro-bay per-rey-dem to al-geb-ra-i-che-skim.
Example 3. Add fractions:.
Solution: as already said above, the addition of al-geb-ra-and-che-fractions is nothing from the zhe-niya usually-but-vein-nyh dro-bay. Therefore, the solution method is the same:.
Example 4. You-honor fractions:.
Solution
You-chi-ta-nie al-geb-ra-and-che-dro-bey from-whether-cha-et-sya from the complication only by the fact that in the number of pi-sy-va-et-sya difference in the number of-li-te-lei is-run-nyh-dro-bay. That's why .
Example 5. You-honor fractions:.
Solution: .
Example 6. Simplify:.
Solution: .
Examples of applying the rule followed by reduction
In a fraction, someone-paradise in-lu-cha-e-sya in re-zul-ta-those com- plications or you-chi-ta-nia, it is possible to co-beautifully niya. In addition, you should not forget about the ODZ al-geb-ra-i-che-dro-bey.
Example 7. Simplify:.
Solution: .
Wherein . In general, if the ODZ of the out-of-hot-shot-bay owls-pa-yes-et with the ODZ of the total-go-howl, then you can not indicate it (after all, a fraction, in a lu-chen- naya in from-ve-those, also will not exist with co-from-vet-stu-u-s-knowing-che-no-yah-re-men-nyh). But if the ODZ is the source of the running dro-bay and from-ve-that does not co-pa-yes-et, then the ODZ indicates the need-ho-di-mo.
Example 8. Simplify:.
Solution: . At the same time, y (ODZ of the outgoing draw-bay does not coincide with the ODZ of re-zul-ta-ta).
Addition and subtraction of ordinary fractions with different denominators
To store and you-chi-tat al-geb-ra-and-che-fractions with different-we-know-me-on-te-la-mi, pro-ve-dem ana-lo -gyu from the usual-but-ven-ny-mi dro-bya-mi and re-re-not-sem it into al-geb-ra-and-che-fractions.
Ras-look at the simplest example for ordinary venous shots.
Example 1. Add fractions:.
Solution:
Let's remember the right-vi-lo-slo-drow-bay. For na-cha-la fractions, it is necessary to add-ve-sti to the common sign-me-to-te-lu. In the role of a general sign-me-on-te-la for ordinary-but-vein-draw-beats, you-stu-pa-et least common multiple(NOK) the source of the signs-me-on-the-lei.
Definition
The smallest-neck-to-tu-ral-number, someone-swarm is de-lit at the same time into numbers and.
To find the NOC, you need to de-lo-live know-me-on-the-whether into simple multipliers, and then choose to take everything pro- there are many, many, some of them are included in the difference between both signs-me-on-the-lei.
; . Then the LCM of numbers should include two twos and two threes:.
After finding the general sign-on-te-la, it is necessary for each of the dro-bays to find an additional multi- zhi-tel (fak-ti-che-ski, in de-pour the common sign-me-on-tel on the sign-me-on-tel co-from-reply-to-th-th fraction).
Then, each fraction is multiplied by a half-beam to-half-no-tel-ny multiplier. Fractions with the same-on-to-you-know-me-on-te-la-mi, warehouses and you-chi-tat someone we are on - studied in the past lessons.
By-lu-cha-eat: 
.
Answer:.
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Addition and subtraction of algebraic fractions with different denominators
Example 2. Add fractions:.
Solution:
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Answer:.
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Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use some object not entirely, but in separate pieces. The beginning of the study of this topic - share. Shares are equal parts into which an object is divided. After all, it is not always possible to express, for example, the length or price of a product as an integer; one should take into account parts or shares of any measure. Formed from the verb "to crush" - to divide into parts, and having Arabic roots, in the VIII century the word "fraction" itself appeared in Russian.
Fractional expressions have long been considered the most difficult section of mathematics. In the 17th century, when first textbooks in mathematics appeared, they were called "broken numbers", which was very difficult to display in people's understanding.
The modern form of simple fractional residues, parts of which are separated precisely by a horizontal line, was first promoted by Fibonacci - Leonardo of Pisa. His writings are dated 1202. But the purpose of this article is to simply and clearly explain to the reader how the multiplication of mixed fractions with different denominators occurs.
Multiplying fractions with different denominators
Initially, it is necessary to determine varieties of fractions:
- correct;
- wrong;
- mixed.
Next, you need to remember how fractional numbers with the same denominators are multiplied. The very rule of this process is easy to formulate independently: the result of multiplying simple fractions with the same denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in fact, the new denominator is the square of one of the existing ones initially.
When multiplying simple fractions with different denominators for two or more factors, the rule does not change:
a/b * c/d = a*c / b*d.
The only difference is that the formed number under the fractional bar will be the product of different numbers and, of course, it cannot be called the square of one numerical expression.
It is worth considering the multiplication of fractions with different denominators using examples:
- 8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
- 4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .
The examples use ways to reduce fractional expressions. You can reduce only the numbers of the numerator with the numbers of the denominator; adjacent factors above or below the fractional bar cannot be reduced.
Along with simple fractional numbers, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:
1 4/ 11 =1 + 4/ 11.
How does multiplication work?
Several examples are provided for consideration.
2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.
The example uses the multiplication of a number by ordinary fractional part, you can write down the rule for this action by the formula:
a* b/c = a*b /c.
In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. Special case:
4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.
There is another option for solving the multiplication of a number by a fractional remainder. You just need to divide the denominator by this number:
d* e/f = e/f: d.
It is useful to use this technique when the denominator is divided by a natural number without a remainder or, as they say, completely.
Convert mixed numbers to improper fractions and get the product in the previously described way:
1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.
This example involves a way to represent a mixed fraction as an improper fraction, it can also be represented as a general formula:
a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the integer part with the denominator and adding it to the numerator of the original fractional remainder, and the denominator remains the same.
This process also works in reverse. To select the integer part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator with a “corner”.
Multiplication of improper fractions produced in the usual way. When the entry goes under a single fractional line, as necessary, you need to reduce the fractions in order to reduce the numbers using this method and it is easier to calculate the result.
There are many assistants on the Internet to solve even complex mathematical problems in various program variations. A sufficient number of such services offer their help in calculating the multiplication of fractions with different numbers in the denominators - the so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions and mixed numbers. It is not difficult to work with it, the corresponding fields are filled in on the site page, the sign of the mathematical action is selected and “calculate” is pressed. The program counts automatically.
The topic of arithmetic operations with fractional numbers is relevant throughout the education of middle and senior schoolchildren. In high school, they are no longer considering the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations, obtained earlier, is applied in its original form. Well-learned basic knowledge gives full confidence in the successful solution of the most complex tasks.
In conclusion, it makes sense to cite the words of Leo Tolstoy, who wrote: “Man is a fraction. It is not in the power of man to increase his numerator - his merits, but everyone can decrease his denominator - his opinion of himself, and by this decrease come closer to his perfection.
The study of the issue of subtracting fractions with different denominators is found in the school subject Algebra in the eighth grade and it sometimes causes difficulties for children in understanding. To subtract fractions with different denominators, use the following formula:
The procedure for subtracting fractions is similar to addition, since it completely copies the principle of operation.
First, we calculate the smallest number that is a multiple of both one and the other denominator.
Secondly, we multiply the numerator and denominator of each fraction by a certain number, which will allow us to bring the denominator to the given minimum common denominator.
Thirdly, the subtraction procedure itself takes place, when, as a result, the denominator is duplicated, and the numerator of the second fraction is subtracted from the first.
Example: 8/3 2/4 = 8/3 1/2 = 16/6 3/6 = 13/6 = 2 integer 1/6
First you need to bring them to the same denominator, and then subtract them. For example, 1/2 - 1/4 = 2/4 - 1/4 = 1/4. Or, harder, 1/3 - 1/5 = 5/15 - 3/15 = 2/15. Do you need to explain how fractions are reduced to a common denominator?
In operations such as adding or subtracting ordinary fractions with different denominators, a simple rule applies - the denominators of these fractions are reduced to one number, and the operation itself is performed with the numbers in the numerator. That is, fractions get a common denominator and seem to be combined into one. Finding a common denominator for arbitrary fractions usually comes down to simply multiplying each of the fractions by the denominator of the other fraction. But in simpler cases, you can immediately find factors that will bring the denominators of fractions to the same number.
Fraction subtraction example: 2/3 - 1/7 = 2*7/3*7 - 1*3/7*3 = 14/21 - 3/21 = (14-3)/21 = 11/21
Many adults have already forgotten how to subtract fractions with different denominators, but this action belongs to elementary mathematics.
To subtract fractions with different denominators, you need to bring them to a common denominator, that is, find the least common multiple of the denominators, then multiply the numerators by additional factors equal to the ratio of the least common multiple and the denominator.
The signs of the fractions are preserved. After the fractions have the same denominators, you can subtract, and then, if possible, reduce the fraction.
Elena, did you decide to repeat the school mathematics course?)))
To subtract fractions with different denominators, they must first be reduced to the same denominator, and then subtracted. The simplest option: Multiply the numerator and denominator of the first fraction by the denominator of the second fraction, and multiply the numerator and denominator of the second fraction by the denominator of the first fraction. Get two fractions with the same denominators. Now we subtract the numerator of the second fraction from the numerator of the first fraction, and they have the same denominator.
For example, three fifths subtract two sevenths is equal to twenty-one thirty-fifths subtracted ten thirty-fifths and this is equal to eleven thirty-fifths.
If the denominators are large numbers, then you can find their least common multiple, i.e. a number that will be divisible by both one and the other denominator. And bring both fractions to a common denominator (least common multiple)
How to subtract fractions with different denominators the task is very simple - we bring the fractions to a common denominator and then do the subtraction in the numerator.
A lot of people face difficulties when there are integers near these fractions, so I wanted to show how to do this with the following example:
subtraction of fractions with an integer part and with different denominators
first we subtract the whole parts 8-5 = 3 (the triple remains near the first fraction);
we bring the fractions to a common denominator 6 (if the numerator of the first fraction is greater than the second, we subtract and write near the integer part, in our case we move on);
we decompose the integer part 3 into 2 and 1;
1 is written as a fraction 6/6;
6/6+3/6-4/6 we write under the common denominator 6 and do the actions in the numerator;
write down the found result 2 5/6.
It is important to remember that fractions are subtracted if they have the same denominator. Therefore, when we have fractions with different denominators in the difference, they need to be brought simply to a common denominator, which is not difficult to do. We just have to factor each fraction's numerator and calculate the least common multiple, which must not be zero. Do not forget to also multiply the numerators by the additional factors obtained, but here is an example for convenience:
If you want to subtract fractions with different denominators, then first you have to find a common denominator for these two fractions. And then subtract the second from the numerator of the first fraction. It turns out a new fraction, with a new value.
As far as I remember from the 3rd grade mathematics course, to subtract fractions with different denominators, you first need to calculate the common denominator and bring it to it, and then the numerators are simply subtracted from each other and the denominator remains that common.
To subtract fractions with different denominators, we first have to find the smallest common denominator of these fractions.
Let's look at an example:
Divide the larger number 25 by the smaller 20. Not divisible. So we multiply the denominator 25 by such a number that the resulting sum can be divided by 20. This number will be 4. 25x4 \u003d 100. 100:20=5. Thus, we found the lowest common denominator - 100.
Now we need to find an additional factor for each fraction. To do this, we divide the new denominator by the old one.
Multiply 9 by 4 = 36. Multiply 7 by 5 = 35.
Having a common denominator, we subtract, as shown in the example, and get the result.
