How to subtract fractions with the same denominators. Addition and subtraction of fractions

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 with-stand with-from-vet-stu-u-th al-geb-ra-i-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:.

Decision

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:.

Decision

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 stated above, the addition of al-geb-ra-and-che-dro-bey is nothing from-is-cha-is-sya from the zhe-niya usually-but-vein-nyh dro-bay. Therefore, the solution method is the same:.

Example 4. You-honor fractions:.

Decision

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. So .

Example 5. You-honor fractions:.

Decision: .

Example 6. Simplify:.

Decision: .

Examples of applying the rule followed by reduction

In a fraction, someone-paradise is in a re-zul-ta-those addition 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:.

Decision: .

Wherein . In general, if the ODZ of the out-of-hot-drow-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:.

Decision: . 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:.

Decision:

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-pouring a common sign-me-on-tel on sign-me-on-tel co-from-rep-to-th-th fraction).

Then, each fraction is multiplied by a semi-chen-ny 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:.

Ras-look-rim now the fold of al-geb-ra-and-che-dro-bey with different signs-me-on-te-la-mi. Sleep-cha-la, we-look at the fractions, know-me-on-the-whether some of them are-la-yut-sya number-la-mi.

Addition and subtraction of algebraic fractions with different denominators

Example 2. Add fractions:.

Decision:

Al-go-rhythm of re-she-niya ab-so-lyut-but ana-lo-gi-chen previous-du-sche-mu p-me-ru. It’s easy to take a common denominator on the given fractions: and add-to-full multipliers for each of them.

Answer:.

So, sfor-mu-li-ru-em al-go-rhythm of complication and you-chi-ta-niya al-geb-ra-and-che-dro-beats with different-we-know-me-on-te-la-mi:

1. Find the smallest common sign-me-on-tel draw-bay.

2. Find additional multipliers for each of the draw-bay fractions).

3. Do-multiply-live numbers-whether-the-whether on the co-ot-vet-stu-u-s-up to-half-no-tel-nye-multiple-those.

4. Add-to-live or you-honor the fractions, use the right-wi-la-mi of the fold and you-chi-ta-niya draw-bay with one-to-you-know -me-on-te-la-mi.

Ras-look-rim now an example with dro-bya-mi, in the know-me-on-the-le-there-are-there-are-there-are-beech-ven-nye you-ra-same -tion.

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

How to subtract fractions whose denominators are the same

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

In order to subtract the second from one fraction, it is necessary to subtract the numerator of the fraction to be subtracted from the numerator of the reduced fraction. 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 reduced fraction "7" subtract the numerator of the subtracted fraction "3", we get "4". We write this number in the numerator of the answer, and put in the denominator the same number that was in the denominators of the first and second fractions - "19".

The picture below shows a few more such examples.

Consider a more complex example where fractions with the same 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 reduced fraction "29" 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 in the numerator of the answer, and in the denominator we write the number that is in the denominators of all these fractions - "47".

Adding fractions with the same denominator

Addition and subtraction of ordinary fractions is carried out according to the same principle.

To add fractions with the same denominators, you need to add the numerators. The resulting number is the numerator of the sum, and the denominator remains the same: k/m + b/m = (k + b)/m.

Let's see how it looks like in an example:

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

To the numerator of the first term of the fraction - "1" - we add the numerator of the second term of the fraction - "2". The result - "3" - is written in the numerator of the amount, and the denominator is left the same as that was present in the fractions - "4".

Fractions with different denominators and their subtraction

We have already considered the action 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 action with fractions that have different denominators? Many high 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 the solution of 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.

Fraction property

In order to reduce several fractions to the same denominator, you need to use the main property of the 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 look like any number that is a multiple of "3". After we multiply the numerator and denominator by "2", we get a fraction of 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 action with the number "4", we get 8/12. In one equation, this can be written as:

2/3 = 4/6 = 6/9 = 8/12…

How to bring multiple fractions to the same denominator

Consider how to reduce several fractions to the same denominator. For example, take the fractions shown in the picture below. First you need to determine what number can become the denominator for all of them. To make it easier, let's decompose the available denominators into factors.

The denominator of the fraction 1/2 and the fraction 2/3 cannot be factored. The denominator of 7/9 has two factors 7/9 = 7/(3 x 3), the denominator of the fraction 5/6 = 5/(2 x 3). Now you 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 triples, which means that they must also be present in the denominator. Given the above, we determine that the denominator consists of three factors: 3, 2, 3 and is equal to 3 x 2 x 3 = 18.

Consider the first fraction - 1/2. Its denominator contains "2", but there is not a single "3", but there should be two. To do this, we multiply the denominator by two triples, but, according to the property of the fraction, we must multiply the numerator by two triples:
1/2 = (1 x 3 x 3)/(2 x 3 x 3) = 9/18.

Similarly, we perform actions 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 two:
7/9 = (7 x 2)/(9 x 2) = 14/18.
5/6 or 5/(2 x 3) - the denominator is missing a triple:
5/6 = (5 x 3)/(6 x 3) = 15/18.

All together it looks like this:

How to subtract and add fractions with different denominators

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

Consider this with an example: 4/18 - 3/15.

Finding multiples of 18 and 15:

The number 18 consists of 3 x 2 x 3.
The number 15 consists of 5 x 3.
The common multiple will consist of the following factors 5 x 3 x 3 x 2 = 90.

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

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 step in our solution is to bring each fraction to the denominator "90".

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

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

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

Similarly produced and having different denominators.

Subtraction and having integer parts

Subtraction of fractions and their addition, we have already analyzed in detail. But how to subtract if the 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 the whole part. To do this, the number of the integer part is multiplied by the denominator of the fraction, the resulting product is added to the numerator. The number that will be obtained after these actions is the numerator of an improper fraction. The denominator remains unchanged.
If fractions have different denominators, they should be reduced to the same.
Perform addition or subtraction with the same denominators.
When receiving an improper fraction, select the whole part.

There is another way by which you can add and subtract fractions with integer parts. For this, actions are performed separately with integer parts, and separately with fractions, and the results are recorded together.

The above example consists of fractions that have the same denominator. In the case when the denominators are different, they must be reduced to the same, and then follow the steps as shown in the example.

Subtracting fractions from a whole number

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

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

The subtraction of fractions given in this article (Grade 6) is the basis for solving more complex examples, which are considered in subsequent classes. Knowledge of this topic is used subsequently to solve functions, derivatives, and so on. Therefore, it is very important to understand and understand the actions with fractions discussed above.

Adding and subtracting fractions with the same denominators
Adding and subtracting fractions with different denominators
The concept of the NOC
Bringing fractions to the same denominator
How to add a whole number and a fraction

1 Adding and subtracting fractions with the same denominators

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

To subtract fractions with the same denominators, subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same, for example:

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

If, when adding the fractional parts, an improper fraction is obtained, we select the integer part from it and add it to the integer part, for example:

2 Adding and subtracting fractions with different denominators

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

3 Least common multiple (LCM)

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

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

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

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

4Reducing fractions to the same denominator

Let's go back to adding fractions with different denominators.

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

Thus, in order to bring fractions to one indicator, you must first find the LCM (that is, the smallest number that is divisible by both denominators) of the denominators of these fractions, then put additional factors on the numerators of the fractions. You can find them by dividing the common denominator (LCD) by the denominator of the corresponding fraction. Then you need to multiply the numerator of each fraction by an additional factor, and put the LCM as the denominator.

5How to add a whole number and a fraction

In order to add a whole number and a fraction, you just need to add this number in front of the fraction, and you get a mixed fraction, for example.

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.

In this lesson, we will consider the addition and subtraction of algebraic fractions with different denominators. We already know how to add and subtract common fractions with different denominators. To do this, the fractions must be reduced to a common denominator. It turns out that algebraic fractions follow the same rules. At the same time, we already know how to reduce algebraic fractions to a common denominator. Adding and subtracting fractions with different denominators is one of the most important and difficult topics in the 8th grade course. Moreover, this topic will be found in many topics of the algebra course, which you will study in the future. As part of the lesson, we will study the rules for adding and subtracting algebraic fractions with different denominators, as well as analyze a number of typical examples.

Consider the simplest example for ordinary fractions.

Example 1 Add fractions: .

Decision:

Remember the rule for adding fractions. To begin with, fractions must be reduced to a common denominator. The common denominator for ordinary fractions is least common multiple(LCM) of the original denominators.

Definition

The smallest natural number that is divisible by both numbers and .

To find the LCM, it is necessary to decompose the denominators into prime factors, and then select all the prime factors that are included in the expansion of both denominators.

; . Then the LCM of numbers must include two 2s and two 3s: .

After finding the common denominator, it is necessary to find an additional factor for each of the fractions (in fact, divide the common denominator by the denominator of the corresponding fraction).

Then each fraction is multiplied by the resulting additional factor. We get fractions with the same denominators, which we learned to add and subtract in previous lessons.

We get: .

Answer:.

Consider now the addition of algebraic fractions with different denominators. First consider fractions whose denominators are numbers.

Example 2 Add fractions: .

Decision:

The solution algorithm is absolutely similar to the previous example. It is easy to find a common denominator for these fractions: and additional factors for each of them.

Answer:.

So let's formulate algorithm for adding and subtracting algebraic fractions with different denominators:

1. Find the smallest common denominator of fractions.

2. Find additional factors for each of the fractions (by dividing the common denominator by the denominator of this fraction).

3. Multiply the numerators by the appropriate additional factors.

4. Add or subtract fractions using the rules for adding and subtracting fractions with the same denominators.

Consider now an example with fractions in the denominator of which there are literal expressions.

Example 3 Add fractions: .

Decision:

Since the literal expressions in both denominators are the same, you should find a common denominator for numbers. The final common denominator will look like: . So the solution to this example is:

Answer:.

Example 4 Subtract fractions: .

Decision:

If you can’t “cheat” when choosing a common denominator (you can’t factor it or use the abbreviated multiplication formulas), then you have to take the product of the denominators of both fractions as a common denominator.

Answer:.

In general, when solving such examples, the most difficult task is to find a common denominator.

Let's look at a more complex example.

Example 5 Simplify: .

Decision:

When finding a common denominator, you must first try to factorize the denominators of the original fractions (to simplify the common denominator).

In this particular case:

Then it is easy to determine the common denominator: .

We determine additional factors and solve this example:

Answer:.

Now we will fix the rules for adding and subtracting fractions with different denominators.

Example 6 Simplify: .

Decision:

Answer:.

Example 7 Simplify: .

Decision:

Answer:.

Consider now an example in which not two, but three fractions are added (after all, the rules for addition and subtraction for more fractions remain the same).

Example 8 Simplify: .